Markov chain

Short description: Random process independent of past history

A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state. Part of a series on statistics

Probability theory

* Probability axioms

* Probability space
* Sample space
* Elementary event
* Event
* Random variable
* Probability measure

* Complementary event
* Joint probability
* Marginal probability
* Conditional probability

* Independence
* Conditional independence
* Law of total probability
* Law of large numbers
* Bayes' theorem
* Boole's inequality

* Venn diagram
* Tree diagram

* v
* t
* e

In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov.

Markov chains have many applications as statistical models of real-world processes.[1] They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance, information theory, physics, signal processing, and speech processing.[1][2][3]

The adjectives Markovian and Markov are used to describe something that is related to a Markov process.[4]

## Contents

* 1 Principles
* 1.1 Definition
* 1.2 Types of Markov chains
* 1.3 Transitions
* 2 History
* 3 Examples
* 3.1 A non-Markov example
* 4 Formal definition
* 4.1 Discrete-time Markov chain
* 4.1.1 Variations
* 4.2 Continuous-time Markov chain
* 4.2.1 Infinitesimal definition
* 4.2.2 Jump chain/holding time definition
* 4.2.3 Transition probability definition
* 4.3 Finite state space
* 4.3.1 Stationary distribution relation to eigenvectors and simplices
* 4.3.2 Time-homogeneous Markov chain with a finite state space
* 4.3.3 Convergence speed to the stationary distribution
* 4.4 Markov processes in continuous time
* 4.5 General state space
* 4.5.1 Harris chains
* 4.5.2 Locally interacting Markov chains
* 5 Properties
* 5.1 Irreducibility
* 5.2 Ergodicity
* 5.2.1 Terminology
* 5.2.2 Index of primitivity
* 5.3 Measure-preserving dynamical system
* 5.4 Markovian representations
* 5.5 Hitting times
* 5.5.1 Expected hitting times
* 5.6 Time reversal
* 5.7 Embedded Markov chain
* 6 Special types of Markov chains
* 6.1 Markov model
* 6.2 Bernoulli scheme
* 6.3 Subshift of finite type
* 7 Applications
* 7.1 Physics
* 7.2 Chemistry
* 7.3 Biology
* 7.4 Testing
* 7.5 Solar irradiance variability
* 7.6 Speech recognition
* 7.7 Information theory
* 7.8 Queueing theory
* 7.9 Internet applications
* 7.10 Statistics
* 7.11 Conflict and combat
* 7.12 Economics and finance
* 7.13 Social sciences
* 7.14 Music
* 7.15 Games and sports
* 7.16 Markov text generators
* 8 See also
* 9 Notes
* 10 References
* 11 External links

## Principles

Russian mathematician Andrey Markov

### Definition

A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history.[5] In other words, conditional on the present state of the system, its future and past states are independent.

A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies.[6] For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time),[7][8][9][10] but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).[6]

### Types of Markov chains

The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time:

| Countable state space  | Continuous or general state space

|
|

Discrete-time  | (discrete-time) Markov chain on a countable or finite state space  | Markov chain on a measurable state space (for example, Harris chain)
Continuous-time  | Continuous-time Markov process or Markov jump process  | Any continuous stochastic process with the Markov property (for example, the Wiener process)

Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC),[11] but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention.[12][13][14] In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see Markov model). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term.

While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.[15] However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see Variations). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.

### Transitions

The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate.

A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are the integers or natural numbers, and the random process is a mapping of these to states. The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps.

Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important.

## History

Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906.[16][17][18] Markov Processes in continuous time were discovered long before his work in the early 20th century in the form of the Poisson process.[19][20][21] Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with Pavel Nekrasov who claimed independence was necessary for the weak law of large numbers to hold.[22] In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption,[16][17][18] which had been commonly regarded as a requirement for such mathematical laws to hold.[18] Markov later used Markov chains to study the distribution of vowels in Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains.[16]

In 1912 Henri Poincaré studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov.[16][17] After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé.[23] Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.[16][24]

Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.[25][26] Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener's work on Einstein's model of Brownian movement.[25][27] He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.[25][28] Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.[29] The differential equations are now called the Kolmogorov equations[30] or the Kolmogorov–Chapman equations.[31] Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in 1930s, and then later Eugene Dynkin, starting in the 1950s.[26]

## Examples

Main article: Examples of Markov chains

* Mark V. Shaney is a third-order Markov chain program, and a Markov text generator. It ingests the sample text (the Tao Te Ching, or the posts of a Usenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on.[32]

* Random walks based on integers and the gambler's ruin problem are examples of Markov processes.[33][34] Some variations of these processes were studied hundreds of years earlier in the context of independent variables.[35][36] Two important examples of Markov processes are the Wiener process, also known as the Brownian motion process, and the Poisson process,[19] which are considered the most important and central stochastic processes in the theory of stochastic processes.[37][38][39] These two processes are Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.[33][34]
* A famous Markov chain is the so-called "drunkard's walk", a random walk on the number line where, at each step, the position may change by +1 or −1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6.
* A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current one.

### A non-Markov example

Suppose that there is a coin purse containing five coins worth 25¢, five coins worth 10¢ and five coins worth 5¢, and one by one, coins are randomly drawn from the purse and are set on a table. If Xn represents the total value of the coins set on the table after n draws, with X0=0, then the sequence {Xn:n∈ℕ} is not a Markov process.

To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. Thus X6=$0.50. If we know not just X6, but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel; so we can determine that X7≥$0.60 with probability 1. But if we do not know the earlier values, then based only on the value X6 we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses about X7 are impacted by our knowledge of values prior to X6.

However, it is possible to model this scenario as a Markov process. Instead of defining Xn to represent the total value of the coins on the table, we could define Xn to represent the count of the various coin types on the table. For instance, X6=1,0,5 could be defined to represent the state where there is one quarter, zero dimes, and five nickels on the table after 6 one-by-one draws. This new model could be represented by 6×6×6=216 possible states, where each state represents the number of coins of each type (from 0 to 5) that are on the table. (Not all of these states are reachable within 6 draws.) Suppose that the first draw results in state X1=0,1,0. The probability of achieving X2 now depends on X1; for example, the state X2=1,0,1 is not possible. After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state (since probabilistically important information has since been added to the scenario). In this way, the likelihood of the Xn=i,j,k state depends exclusively on the outcome of the Xn−1=ℓ,m,p state.

## Formal definition

### Discrete-time Markov chain

Main article: Discrete-time Markov chain

A discrete-time Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states:

Pr⁡(Xn+1=x∣X1=x1,X2=x2,…,Xn=xn)=Pr⁡(Xn+1=x∣Xn=xn), if both conditional probabilities are well defined, that is, if Pr⁡(X1=x1,…,Xn=xn)>0.

The possible values of Xi form a countable set S called the state space of the chain.

#### Variations

* Time-homogeneous Markov chains are processes where Pr⁡(Xn+1=x∣Xn=y)=Pr⁡(Xn=x∣Xn−1=y) for all n. The probability of the transition is independent of n.
* Stationary Markov chains are processes where Pr⁡(X0=x0,X1=x1,…,Xk=xk)=Pr⁡(Xn=x0,Xn+1=x1,…,Xn+k=xk) for all n and k. Every stationary chain can be proved to be time-homogeneous by Bayes' rule.

A necessary and sufficient condition for a time-homogeneous Markov chain to be stationary is that the distribution of X0 is a stationary distribution of the Markov chain.
* A Markov chain with memory (or a Markov chain of order m) where m is finite, is a process satisfying Pr⁡(Xn=xn∣Xn−1=xn−1,Xn−2=xn−2,…,X1=x1)=Pr⁡(Xn=xn∣Xn−1=xn−1,Xn−2=xn−2,…,Xn−m=xn−m) for n>m In other words, the future state depends on the past m states. It is possible to construct a chain (Yn) from (Xn) which has the 'classical' Markov property by taking as state space the ordered m-tuples of X values, i.e., Yn=(Xn,Xn−1,…,Xn−m+1).

### Continuous-time Markov chain

Main article: Continuous-time Markov chain

A continuous-time Markov chain (Xt)t≥0 is defined by a finite or countable state space S, a transition rate matrix Q with dimensions equal to that of the state space and initial probability distribution defined on the state space. For i ≠ j, the elements qij are non-negative and describe the rate of the process transitions from state i to state j. The elements qii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one.

There are three equivalent definitions of the process.[40]

#### Infinitesimal definition

The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j.

