Probability theory

2020 Mathematics Subject Classification: Primary: 60-XX [MSN][ZBL]

A mathematical science in which the probabilities (cf. Probability) of certain random events are used to deduce the probabilities of other random events which are connected with the former events in some manner.

A statement to the effect that the probability of occurrence of a certain event is, say, 1/2, is not in itself valuable, since one is interested in reliable knowledge. Only results which state that the probability of occurrence of a certain event $A$ is quite near to one or (which is the same thing) that the probability of the event not occurring is very small, represent ultimately valuable information. In accordance with the principle of "discarding sufficiently small probabilities" , such an event is considered to be practically certain. It will be shown below (cf. the section: Limit theorems) that conclusions of scientific and practical interest are usually based on the assumption that the occurrence or non-occurrence of an event $A$ depends on a large number of random factors, which are interconnected only to a minor extent (cf. Law of large numbers in connection with this subject). It may also be said, accordingly, that probability theory is the mathematical science of the laws governing the interaction of a large number of random factors.

The subject matter of probability theory.[edit]

In order to describe a regular connection between certain conditions $S$ and an event $A$, the occurrence or non-occurrence of which can be established exactly, one of the following two schemes are usually employed in science.

1) The occurrence of event $A$ follows each realization of the conditions $S$. This is the form of, say, all the laws of classical mechanics which state that under given initial conditions and forces acting on a body or a system of bodies, the motion will proceed in a uniquely determined manner.

2) Under the conditions $S$ the occurrence of event $A$ has a definite probability $\mathsf{P}(A\mid S)$ which is equal to $p$. For instance, the laws governing ionizing radiation say that, for each radioactive substance there is a definite probability that, in a given period of time, some number $N$ of the atoms of the substance will decay.

The frequency of occurrence of event $A$ in a given sequence of $n$ trials (i.e. $n$ repeated realizations of the conditions $S$) is the ratio $p=m/n$ between the number $m$ of trials in which $A$ has occurred to the total number of trials $n$. That there is in fact a definite probability $p$ for $A$ to occur, under the conditions $S$, is manifested by the fact that in almost-all sufficiently large sequences of trials the frequency of occurrence of $A$ is approximately equal to $p$. Any mathematical model which is intended to be a schematic description of the connection between conditions $S$ and a random event $A$, usually also contains certain assumptions about the nature and the degree of dependence of the trials. After these additional assumptions (of which the most frequent one is mutual independence of the trials; see the section: Fundamental concepts in probability theory) have been made, it is possible to give a quantitative, more precise expression of the somewhat vague statement made above to the effect that the frequency is close to the probability.

Statistical relationships, i.e. relationships which may be described by a scheme of type 2) above, were first noted for games of chance such as throwing a die. Statistical relationships concerning births and deaths have been known for a very long time (e.g. the probability of a newborn (human) baby being a boy is 0.515). The end of the 19th century and the first half of the 20th century have witnessed the discovery of a large number of statistical laws in physics, chemistry, biology, and other sciences. It should be noted that statistical laws are also involved in schemes not directly related to the concept of randomness, e.g. in the distribution of digits in tables of functions, etc. (cf. Random and pseudo-random numbers). This fact is utilized, in particular, in the "simulation" of random phenomena (see Statistical experiments, method of).

That methods of probability theory can be used in studying the relationships prevailing in a large number of sciences apparently unrelated to each other is due to the fact that probabilities of occurrence of events invariably satisfy certain simple laws, which will be discussed below (cf. the section: Fundamental concepts in probability theory). The study of the properties of the probability of occurrence of events, based on these simple laws, forms the subject matter of probability theory.

Fundamental concepts in probability theory.[edit]

The fundamental concepts in probability theory, as a mathematical discipline, are most simply exemplified within the framework of so-called elementary probability theory. Each trial $T$ considered in elementary probability theory is such that it yields one and only one outcome or, as it is called, one of the elementary events ${\omega }_1\dots {\omega }_s$, which are supposed to be finite in number. To each outcome ${\omega }_k$ a non-negative number $p_k$ is connected — the probability of this outcome. The sum of the numbers $p_k$ must be one. Consider events $A$ which are characterized by the condition

"either wi or wj… or wk occurs."

The outcomes ${\omega }_i,{\omega }_j\dots {\omega }_k$ are said to be favourable to $A$ and, by definition, one says that the probability $\mathsf{P}(A)$ of $A$ is equal to the sum of the probabilities of the outcomes favourable to this event:

$$\begin{array}{}\text{(1)}& \mathsf{P}(A)=p_i+p_j+\cdots +p_k.\end{array}$$

If there are $r$ outcomes favourable to $A$, then the special case $p_1=\cdots =p_s=1/s$ yields the formula

$$\begin{array}{}\text{(2)}& \mathsf{P}(A)=\frac{r}{s}.\end{array}$$

Formula (2) expresses the so-called classical concept of probability, according to which the probability of some event $A$ is equal to the ratio between the number $r$ of outcomes favourable to $A$ and the number $s$ of all "equally probable" outcomes. The computation of probabilities is thus reduced to counting the number of outcomes favourable to $A$ and often proves to be a difficult problem in combinatorics.