Let Xt be the random variable describing the state of the process at time t, and assume the process is in a state i at time t. Then, knowing Xt=i, Xt+h=j is independent of previous values (Xs:s<t), and as h → 0 for all j and for all t, Pr⁡(X(t+h)=j∣X(t)=i)=δij+qijh+o(h), where δij is the Kronecker delta, using the little-o notation. The qij can be seen as measuring how quickly the transition from i to j happens.

#### Jump chain/holding time definition

Define a discrete-time Markov chain Yn to describe the nth jump of the process and variables S1, S2, S3, ... to describe holding times in each of the states where Si follows the exponential distribution with rate parameter −qYiYi.

#### Transition probability definition

For any value n = 0, 1, 2, 3, ... and times indexed up to this value of n: t0, t1, t2, ... and all states recorded at these times i0, i1, i2, i3, ... it holds that

Pr⁡(Xtn+1=in+1∣Xt0=i0,Xt1=i1,…,Xtn=in)=pinin+1(tn+1−tn)

where pij is the solution of the forward equation (a first-order differential equation)

P′(t)=P(t)Q

with initial condition P(0) is the identity matrix.

### Finite state space

If the state space is finite, the transition probability distribution can be represented by a matrix, called the transition matrix, with the (i, j)th element of P equal to

pij=Pr⁡(Xn+1=j∣Xn=i).

Since each row of P sums to one and all elements are non-negative, P is a right stochastic matrix.

#### Stationary distribution relation to eigenvectors and simplices

A stationary distribution π is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrix P on it and so is defined by

π𝐏=π.

By comparing this definition with that of an eigenvector we see that the two concepts are related and that

π=e∑iei

is a normalized (∑iπi=1) multiple of a left eigenvector e of the transition matrix P with an eigenvalue of 1. If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution.

The values of a stationary distribution πi are associated with the state space of P and its eigenvectors have their relative proportions preserved. Since the components of π are positive and the constraint that their sum is unity can be rewritten as ∑i1⋅πi=1 we see that the dot product of π with a vector whose components are all 1 is unity and that π lies on a simplex.

#### Time-homogeneous Markov chain with a finite state space

If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, Pk.

If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π.[41] Additionally, in this case Pk converges to a rank-one matrix in which each row is the stationary distribution π:

limk->∞𝐏k=𝟏π

where 1 is the column vector with all entries equal to 1. This is stated by the Perron–Frobenius theorem. If, by whatever means, limk->∞𝐏k is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below.

For some stochastic matrices P, the limit limk->∞𝐏k does not exist while the stationary distribution does, as shown by this example:

𝐏=(0110)𝐏2k=I𝐏2k+1=𝐏
(1212)(0110)=(1212)

(This example illustrates a periodic Markov chain.)

Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task. However, there are many techniques that can assist in finding this limit. Let P be an n×n matrix, and define 𝐐=limk->∞𝐏k.

It is always true that

𝐐𝐏=𝐐.

Subtracting Q from both sides and factoring then yields

𝐐(𝐏−𝐈n)=𝟎n,n,

where In is the identity matrix of size n, and 0n,n is the zero matrix of size n×n. Multiplying together stochastic matrices always yields another stochastic matrix, so Q must be a stochastic matrix (see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that Q is a stochastic matrix to solve for Q. Including the fact that the sum of each the rows in P is 1, there are n+1 equations for determining n unknowns, so it is computationally easier if on the one hand one selects one row in Q and substitutes each of its elements by one, and on the other one substitutes the corresponding element (the one in the same column) in the vector 0, and next left-multiplies this latter vector by the inverse of transformed former matrix to find Q.

Here is one method for doing so: first, define the function f(A) to return the matrix A with its right-most column replaced with all 1's. If [f(P − In)]−1 exists then[42][41]

𝐐=f(𝟎n,n)[f(𝐏−𝐈n)]−1.
Explain: The original matrix equation is equivalent to a system of n×n linear equations in n×n variables. And there are n more linear equations from the fact that Q is a right stochastic matrix whose each row sums to 1. So it needs any n×n independent linear equations of the (n×n+n) equations to solve for the n×n variables. In this example, the n equations from "Q multiplied by the right-most column of (P-In)" have been replaced by the n stochastic ones.

One thing to notice is that if P has an element Pi,i on its main diagonal that is equal to 1 and the ith row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers Pk. Hence, the ith row or column of Q will have the 1 and the 0's in the same positions as in P.

#### Convergence speed to the stationary distribution

As stated earlier, from the equation π=π𝐏, (if exists) the stationary (or steady state) distribution π is a left eigenvector of row stochastic matrix P. Then assuming that P is diagonalizable or equivalently that P has n linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is, defective matrices, one may start with the Jordan normal form of P and proceed with a bit more involved set of arguments in a similar way.[43])

Let U be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector of P and let Σ be the diagonal matrix of left eigenvalues of P, that is, Σ = diag(λ1,λ2,λ3,...,λn). Then by eigendecomposition

𝐏=𝐔Σ𝐔−1.

Let the eigenvalues be enumerated such that:

1=|λ1|>|λ2|≥|λ3|≥⋯≥|λn|.

Since P is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector is unique too (because there is no other π which solves the stationary distribution equation above). Let ui be the i-th column of U matrix, that is, ui is the left eigenvector of P corresponding to λi. Also let x be a length n row vector that represents a valid probability distribution; since the eigenvectors ui span ℝn, we can write

𝐱T=∑i=1nai𝐮i,ai∈ℝ.

If we multiply x with P from right and continue this operation with the results, in the end we get the stationary distribution π. In other words, π = a1 u1 ← xPP...P = xPk as k → ∞. That means

π(k)=𝐱(𝐔Σ𝐔−1)(𝐔Σ𝐔−1)⋯(𝐔Σ𝐔−1)=𝐱𝐔Σk𝐔−1=(a1𝐮1T+a2𝐮2T+⋯+an𝐮nT)𝐔Σk𝐔−1=a1λ1k𝐮1T+a2λ2k𝐮2T+⋯+anλnk𝐮nTui⊥uj for i≠j=λ1k{a1𝐮1T+a2(λ2λ1)k𝐮2T+a3(λ3λ1)k𝐮3T+⋯+an(λnλ1)k𝐮nT}

Since π is parallel to u1(normalized by L2 norm) and π(k) is a probability vector, π(k) approaches to a1 u1 = π as k → ∞ with a speed in the order of λ2/λ1 exponentially. This follows because |λ2|≥⋯≥|λn|, hence λ2/λ1 is the dominant term. The smaller the ratio is, the faster the convergence is.[44] Random noise in the state distribution π can also speed up this convergence to the stationary distribution.[45]

### Markov processes in continuous time

Let (Ω,ℱ,𝔽,P) be a filtered probability space with 𝔽=(ℱt)t≥0 and X an adapted stochastic process with values in a Polish space (E,ℬ(E)) (e.g., (ℝ,ℬ(ℝ))). Then X is called a Markov process, if it satisfies the Markov property, i.e., for all t≥s≥0 and A∈ℬ(E)[5]

P(Xt∈A∣ℱs)=P(Xt∈A∣Xs).

Moreover, X is called time-homogeneous, if it satisfies the weak Markov property for all t≥s≥0:

P(Xt+s∈A∣ℱs)=P(Xt∈A∣X0=x)|x=Xs=:Pt(Xs,A).

The function (t,x,A)↦Pt(x,A) is the so-called transition function of X and (Pt)t≥0 the transition semigroup of the process. Transition functions are generalizations of the transition matrices used in the setting with finite state space.

In a more abstract way, Markov processes can also be defined or constructed the other way around: Let (Pt)t≥0 be a transition semigroup, i.e.,

1. Pt is Markov kernel for all t≥0,
2. Pt+s(x,A)=∫EPt(y,A)Ps(x,dy)∀t,s≥0,x∈ℝ,A∈ℬ(E) (Chapman-Kolmogorov-equation),
3. P0(x,⋅)=δx,

where δx is the Dirac-measure in x, and X:Ω×[0,∞)->E. Then X is a homogeneous Markov process w.r.t. the natural filtration 𝔽X=(σ(Xs:0≤s≤t))t≥0, if for all 0≤t1<...<tn, A1,...,An∈ℬ(E) the underlying probability measure P satisfies

P(Xt1∈A1,...,Xtn∈An∣X0=x)=∫A1...∫An−1Ptn−tn−1(xn−1,An)⋯Pt1(x,dx1).

Or, if no probability measure P has been specified, the above equation defines a measure Px:=P(⋅∣X0=x) on σ(Xs:s≥0) under which the process X started in x is a Markov process by construction.

In other words, Markov processes can be defined either as stochastic processes X on a filtered probability space, or indirectly in terms of a transition semigroup (i.e., the transition probabilities of the process), which induces a probability space under which X has the Markov property.

### General state space

Main article: Markov chains on a measurable state space

#### Harris chains

Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through Harris chains.

The use of Markov chains in Markov chain Monte Carlo methods covers cases where the process follows a continuous state space.

#### Locally interacting Markov chains

"Locally interacting Markov chains" are Markov chains with an evolution that takes into account the state of other Markov chains. This corresponds to the situation when the state space has a (Cartesian-) product form. See interacting particle system and stochastic cellular automata (probabilistic cellular automata). See for instance Interaction of Markov Processes[46] or.[47]

## Properties

Two states are said to communicate with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class is closed if the probability of leaving the class is zero. A Markov chain is irreducible if there is one communicating class, the state space.

A state i has period k if k is the greatest common divisor of the number of transitions by which i can be reached, starting from i. That is:

k=gcd⁡{n>0:Pr⁡(Xn=i∣X0=i)>0}

The state is periodic if k>1; otherwise k=1 and the state is aperiodic.

A state i is said to be transient if, starting from i, there is a non-zero probability that the chain will never return to i. It is called recurrent (or persistent) otherwise.[48] For a recurrent state i, the mean hitting time is defined as:

Mi=E[Ti]=∑n=1∞n⋅fii(n).

State i is positive recurrent if Mi is finite and null recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties — that is, if one state has the property then all states in its communicating class have the property.[49]

A state i is called absorbing if there are no outgoing transitions from the state.

### Irreducibility

Since periodicity is a class property, if a Markov chain is irreducible, then all its states have the same period. In particular, if one state is aperiodic, then the whole Markov chain is aperiodic.[50]

If a finite Markov chain is irreducible, then all states are positive recurrent, and it has a unique stationary distribution given by πi=1/E[Ti].

### Ergodicity

A state i is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state i is ergodic if it is recurrent, has a period of 1, and has finite mean recurrence time.

If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. Equivalently, there exists some integer k such that all entries of Mk are positive.

It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. More generally, a Markov chain is ergodic if there is a number N such that any state can be reached from any other state in any number of steps less or equal to a number N. In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled with N = 1\.

A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic.

#### Terminology

Some authors call any irreducible, positive recurrent Markov chains ergodic, even periodic ones.[51] In fact, merely irreducible Markov chains correspond to ergodic processes, defined according to ergodic theory.[52]

Some authors call a matrix primitive if there exists some integer k such that all entries of Mk are positive.[53] Some authors call it regular.[54]

#### Index of primitivity

The index of primitivity, or exponent, of a regular matrix, is the smallest k such that all entries of Mk are positive. The exponent is purely a graph-theoretic property, since it depends only on whether each entry of M is zero or positive, and therefore can be found on a directed graph with sign(M) as its adjacency matrix.

There are several combinatorial results about the exponent when there are finitely many states. Let n be the number of states, then[55]

* The exponent is ≤(n−1)2+1. The only case where it is an equality is when the graph of M goes like 1->2->…->n->1 and 2.
* If M has k≥1 diagonal entries, then its exponent is ≤2n−k−1.
* If sign(M) is symmetric, then M2 has positive diagonal entries, which by previous proposition means its exponent is ≤2n−2.
* (Dulmage-Mendelsohn theorem) The exponent is ≤n+s(n−2) where s is the girth of the graph. It can be improved to ≤(d+1)+s(d+1−2), where d is the diameter of the graph.[56]

### Measure-preserving dynamical system

If a Markov chain has a stationary distribution, then it can be converted to a measure-preserving dynamical system: Let the probability space be Ω=Σℕ, where Σ is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability measure be generated by the stationary distribution, and the Markov chain transition. Let T:Ω->Ω be the shift operator: T(X0,X1,…)=(X1,…). Similarly we can construct such a dynamical system with Ω=Σℤ instead.[57]

Since irreducible Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains.

In ergodic theory, a measure-preserving dynamical system is called ergodic if any measurable subset S such that T−1(S)=S implies S=∅ or Ω (up to a null set).

The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain is irreducible if its corresponding measure-preserving dynamical system is ergodic.[52]

### Markovian representations

In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the "current" and "future" states. For example, let X be a non-Markovian process. Then define a process Y, such that each state of Y represents a time-interval of states of X. Mathematically, this takes the form:

Y(t)={X(s):s∈[a(t),b(t)]}.

If Y has the Markov property, then it is a Markovian representation of X.

An example of a non-Markovian process with a Markovian representation is an autoregressive time series of order greater than one.[58]

### Hitting times

Main article: Phase-type distribution

The hitting time is the time, starting in a given set of states, until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution. The simplest such distribution is that of a single exponentially distributed transition.

#### Expected hitting times

For a subset of states A ⊆ S, the vector kA of hitting times (where element kiA represents the expected value, starting in state i that the chain enters one of the states in the set A) is the minimal non-negative solution to[59]

kiA=0 for i∈A−∑j∈SqijkjA=1 for i∉A.

### Time reversal

For a CTMC Xt, the time-reversed process is defined to be X^t=XT−t. By Kelly's lemma this process has the same stationary distribution as the forward process.

A chain is said to be reversible if the reversed process is the same as the forward process. Kolmogorov's criterion states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.

### Embedded Markov chain

One method of finding the stationary probability distribution, π, of an ergodic continuous-time Markov chain, Q, is by first finding its embedded Markov chain (EMC). Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a jump process. Each element of the one-step transition probability matrix of the EMC, S, is denoted by sij, and represents the conditional probability of transitioning from state i into state j. These conditional probabilities may be found by

sij={qij∑k≠iqikif i≠j0otherwise.

From this, S may be written as

S=I−(diag⁡(Q))−1Q

where I is the identity matrix and diag(Q) is the diagonal matrix formed by selecting the main diagonal from the matrix Q and setting all other elements to zero.

To find the stationary probability distribution vector, we must next find φ such that

φS=φ,

with φ being a row vector, such that all elements in φ are greater than 0 and ‖φ‖1 = 1. From this, π may be found as

π=−φ(diag⁡(Q))−1‖φ(diag⁡(Q))−1‖1.

(S may be periodic, even if Q is not. Once π is found, it must be normalized to a unit vector.)

Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton--the (discrete-time) Markov chain formed by observing X(t) at intervals of δ units of time. The random variables X(0), X(δ), X(2δ), ... give the sequence of states visited by the δ-skeleton.

## Special types of Markov chains

### Markov model

Main article: Markov model

Markov models are used to model changing systems. There are 4 main types of models, that generalize Markov chains depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:

| System state is fully observable  | System state is partially observable

|
|

System is autonomous  | Markov chain  | Hidden Markov model
System is controlled  | Markov decision process | Partially observable Markov decision process

### Bernoulli scheme

Main article: Bernoulli scheme

A Bernoulli scheme is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is independent of even the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as a Bernoulli process.

Note, however, by the Ornstein isomorphism theorem, that every aperiodic and irreducible Markov chain is isomorphic to a Bernoulli scheme;[60] thus, one might equally claim that Markov chains are a "special case" of Bernoulli schemes. The isomorphism generally requires a complicated recoding. The isomorphism theorem is even a bit stronger: it states that any stationary stochastic process is isomorphic to a Bernoulli scheme; the Markov chain is just one such example.

### Subshift of finite type

Main article: Subshift of finite type

When the Markov matrix is replaced by the adjacency matrix of a finite graph, the resulting shift is termed a topological Markov chain or a subshift of finite type.[60] A Markov matrix that is compatible with the adjacency matrix can then provide a measure on the subshift. Many chaotic dynamical systems are isomorphic to topological Markov chains; examples include diffeomorphisms of closed manifolds, the Prouhet–Thue–Morse system, the Chacon system, sofic systems, context-free systems and block-coding systems.[60]

## Applications

Markov chains have been employed in a wide range of topics across the natural and social sciences, and in technological applications. They have been used for forecasting in several areas: for example, price trends,[61] wind power,[62] terrorism risk,[63][64] and solar irradiance.[65] The Markov chain forecasting models utilize a variety of settings, from discretizing the time series,[62] to hidden Markov models combined with wavelets,[61] and the Markov chain mixture distribution model (MCM).[65]

### Physics

Markovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.[66][67] For example, a thermodynamic state operates under a probability distribution that is difficult or expensive to acquire. Therefore, Markov Chain Monte Carlo method can be used to draw samples randomly from a black-box to approximate the probability distribution of attributes over a range of objects.[67]

Markov chains are used in lattice QCD simulations.[68]

### Chemistry

E+S↽−−⇀ESubstratebindingS⟶ECatalyticstep+P

Michaelis-Menten kinetics. The enzyme (E) binds a substrate (S) and produces a product (P). Each reaction is a state transition in a Markov chain.