Example. Each one of the 36 possible outcomes of throwing a pair of dice may be denoted by $(i;j)$, where $i$ is the number of dots shown by the first die, while $j$ is the number of dots shown by the second. Event $A$— "the sum of the dots is 4" — is favoured by three outcomes: $(1;3)$, $(2;2)$, $(3;1)$. Thus, $\mathsf{P}(A)=3/36=1/12$.

The problem of determining the numerical values of the probabilities $p_k$ in a given specific problem lies, strictly speaking, outside the scope of probability theory as a discipline of pure mathematics. In some cases these values are established as a result of processing the results of a large number of observations. In other cases it is possible to predict the probabilities of encountering given events in a given trial theoretically. Such a prediction is frequently based on an objective symmetry of the connections between the conditions under which the trial is conducted and the outcomes of the trials, and in such cases leads to a formula like (2). Let, for instance, the trial consist in throwing a die in the form of a cube made of a homogeneous material. One may then assume that each side of the die has a probability of 1/6 of "coming out" . In this case the assumption that all outcomes are equally probable is confirmed by experiment. Examples of this kind in fact form the basis of the classical definition of a probability.

A more detailed and thorough explanation for the causes of equal probabilities of individual outcomes in some special cases may be given by the so-called method of arbitrary functions. The method is explained below by taking again dice throwing as an example. Let the conditions of the trials be such that accidental effects of air on the die are negligible. In such a case, if the initial position, the initial velocity and the mechanical properties of the die are known exactly, the motion of the die may be calculated by the methods of classical mechanics, and the result of the trial may be reliably predicted. In practice, the initial conditions can never be determined with absolute accuracy and even very small changes in the initial velocity will produce a different result, provided the period of time $t$ between the throw and the fall of the die is sufficiently long. It has been found that, under very general assumptions with respect to the probability distribution of the initial values (hence the name of the method), the probability of each one of the six possible outcomes tends to 1/6 as $t\to \mathrm{\infty }$.

A second $example$ consists of the shuffling of a pack of cards in order to ensure that all possible distributions are equally probable. Here, the transition from one distribution of the cards to the next as a result of two successive shuffles is usually random. The tendency to equi-probability is established by methods of the theory of Markov chains (cf. Markov chain).

Both these cases can be seen as part of general ergodic theory.

Given a certain number of events, two new events may be defined: their union (sum) and combination (product, intersection). The event $B$: "at least one of A1…Ar occurs" , is said to be the union of events $A_1\dots A_r$.

The event $C$: "A1… and Ar occur" , is said to be the combination or intersection of events $A_1\dots A_r$.

The symbols for union and intersection of events are $\cup$ and $\cap$, respectively. Thus:

$$B=A_1\cup \cdots \cup A_r,C=A_1\cap \cdots \cap A_r.$$

Two events $A$ and $B$ are said to be mutually exclusive if their joint occurrence is impossible, i.e. if none of the possible results of a trial favours both $A$ and $B$. If the events $A_i$ are identified with the sets of their favourable outcomes, events $B$ and $C$ will be identical with the union and the intersection of the respective sets.

Two fundamental theorems in probability theory — theorems on addition and multiplication of probabilities — are connected with the operations just introduced.

The theorem on addition of probabilities. If the events $A_1\dots A_r$ are such that any two of them are mutually exclusive, the probability of their union is equal to the sum of their probabilities.

Thus, in the example mentioned above — throwing a pair of dice, "the sum of the dots is 4 or less" is the sum of the three mutually exclusive events $A_2,A_3,A_4$ in which the sum of the dots is 2, 3 and 4, respectively. The probabilities of these events are 1/36, 2/36 and 3/36, respectively; in accordance with the addition theorem, $\mathsf{P}(B)$ is equal to

$$\frac{1}{36}+\frac{2}{36}+\frac{3}{36}=\frac{6}{36}=\frac{1}{6}.$$

The conditional probability of event $B$ occurring if condition $A$ is met is defined by the formula

$$\mathsf{P}(B\mid A)=\frac{\mathsf{P}(A\cap B)}{\mathsf{P}(A)},$$

which may be shown to be in complete agreement with the properties of the frequencies of occurrence. Events $A_1\dots A_r$ are said to be independent if the conditional probability of any one of the events occurring under the condition that some of the other events have also occurred is equal to its "unconditional" probability (see also Independence in probability theory).

The theorem on multiplication of probabilities. The probability of joint occurrence of events $A_1\dots A_r$ is equal to the probability of occurrence of event $A_1$ multiplied by the probability of occurrence of event $A_2$ on the condition that $A_1$ has in fact occurred $\dots$ multiplied by the probability of occurrence of event $A_r$ on the condition that the events $A_1\dots A_{r-}1$ have in fact occurred. If the events are independent, the multiplication theorem yields the formula

$$\begin{array}{}\text{(3)}& \mathsf{P}(A_1\cap \cdots \cap A_r)=\mathsf{P}(A_1)\dots \mathsf{P}(A_r),\end{array}$$

i.e. the probability of joint occurrence of independent events is equal to the product of the probabilities of these events. Formula (3) remains valid if some of the events are replaced in both its parts by the complementary events.