A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain.[69] Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large number n of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially independent of each other, the number of molecules in state A or B at a time is n times the probability a given molecule is in that state.

The classical model of enzyme activity, Michaelis–Menten kinetics, can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains.[70]

An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals in silico towards a desired class of compounds such as drugs or natural products.[71] As a molecule is grown, a fragment is selected from the nascent molecule as the "current" state. It is not aware of its past (that is, it is not aware of what is already bonded to it). It then transitions to the next state when a fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds.[72]

Also, the growth (and composition) of copolymers may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (for example, whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due to steric effects, second-order Markov effects may also play a role in the growth of some polymer chains.

Similarly, it has been suggested that the crystallization and growth of some epitaxial superlattice oxide materials can be accurately described by Markov chains.[73]

### Biology

Markov chains are used in various areas of biology. Notable examples include:

* Phylogenetics and bioinformatics, where most models of DNA evolution use continuous-time Markov chains to describe the nucleotide present at a given site in the genome.
* Population dynamics, where Markov chains are in particular a central tool in the theoretical study of matrix population models.
* Neurobiology, where Markov chains have been used, e.g., to simulate the mammalian neocortex.[74]
* Systems biology, for instance with the modeling of viral infection of single cells.[75]
* Compartmental models for disease outbreak and epidemic modeling.

### Testing

Several theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoining Markov chains to form a "Markov blanket", arranging these chains in several recursive layers ("wafering") and producing more efficient test sets—samples—as a replacement for exhaustive testing.[citation needed]

### Solar irradiance variability

Solar irradiance variability assessments are useful for solar power applications. Solar irradiance variability at any location over time is mainly a consequence of the deterministic variability of the sun's path across the sky dome and the variability in cloudiness. The variability of accessible solar irradiance on Earth's surface has been modeled using Markov chains,[76][77][78][79] also including modeling the two states of clear and cloudiness as a two-state Markov chain.[80][81]

### Speech recognition

Hidden Markov models have been used in automatic speech recognition systems.[82]

### Information theory

Markov chains are used throughout information processing. Claude Shannon's famous 1948 paper A Mathematical Theory of Communication, which in a single step created the field of information theory, opens by introducing the concept of entropy by modeling texts in a natural language (such as English) as generated by an ergodic Markov process, where each letter may depend statistically on previous letters.[83] Such idealized models can capture many of the statistical regularities of systems. Even without describing the full structure of the system perfectly, such signal models can make possible very effective data compression through entropy encoding techniques such as arithmetic coding. They also allow effective state estimation and pattern recognition. Markov chains also play an important role in reinforcement learning.

Markov chains are also the basis for hidden Markov models, which are an important tool in such diverse fields as telephone networks (which use the Viterbi algorithm for error correction), speech recognition and bioinformatics (such as in rearrangements detection[84]).

The LZMA lossless data compression algorithm combines Markov chains with Lempel-Ziv compression to achieve very high compression ratios.

### Queueing theory

Main article: Queueing theory

Markov chains are the basis for the analytical treatment of queues (queueing theory). Agner Krarup Erlang initiated the subject in 1917.[85] This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).[86]

Numerous queueing models use continuous-time Markov chains. For example, an M/M/1 queue is a CTMC on the non-negative integers where upward transitions from i to i + 1 occur at rate λ according to a Poisson process and describe job arrivals, while transitions from i to i – 1 (for i > 1) occur at rate μ (job service times are exponentially distributed) and describe completed services (departures) from the queue.

### Internet applications

A state diagram that represents the PageRank algorithm with a transitional probability of M, or αki+1−αN

The PageRank of a webpage as used by Google is defined by a Markov chain.[87][88][89] It is the probability to be at page i in the stationary distribution on the following Markov chain on all (known) webpages. If N is the number of known webpages, and a page i has ki links to it then it has transition probability αki+1−αN for all pages that are linked to and 1−αN for all pages that are not linked to. The parameter α is taken to be about 0.15.[90]

Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.[citation needed]

### Statistics

Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC). In recent years this has revolutionized the practicability of Bayesian inference methods, allowing a wide range of posterior distributions to be simulated and their parameters found numerically.[citation needed]

### Conflict and combat

In 1971 a Naval Postgraduate School Master's thesis proposed to model a variety of combat between adversaries as a Markov chain "with states reflecting the control, maneuver, target acquisition, and target destruction actions of a weapons system" and discussed the parallels between the resulting Markov chain and Lanchester's laws.[91]

In 1975 Duncan and Siverson remarked that Markov chains could be used to model conflict between state actors, and thought that their analysis would help understand "the behavior of social and political organizations in situations of conflict."[92]

### Economics and finance

Markov chains are used in finance and economics to model a variety of different phenomena, including the distribution of income, the size distribution of firms, asset prices and market crashes. D. G. Champernowne built a Markov chain model of the distribution of income in 1953.[93] Herbert A. Simon and co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes.[94] Louis Bachelier was the first to observe that stock prices followed a random walk.[95] The random walk was later seen as evidence in favor of the efficient-market hypothesis and random walk models were popular in the literature of the 1960s.[96] Regime-switching models of business cycles were popularized by James D. Hamilton (1989), who used a Markov chain to model switches between periods of high and low GDP growth (or, alternatively, economic expansions and recessions).[97] A more recent example is the Markov switching multifractal model of Laurent E. Calvet and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models.[98][99] It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.

Dynamic macroeconomics makes heavy use of Markov chains. An example is using Markov chains to exogenously model prices of equity (stock) in a general equilibrium setting.[100]

Credit rating agencies produce annual tables of the transition probabilities for bonds of different credit ratings.[101]

### Social sciences

Markov chains are generally used in describing path-dependent arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due to Karl Marx's Das Kapital, tying economic development to the rise of capitalism. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of the middle class, the ratio of urban to rural residence, the rate of political mobilization, etc., will generate a higher probability of transitioning from authoritarian to democratic regime.[102]

### Music

Markov chains are employed in algorithmic music composition, particularly in software such as Csound, Max, and SuperCollider. In a first-order chain, the states of the system become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce output note values based on the transition matrix weightings, which could be MIDI note values, frequency (Hz), or any other desirable metric.[103]

1st-order matrix  Note | A | C♯ | E♭

|
|
|

A  | 0.1 | 0.6 | 0.3
C♯ | 0.25 | 0.05 | 0.7
E♭ | 0.7 | 0.3 | 0
2nd-order matrix  Notes | A | D | G

|
|
|

AA  | 0.18 | 0.6 | 0.22
AD  | 0.5 | 0.5 | 0
AG  | 0.15 | 0.75 | 0.1
DD  | 0 | 0 | 1
DA  | 0.25 | 0 | 0.75
DG  | 0.9 | 0.1 | 0
GG  | 0.4 | 0.4 | 0.2
GA  | 0.5 | 0.25 | 0.25
GD  | 1 | 0 | 0

A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table. Higher, nth-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of phrasal structure, rather than the 'aimless wandering' produced by a first-order system.[104]

Markov chains can be used structurally, as in Xenakis's Analogique A and B.[105] Markov chains are also used in systems which use a Markov model to react interactively to music input.[106]

Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this limitation, a new approach has been proposed.[107]

### Games and sports

Markov chains can be used to model many games of chance. The children's games Snakes and Ladders and "Hi Ho! Cherry-O", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares).[citation needed]

Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs and position of the runners. Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team.[108] He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such as bunting and base stealing and differences when playing on grass vs. AstroTurf.[109]

### Markov text generators

Markov processes can also be used to generate superficially real-looking text given a sample document. Markov processes are used in a variety of recreational "parody generator" software (see dissociated press, Jeff Harrison,[110] Mark V. Shaney,[111][112] and Academias Neutronium). Several open-source text generation libraries using Markov chains exist.

## See also

* Dynamics of Markovian particles
* Gauss–Markov process
* Markov chain approximation method
* Markov chain geostatistics
* Markov chain mixing time
* Markov chain tree theorem
* Markov decision process
* Markov information source
* Markov odometer
* Markov operator
* Markov random field
* Master equation
* Quantum Markov chain
* Semi-Markov process
* Stochastic cellular automaton
* Telescoping Markov chain
* Variable-order Markov model

## Notes

1. ↑ 1.0 1.1 Sean Meyn; Richard L. Tweedie (2 April 2009). Markov Chains and Stochastic Stability. Cambridge University Press. p. 3. ISBN 978-0-521-73182-9. https://books.google.com/books?id=Md7RnYEPkJwC.
2. ↑ Reuven Y. Rubinstein; Dirk P. Kroese (20 September 2011). Simulation and the Monte Carlo Method. John Wiley & Sons. p. 225. ISBN 978-1-118-21052-9. https://books.google.com/books?id=yWcvT80gQK4C.
3. ↑ Dani Gamerman; Hedibert F. Lopes (10 May 2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition. CRC Press. ISBN 978-1-58488-587-0. https://books.google.com/books?id=yPvECi_L3bwC.
4. ↑ Markovian (3rd ed.), Oxford University Press, September 2005, http://oed.com/search?searchType=dictionary&q=Markovian (Subscription or UK public library membership required.)
5. ↑ 5.0 5.1 Øksendal, B. K. (Bernt Karsten) (2003). Stochastic differential equations: an introduction with applications (6th ed.). Berlin: Springer. ISBN 3-540-04758-1. OCLC 52203046.
6. ↑ 6.0 6.1 Søren Asmussen (15 May 2003). Applied Probability and Queues. Springer Science & Business Media. p. 7. ISBN 978-0-387-00211-8. https://books.google.com/books?id=BeYaTxesKy0C.
7. ↑ Emanuel Parzen (17 June 2015). Stochastic Processes. Courier Dover Publications. p. 188. ISBN 978-0-486-79688-8. https://books.google.com/books?id=0mB2CQAAQBAJ.
8. ↑ Samuel Karlin; Howard E. Taylor (2 December 2012). A First Course in Stochastic Processes. Academic Press. pp. 29 and 30. ISBN 978-0-08-057041-9. https://books.google.com/books?id=dSDxjX9nmmMC.
9. ↑ John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 106–121. ISBN 978-3-540-90275-1. https://books.google.com/books?id=Pd4cvgAACAAJ.
10. ↑ Sheldon M. Ross (1996). Stochastic processes. Wiley. pp. 174 and 231. ISBN 978-0-471-12062-9. https://books.google.com/books?id=ImUPAQAAMAAJ.
11. ↑ Everitt, B.S. (2002) The Cambridge Dictionary of Statistics. CUP. ISBN 0-521-81099-X
12. ↑ Parzen, E. (1962) Stochastic Processes, Holden-Day. ISBN 0-8162-6664-6 (Table 6.1)
13. ↑ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entry for "Markov chain")
14. ↑ Dodge, Y. The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
15. ↑ Meyn, S. Sean P., and Richard L. Tweedie. (2009) Markov chains and stochastic stability. Cambridge University Press. (Preface, p. iii)
16. ↑ 16.0 16.1 16.2 16.3 16.4 Charles Miller Grinstead; James Laurie Snell (1997). Introduction to Probability. American Mathematical Soc.. pp. 464–466. ISBN 978-0-8218-0749-1. https://archive.org/details/flooved3489.
17. ↑ 17.0 17.1 17.2 Pierre Bremaud (9 March 2013). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer Science & Business Media. p. ix. ISBN 978-1-4757-3124-8. https://books.google.com/books?id=jrPVBwAAQBAJ.
18. ↑ 18.0 18.1 18.2 Hayes, Brian (2013). "First links in the Markov chain". American Scientist 101 (2): 92–96. doi:10.1511/2013.101.92.
19. ↑ 19.0 19.1 Sheldon M. Ross (1996). Stochastic processes. Wiley. pp. 235 and 358. ISBN 978-0-471-12062-9. https://books.google.com/books?id=ImUPAQAAMAAJ.
20. ↑ Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: The early years, 1880–1970". A Festschrift for Herman Rubin. pp. 75–91. doi:10.1214/lnms/1196285381. ISBN 978-0-940600-61-4.
21. ↑ Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes". International Statistical Review 80 (2): 253–268. doi:10.1111/j.1751-5823.2012.00181.x.
22. ↑ Seneta, E. (1996). "Markov and the Birth of Chain Dependence Theory". International Statistical Review 64 (3): 255–257. doi:10.2307/1403785.
23. ↑ Seneta, E. (1998). "I.J. Bienaymé [1796–1878]: Criticality, Inequality, and Internationalization". International Statistical Review 66 (3): 291–292. doi:10.2307/1403518.
24. ↑ "Maurice Fréchet". Statisticians of the Centuries. New York, NY: Springer. 2001. pp. 331–334. doi:10.1007/978-1-4613-0179-0_71. ISBN 978-0-387-95283-3.
25. ↑ 25.0 25.1 25.2 Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W. et al. (1990). "Andrei Nikolaevich Kolmogorov (1903–1987)". Bulletin of the London Mathematical Society 22 (1): 33. doi:10.1112/blms/22.1.31.
26. ↑ 26.0 26.1 Cramér, Harald (1976). "Half a Century with Probability Theory: Some Personal Recollections". The Annals of Probability 4 (4): 509–546. doi:10.1214/aop/1176996025.
27. ↑ Marc Barbut; Bernard Locker; Laurent Mazliak (23 August 2016). Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters. Springer London. p. 5. ISBN 978-1-4471-7262-8. https://books.google.com/books?id=lSz_vQAACAAJ.
28. ↑ Valeriy Skorokhod (5 December 2005). Basic Principles and Applications of Probability Theory. Springer Science & Business Media. p. 146. ISBN 978-3-540-26312-8. https://books.google.com/books?id=dQkYMjRK3fYC.
29. ↑ Bernstein, Jeremy (2005). "Bachelier". American Journal of Physics 73 (5): 395–398. doi:10.1119/1.1848117. Bibcode: 2005AmJPh..73..395B.
30. ↑ William J. Anderson (6 December 2012). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer Science & Business Media. p. vii. ISBN 978-1-4612-3038-0. https://books.google.com/books?id=YpHfBwAAQBAJ&pg=PR8.
31. ↑ Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W. et al. (1990). "Andrei Nikolaevich Kolmogorov (1903–1987)". Bulletin of the London Mathematical Society 22 (1): 57. doi:10.1112/blms/22.1.31.
32. ↑ Subramanian, Devika (Fall 2008). "The curious case of Mark V. Shaney". Comp 140 course notes, Fall 2008. William Marsh Rice University. https://www.cs.rice.edu/~devika/comp140/Shaney.pdf.
33. ↑ 33.0 33.1 Ionut Florescu (7 November 2014). Probability and Stochastic Processes. John Wiley & Sons. pp. 373 and 374. ISBN 978-1-118-59320-2. https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22.
34. ↑ 34.0 34.1 Samuel Karlin; Howard E. Taylor (2 December 2012). A First Course in Stochastic Processes. Academic Press. p. 49. ISBN 978-0-08-057041-9. https://books.google.com/books?id=dSDxjX9nmmMC.
35. ↑ Weiss, George H. (2006). "Random Walks". Encyclopedia of Statistical Sciences. p. 1. doi:10.1002/0471667196.ess2180.pub2. ISBN 978-0-471-66719-3.
36. ↑ Michael F. Shlesinger (1985). The Wonderful world of stochastics: a tribute to Elliott W. Montroll. North-Holland. pp. 8–10. ISBN 978-0-444-86937-1. https://books.google.com/books?id=p6fvAAAAMAAJ.
37. ↑ Emanuel Parzen (17 June 2015). Stochastic Processes. Courier Dover Publications. p. 7, 8. ISBN 978-0-486-79688-8. https://books.google.com/books?id=0mB2CQAAQBAJ.
38. ↑ Joseph L. Doob (1990). Stochastic processes. Wiley. p. 46, 47. https://books.google.com/books?id=7Bu8jgEACAAJ.
39. ↑ Donald L. Snyder; Michael I. Miller (6 December 2012). Random Point Processes in Time and Space. Springer Science & Business Media. p. 32. ISBN 978-1-4612-3166-0. https://books.google.com/books?id=c_3UBwAAQBAJ.
40. ↑ Norris, J. R. (1997). "Continuous-time Markov chains I". Markov Chains. pp. 60–107. doi:10.1017/CBO9780511810633.004. ISBN 978-0-511-81063-3.
41. ↑ 41.0 41.1 Serfozo, Richard (2009). Basics of Applied Stochastic Processes. Probability and Its Applications. Berlin: Springer. doi:10.1007/978-3-540-89332-5. ISBN 978-3-540-89331-8.
42. ↑ "Chapter 11 "Markov Chains"". https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf.
43. ↑ Schmitt, Florian; Rothlauf, Franz (2001). "On the Importance of the Second Largest Eigenvalue on the Convergence Rate of Genetic Algorithms". Proceedings of the 14th Symposium on Reliable Distributed Systems.
44. ↑ Rosenthal, Jeffrey S. (1995). "Convergence Rates for Markov Chains". SIAM Review 37 (3): 387–405. doi:10.1137/1037083.
45. ↑ Franzke, Brandon; Kosko, Bart (1 October 2011). "Noise can speed convergence in Markov chains". Physical Review E 84 (4). doi:10.1103/PhysRevE.84.041112. PMID 22181092. Bibcode: 2011PhRvE..84d1112F.
46. ↑ Spitzer, Frank (1970). "Interaction of Markov Processes". Advances in Mathematics 5 (2): 246–290. doi:10.1016/0001-8708(70)90034-4.
47. ↑ Dobrushin, R. L.; Kryukov, V.I.; Toom, A. L. (1978). Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. Manchester University Press. ISBN 978-0-7190-2206-7. https://books.google.com/books?id=0Wa7AAAAIAAJ&q=locally+interacting+markov+chains+toom+Dobrushin&pg=PA181. Retrieved 2016-03-04.
48. ↑ Heyman, Daniel P.; Sobel, Mathew J. (1982). Stochastic Models in Operations Research, Volume 1. New York: McGraw-Hill. p. 230. ISBN 0-07-028631-0.
49. ↑ Peres, Yuval. "Show that positive recurrence is a class property" (in en). https://math.stackexchange.com/questions/4572155/show-that-positive-recurrence-is-a-class-property.
50. ↑ Lalley, Steve (2016). "Markov Chains: Basic Theory". http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf.
51. ↑ Parzen, Emanuel (1962). Stochastic Processes. San Francisco: Holden-Day. p. 145. ISBN 0-8162-6664-6.
52. ↑ 52.0 52.1 Shalizi, Cosma (1 Dec 2023). "Ergodic Theory". http://bactra.org/notebooks/ergodic-theory.html.
53. ↑ Seneta, E. (Eugene) (1973). Non-negative matrices; an introduction to theory and applications. Internet Archive. New York, Wiley. ISBN 978-0-470-77605-6. http://archive.org/details/nonnegativematri00esen_0.
54. ↑ "10.3: Regular Markov Chains" (in en). 2020-03-22. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Applied_Finite_Mathematics_(Sekhon_and_Bloom)/10:_Markov_Chains/10.03:_Regular_Markov_Chains.
55. ↑ Seneta, E. (Eugene) (1973). "2.4. Combinatorial properties". Non-negative matrices; an introduction to theory and applications. Internet Archive. New York, Wiley. ISBN 978-0-470-77605-6. http://archive.org/details/nonnegativematri00esen_0.
56. ↑ Shen, Jian (1996-10-15). "An improvement of the Dulmage-Mendelsohn theorem". Discrete Mathematics 158 (1): 295–297. doi:10.1016/0012-365X(95)00060-A.
57. ↑ Kallenberg, Olav (2002). Foundations of modern probability. Probability and its applications (2. ed., [Nachdr.] ed.). New York, NY Berlin Heidelberg: Springer. Proposition 8.6 (page 145). ISBN 978-0-387-95313-7.
58. ↑ Doblinger, G. (September 1998). "Smoothing of noisy AR signals using an adaptive Kalman filter". 9th European Signal Processing Conference (EUSIPCO 1998): 781–784. https://publik.tuwien.ac.at/files/pub-et_3285.pdf.
59. ↑ Norris, J. R. (1997). "Continuous-time Markov chains II". Markov Chains. pp. 108–127. doi:10.1017/CBO9780511810633.005. ISBN 978-0-511-81063-3.
60. ↑ 60.0 60.1 60.2 Matthew Nicol and Karl Petersen, (2009) "Ergodic Theory: Basic Examples and Constructions", Encyclopedia of Complexity and Systems Science, Springer https://doi.org/10.1007/978-0-387-30440-3_177
61. ↑ 61.0 61.1 de Souza e Silva, E.G.; Legey, L.F.L.; de Souza e Silva, E.A. (2010). "Forecasting oil price trends using wavelets and hidden Markov models". Energy Economics 32 (6): 1507. doi:10.1016/j.eneco.2010.08.006. Bibcode: 2010EneEc..32.1507D. https://www.sciencedirect.com/science/article/pii/S0140988310001271.
62. ↑ 62.0 62.1 Carpinone, A; Giorgio, M; Langella, R.; Testa, A. (2015). "Markov chain modeling for very-short-term wind power forecasting". Electric Power Systems Research 122: 152–158. doi:10.1016/j.epsr.2014.12.025. Bibcode: 2015EPSR..122..152C.
63. ↑ Woo, Gordon (2002-04-01). "Quantitative Terrorism Risk Assessment". The Journal of Risk Finance 4 (1): 7–14. doi:10.1108/eb022949. https://www.emerald.com/insight/content/doi/10.1108/eb022949/full/html. Retrieved 5 October 2023.
64. ↑ Woo, Gordon (December 2003). "Insuring Against Al-Quaeda". Cambridge: National Bureau of Economic Research. https://conference.nber.org/confer/2003/insurance03/woo.pdf. Retrieved 26 March 2024.
65. ↑ 65.0 65.1 Munkhammar, J.; van der Meer, D.W.; Widén, J. (2019). "Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model". Solar Energy 184: 688–695. doi:10.1016/j.solener.2019.04.014. Bibcode: 2019SoEn..184..688M.
66. ↑ Fitzpatrick, Richard. "Thermodynamics and Statistical Mechanics". https://farside.ph.utexas.edu/teaching/sm1/Thermal.pdf.
67. ↑ 67.0 67.1 van Ravenzwaaij, Don; Cassey, Pete; Brown, Scott D. (2016-03-11). "A simple introduction to Markov Chain Monte–Carlo sampling". Psychonomic Bulletin & Review 25 (1): 143–154. doi:10.3758/s13423-016-1015-8. PMID 26968853.
68. ↑ Gattringer, Christof; Lang, Christian B (2010). Quantum Chromodynamics on the Lattice. Lecture Notes in Physics. 788. Springer-Verlag Berlin Heidelberg. doi:10.1007/978-3-642-01850-3. ISBN 978-3-642-01849-7. https://www.springer.com/gb/book/9783642018497.
69. ↑ Anderson, David F.; Kurtz, Thomas G. (2011), "Continuous Time Markov Chain Models for Chemical Reaction Networks", Design and Analysis of Biomolecular Circuits (Springer New York): pp. 3–42, doi:10.1007/978-1-4419-6766-4_1, ISBN 978-1-4419-6765-7
70. ↑ Du, Chao; Kou, S. C. (September 2012). "Correlation analysis of enzymatic reaction of a single protein molecule". The Annals of Applied Statistics 6 (3): 950–976. doi:10.1214/12-aoas541. PMID 23408514. Bibcode: 2012arXiv1209.6210D.
71. ↑ Kutchukian, Peter; Lou, David; Shakhnovich, Eugene (2009). "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of Molecules occupying Druglike Chemical". Journal of Chemical Information and Modeling 49 (7): 1630–1642. doi:10.1021/ci9000458. PMID 19527020.
72. ↑ Kutchukian, P.S.; Lou, D.; Shakhnovich, Eugene I. (2009-06-15). "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of Molecules Occupying Druglike Chemical Space". Journal of Chemical Information and Modeling 49 (7): 1630–1642. doi:10.1021/ci9000458. PMID 19527020.
73. ↑ Kopp, V. S.; Kaganer, V. M.; Schwarzkopf, J.; Waidick, F.; Remmele, T.; Kwasniewski, A.; Schmidbauer, M. (2011). "X-ray diffraction from nonperiodic layered structures with correlations: Analytical calculation and experiment on mixed Aurivillius films". Acta Crystallographica Section A 68 (Pt 1): 148–155. doi:10.1107/S0108767311044874. PMID 22186291. Bibcode: 2012AcCrA..68..148K.
74. ↑ George, Dileep; Hawkins, Jeff (2009). Friston, Karl J.. ed. "Towards a Mathematical Theory of Cortical Micro-circuits". PLOS Comput Biol 5 (10). doi:10.1371/journal.pcbi.1000532. PMID 19816557. Bibcode: 2009PLSCB...5E0532G.
75. ↑ Gupta, Ankur; Rawlings, James B. (April 2014). "Comparison of Parameter Estimation Methods in Stochastic Chemical Kinetic Models: Examples in Systems Biology". AIChE Journal 60 (4): 1253–1268. doi:10.1002/aic.14409. PMID 27429455. Bibcode: 2014AIChE..60.1253G.
76. ↑ Aguiar, R. J.; Collares-Pereira, M.; Conde, J. P. (1988). "Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices". Solar Energy 40 (3): 269–279. doi:10.1016/0038-092X(88)90049-7. Bibcode: 1988SoEn...40..269A.
77. ↑ Ngoko, B. O.; Sugihara, H.; Funaki, T. (2014). "Synthetic generation of high temporal resolution solar radiation data using Markov models". Solar Energy 103: 160–170. doi:10.1016/j.solener.2014.02.026. Bibcode: 2014SoEn..103..160N.
78. ↑ Bright, J. M.; Smith, C. I.; Taylor, P. G.; Crook, R. (2015). "Stochastic generation of synthetic minutely irradiance time series derived from mean hourly weather observation data". Solar Energy 115: 229–242. doi:10.1016/j.solener.2015.02.032. Bibcode: 2015SoEn..115..229B.
79. ↑ Munkhammar, J.; Widén, J. (2018). "An N-state Markov-chain mixture distribution model of the clear-sky index". Solar Energy 173: 487–495. doi:10.1016/j.solener.2018.07.056. Bibcode: 2018SoEn..173..487M.
80. ↑ Morf, H. (1998). "The stochastic two-state solar irradiance model (STSIM)". Solar Energy 62 (2): 101–112. doi:10.1016/S0038-092X(98)00004-8. Bibcode: 1998SoEn...62..101M.
81. ↑ Munkhammar, J.; Widén, J. (2018). "A Markov-chain probability distribution mixture approach to the clear-sky index". Solar Energy 170: 174–183. doi:10.1016/j.solener.2018.05.055. Bibcode: 2018SoEn..170..174M.
82. ↑ Mor, Bhavya; Garhwal, Sunita; Kumar, Ajay (May 2021). "A Systematic Review of Hidden Markov Models and Their Applications" (in en). Archives of Computational Methods in Engineering 28 (3): 1429–1448. doi:10.1007/s11831-020-09422-4. ISSN 1134-3060. https://link.springer.com/10.1007/s11831-020-09422-4.
83. ↑ Thomsen, Samuel W. (2009), "Some evidence concerning the genesis of Shannon's information theory", Studies in History and Philosophy of Science 40 (1): 81–91, doi:10.1016/j.shpsa.2008.12.011, Bibcode: 2009SHPSA..40...81T
84. ↑ Pratas, D; Silva, R; Pinho, A; Ferreira, P (May 18, 2015). "An alignment-free method to find and visualise rearrangements between pairs of DNA sequences". Scientific Reports 5 (10203). doi:10.1038/srep10203. PMID 25984837. Bibcode: 2015NatSR...510203P.
85. ↑ O'Connor, John J.; Robertson, Edmund F., "Markov chain", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Erlang.html .
86. ↑ S. P. Meyn, 2007. Control Techniques for Complex Networks , Cambridge University Press, 2007.
87. ↑ U.S. Patent 6,285,999
88. ↑ Gupta, Brij; Agrawal, Dharma P.; Yamaguchi, Shingo (16 May 2016). Handbook of Research on Modern Cryptographic Solutions for Computer and Cyber Security. IGI Global. pp. 448–. ISBN 978-1-5225-0106-0. https://books.google.com/books?id=Ctk6DAAAQBAJ&pg=PA448.
89. ↑ Langville, Amy N.; Meyer, Carl D. (2006). "A Reordering for the PageRank Problem". SIAM Journal on Scientific Computing 27 (6): 2112–2113. doi:10.1137/040607551. Bibcode: 2006SJSC...27.2112L. http://meyer.math.ncsu.edu/Meyer/PS_Files/ReorderingPageRank.pdf.
90. ↑ Page, Lawrence; Brin, Sergey; Motwani, Rajeev; Winograd, Terry (1999). The PageRank Citation Ranking: Bringing Order to the Web (Technical report). CiteSeerX 10.1.1.31.1768.
91. ↑ Reese, Thomas Fred (September 1971). "A Finite Markov Chain Model of the Combat Process". Naval Postgraduate School. https://apps.dtic.mil/sti/citations/AD0736113.
92. ↑ Duncan, George T.; Siverson, Randolph M. (1975). "Markov Chain Models for Conflict Analysis: Results from Sino-Indian Relations, 1959-1964". International Studies Quarterly 19 (3): 344–374. doi:10.2307/2600315.
93. ↑ Champernowne, D (1953). "A model of income distribution". The Economic Journal 63 (250): 318–51. doi:10.2307/2227127.
94. ↑ Simon, Herbert; C Bonini (1958). "The size distribution of business firms". Am. Econ. Rev. 42: 425–40.
95. ↑ Bachelier, Louis (1900). "Théorie de la spéculation". Annales Scientifiques de l'École Normale Supérieure 3: 21–86. doi:10.24033/asens.476.
96. ↑ e.g.Fama, E (1965). "The behavior of stock market prices". Journal of Business 38.
97. ↑ Hamilton, James (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle". Econometrica 57 (2): 357–84. doi:10.2307/1912559.
98. ↑ Calvet, Laurent E.; Fisher, Adlai J. (2001). "Forecasting Multifractal Volatility". Journal of Econometrics 105 (1): 27–58. doi:10.1016/S0304-4076(01)00069-0. http://archive.nyu.edu/handle/2451/26894.
99. ↑ Calvet, Laurent; Adlai Fisher (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes". Journal of Financial Econometrics 2: 49–83. doi:10.1093/jjfinec/nbh003.
100. ↑ Brennan, Michael; Xiab, Yihong. "Stock Price Volatility and the Equity Premium". http://bbs.cenet.org.cn/uploadImages/200352118122167693.pdf.
101. ↑ "A Markov Chain Example in Credit Risk Modelling". http://www.columbia.edu/~ww2040/4106S11/MC_BondRating.pdf.
102. ↑ Acemoglu, Daron; Georgy Egorov; Konstantin Sonin (2011). "Political model of social evolution". Proceedings of the National Academy of Sciences 108 (Suppl 4): 21292–21296. doi:10.1073/pnas.1019454108. PMID 22198760. Bibcode: 2011PNAS..10821292A.
103. ↑ K McAlpine; E Miranda; S Hoggar (1999). "Making Music with Algorithms: A Case-Study System". Computer Music Journal 23 (2): 19–30. doi:10.1162/014892699559733.
104. ↑ Curtis Roads, ed (1996). The Computer Music Tutorial. MIT Press. ISBN 978-0-262-18158-7.
105. ↑ Xenakis, Iannis; Kanach, Sharon (1992) Formalized Music: Mathematics and Thought in Composition, Pendragon Press. ISBN 1576470792
106. ↑ "Continuator". http://www.csl.sony.fr/~pachet/.
107. ↑ Pachet, F.; Roy, P.; Barbieri, G. (2011) "Finite-Length Markov Processes with Constraints" , Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI, pages 635–642, Barcelona, Spain, July 2011
108. ↑ Pankin, Mark D.. "MARKOV CHAIN MODELS: THEORETICAL BACKGROUND". http://www.pankin.com/markov/theory.htm.
109. ↑ Pankin, Mark D.. "BASEBALL AS A MARKOV CHAIN". http://www.pankin.com/markov/intro.htm.
110. ↑ "Poet's Corner – Fieralingue". http://www.fieralingue.it/modules.php?name=Content&pa=list_pages_categories&cid=111.
111. ↑ Kenner, Hugh; O'Rourke (November 1984). "A Travesty Generator for Micros". BYTE 9 (12): 129–131, 449–469.
112. ↑ Hartman, Charles (1996). Virtual Muse: Experiments in Computer Poetry. Hanover, NH: Wesleyan University Press. ISBN 978-0-8195-2239-9. https://archive.org/details/virtualmuseexper00hart.

## References

* A. A. Markov (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga". Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya, tom 15, pp. 135–156.
* A. A. Markov (1971). "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons.
* Classical Text in Translation: Markov, A. A. (2006). "An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains". Science in Context 19 (4): 591–600. doi:10.1017/s0269889706001074.
* Leo Breiman (1992) [1968] Probability. Original edition published by Addison-Wesley; reprinted by Society for Industrial and Applied Mathematics ISBN 0-89871-296-3. (See Chapter 7)
* J. L. Doob (1953) Stochastic Processes. New York: John Wiley and Sons ISBN 0-471-52369-0.
* S. P. Meyn and R. L. Tweedie (1993) Markov Chains and Stochastic Stability. London: Springer-Verlag ISBN 0-387-19832-6. online: MCSS . Second edition to appear, Cambridge University Press, 2009.
* Markov Processes. Grundlehren der mathematischen Wissenschaften. I (121). Berlin: Springer-Verlag. 1965. doi:10.1007/978-3-662-00031-1. Title-No. 5104. ISBN 978-3-662-00033-5. https://archive.org/details/markovprocesses0001dynk. ;  Markov Processes. Grundlehren der mathematischen Wissenschaften. II (122). 1965. doi:10.1007/978-3-662-25360-1. Title-No. 5105. ISBN 978-3-662-23320-7. https://archive.org/details/markovprocesses0002dynk. (NB. This was originally published in Russian as Марковские процессы (Markovskiye protsessy) by Fizmatgiz in 1963 and translated to English with the assistance of the author.)
* S. P. Meyn. Control Techniques for Complex Networks. Cambridge University Press, 2007. ISBN 978-0-521-88441-9. Appendix contains abridged Meyn & Tweedie. online: CTCN
* Booth, Taylor L. (1967). Sequential Machines and Automata Theory (1st ed.). New York, NY: John Wiley and Sons, Inc.. Library of Congress Card Catalog Number 67-25924. ] Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes pp. 449ff. Discusses Z-transforms, D transforms in their context.
* Kemeny, John G.; Hazleton Mirkil; J. Laurie Snell; Gerald L. Thompson (1959). Finite Mathematical Structures (1st ed.). Englewood Cliffs, NJ: Prentice-Hall, Inc.. Library of Congress Card Catalog Number 59-12841. https://archive.org/details/finitemathematic0000keme_h5g0. Classical text. cf Chapter 6 Finite Markov Chains pp. 384ff.
* John G. Kemeny & J. Laurie Snell (1960) Finite Markov Chains, D. van Nostrand Company ISBN 0-442-04328-7
* E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press, 1984, 2004. ISBN 0-521-60494-X
* Seneta, E. Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) ISBN 978-0-387-29765-1
* Kishor S. Trivedi, Probability and Statistics with Reliability, Queueing, and Computer Science Applications, John Wiley & Sons, Inc. New York, 2002. ISBN 0-471-33341-7.
* K. S. Trivedi and R.A.Sahner, SHARPE at the age of twenty-two, vol. 36, no. 4, pp. 52–57, ACM SIGMETRICS Performance Evaluation Review, 2009.
* R. A. Sahner, K. S. Trivedi and A. Puliafito, Performance and reliability analysis of computer systems: an example-based approach using the SHARPE software package, Kluwer Academic Publishers, 1996. ISBN 0-7923-9650-2.
* G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, Queueing Networks and Markov Chains, John Wiley, 2nd edition, 2006. ISBN 978-0-7923-9650-5.

## External links

* Hazewinkel, Michiel, ed. (2001), "Markov chain", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/m062350
* Markov Chains chapter in American Mathematical Society's introductory probability book
* Introduction to Markov Chains on YouTube
* A visual explanation of Markov Chains
* Original paper by A.A Markov (1913): An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains (translated from Russian)

* v
* t
* e

Stochastic processes

Discrete time|

* Bernoulli process
* Branching process
* Chinese restaurant process
* Galton–Watson process
* Independent and identically distributed random variables
* Markov chain
* Moran process
* Random walk
* Loop-erased
* Self-avoiding
* Biased
* Maximal entropy

Continuous time|

* Bessel process
* Birth–death process
* Brownian motion
* Bridge
* Excursion
* Fractional
* Geometric
* Meander
* Cauchy process
* Contact process
* Continuous-time random walk
* Cox process
* Diffusion process
* Empirical process
* Feller process
* Fleming–Viot process
* Gamma process
* Geometric process
* Hunt process
* Interacting particle systems
* Itô diffusion
* Itô process
* Jump diffusion
* Jump process
* Lévy process
* Local time
* Markov additive process
* McKean–Vlasov process
* Ornstein–Uhlenbeck process
* Poisson process
* Compound
* Non-homogeneous
* Point process
* Schramm–Loewner evolution
* Semimartingale
* Sigma-martingale
* Stable process
* Superprocess
* Telegraph process
* Variance gamma process
* Wiener process
* Wiener sausage

Both|

* Branching process
* Galves–Löcherbach model
* Gaussian process
* Hidden Markov model (HMM)
* Markov process
* Martingale
* Differences
* Local
* Sub-
* Super-
* Random dynamical system
* Regenerative process
* Renewal process
* Stochastic chains with memory of variable length
* White noise

Fields and other|

* Dirichlet process
* Gaussian random field
* Gibbs measure
* Hopfield model
* Ising model
* Potts model
* Boolean network
* Markov random field
* Percolation
* Pitman–Yor process
* Point process
* Cox
* Poisson
* Random field
* Random graph

Time series models|

* Autoregressive conditional heteroskedasticity (ARCH) model
* Autoregressive integrated moving average (ARIMA) model
* Autoregressive (AR) model
* Autoregressive–moving-average (ARMA) model
* Generalized autoregressive conditional heteroskedasticity (GARCH) model
* Moving-average (MA) model

Financial models|

* Black–Derman–Toy
* Black–Karasinski
* Black–Scholes
* Chen
* Constant elasticity of variance (CEV)
* Cox–Ingersoll–Ross (CIR)
* Garman–Kohlhagen
* Heath–Jarrow–Morton (HJM)
* Heston
* Ho–Lee
* Hull–White
* LIBOR market
* Rendleman–Bartter
* SABR volatility
* Vašíček
* Wilkie

Actuarial models|

* Bühlmann
* Cramér–Lundberg
* Risk process
* Sparre–Anderson

Queueing models|

* Bulk
* Fluid
* Generalized queueing network
* M/G/1
* M/M/1
* M/M/c

Properties|

* Càdlàg paths
* Continuous
* Continuous paths
* Ergodic
* Exchangeable
* Feller-continuous
* Gauss–Markov
* Markov
* Mixing
* Piecewise deterministic
* Predictable
* Progressively measurable
* Self-similar
* Stationary
* Time-reversible

Limit theorems|

* Central limit theorem
* Donsker's theorem
* Doob's martingale convergence theorems
* Ergodic theorem
* Fisher–Tippett–Gnedenko theorem
* Large deviation principle
* Law of large numbers (weak/strong)
* Law of the iterated logarithm
* Maximal ergodic theorem
* Sanov's theorem

Inequalities|

* Burkholder–Davis–Gundy
* Doob's martingale
* Kunita–Watanabe

Tools|

* Cameron–Martin formula
* Convergence of random variables
* Doléans-Dade exponential
* Doob decomposition theorem
* Doob–Meyer decomposition theorem
* Doob's optional stopping theorem
* Dynkin's formula
* Feynman–Kac formula
* Filtration
* Girsanov theorem
* Infinitesimal generator
* Itô integral
* Itô's lemma
* Karhunen–Loève_theorem
* Kolmogorov continuity theorem
* Kolmogorov extension theorem
* Lévy–Prokhorov metric
* Malliavin calculus
* Martingale representation theorem
* Optional stopping theorem
* Prokhorov's theorem
* Quadratic variation
* Reflection principle
* Skorokhod integral
* Skorokhod's representation theorem
* Skorokhod space
* Snell envelope
* Stochastic differential equation
* Tanaka
* Stopping time
* Stratonovich integral
* Uniform integrability
* Usual hypotheses
* Wiener space
* Classical
* Abstract

Disciplines|

* Actuarial mathematics
* Econometrics
* Ergodic theory
* Extreme value theory (EVT)
* Large deviations theory
* Mathematical finance
* Mathematical statistics
* Probability theory
* Queueing theory
* Renewal theory
* Ruin theory
* Statistics
* System on Chip design
* Stochastic analysis
* Time series analysis
* Machine learning

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