Example. Four shots are fired at a target, the probability of hitting the target being 0.2 with each shot. The hits scored in different shots are considered to be independent events. What will be the probability of hitting the target exactly three times?

Each outcome of a trial can be symbolized by a sequence of four letters (e.g. $(h,m,m,h)$ means that the first and fourth shots were hits, while the second and the third shots were misses). The total number of outcomes will be $2\times 2\times 2\times 2=16$. Since the results of individual shots are assumed to be independent, the probability of the outcomes must be determined with the aid of formula (3) including the comment which accompanies it. Thus, the probability of the outcome $(h,m,m,m)$ will be

$$0.2\cdot 0.8\cdot 0.8\cdot 0.8=0.1024;$$

where $0.8=1-0.2$ is the probability of miss in a single shot. The outcomes favouring the event "the target is hit three times" are $(h,h,h,m)$, $(h,h,m,h)$, $(h,m,h,h)$, and $(m,h,h,h)$. The probabilities of all four outcomes are equal:

$$0.2\cdot 0.2\cdot 0.2\cdot 0.8=\cdots =0.8\cdot 0.2\cdot 0.2\cdot 0.2=0.0064,$$

so that the probability of the event is

$$4\cdot 0.0064=0.0256.$$

A generalization of the above reasoning leads to one of the fundamental formulas in probability theory: If the events $A_1\dots A_n$ are independent and if the probability of each individual event occurring is $p$, then the probability of occurrence of exactly $m$ such events is

$$\begin{array}{}\text{(4)}& {\mathsf{P}}_n(m)=C_n^m{p}^m(1-p{)}^{n-}m,\end{array}$$

where $C_n^m=({}_m^n)$ denotes the number of combinations of $m$ elements out of $n$ elements (see Binomial distribution). If $n$ is large, computations according to formula (4) become laborious. In the above example, let the number of shots be 100; one has to find the probability $x$ of the number of hits being between 8 and 32. The use of formula (4) and of the addition theorem yields an accurate, but unwieldy expression for the probability value sought, namely:

$$x=\sum _{m=}{8}^{32}\left(\begin{array}{c}100\\ m\end{array}\right)(0.2{)}^m(0.8{)}^{100-}m.$$

An approximate value of the probability $x$ may be found by the use of the Laplace theorem:

$$x\approx \frac{1}{\sqrt{2\pi }}\underset{-}{\int }{3}^{+}3e^{-z^2/2}dz=0.9973,$$

the error not exceeding 0.0009. This result shows that the occurrence of the event $8\le m\le 32$ is practically certain. This is a very simple, but typical, example of the use of limit theorems in probability theory.

Another fundamental formula in elementary probability theory is the so-called formula of total probability: If events $A_1\dots A_r$ are pairwise mutually exclusive and if their union is the sure event, the probability of any single event $B$ is equal to the sum

$$\mathsf{P}(B)=\sum _{k=}{1}^r\mathsf{P}(B\mid A_k)\mathsf{P}(A_k).$$

The theorem on multiplication of probabilities is particularly useful when compound trials are considered. One says that a trial $T$ is composed of trials $T_1\dots$ if each outcome of $T$ is a combination of certain outcomes $A_i,B_j\dots X_k,Y_l$ of the respective trials $T_1,T_2\dots T_{n-}1,T_n$. Frequently one is in the situation where the probabilities

$$\begin{array}{}\text{(5)}& \mathsf{P}(A_i),\mathsf{P}(B_j|A_i)\dots \mathsf{P}(Y_l|A_i\cap B_j\cap \cdots \cap X_k)\end{array}$$

are, for some reason, known. The data in (5) together with the multiplication theorem may then be used to determine the probabilities $\mathsf{P}(E)$ for all outcomes $E$ of the compound trial, as well as the probabilities of all events connected with this trial (as was done in the example discussed above). Two types of compound trials are especially important in practice: A) the individual trials are independent, i.e. the probabilities in (5) are equal to the unconditional probabilities $\mathsf{P}(A_i),\mathsf{P}(B_j)\dots \mathsf{P}(X_k),\mathsf{P}(Y_l)$; B) the probabilities of the outcomes of a given trial are only affected by the outcomes of the immediately preceding trial, i.e. the probabilities in (5) are equal, respectively, to $\mathsf{P}(A_i),\mathsf{P}(B_j|A_i)\dots \mathsf{P}(Y_l|X_k)$. One then says that the trials are connected in a Markov chain. The probabilities of all events connected with a compound trial are here fully determined by the initial probabilities $\mathsf{P}(A_i)$ and by the intermediate probabilities $\mathsf{P}(B_j|A_i)\dots \mathsf{P}(Y_l|X_k)$( cf. Markov process).

Random variables. If each outcome of a trial $T$ is put into correspondence with a number $x_r$, one says that a random variable $X$ has been specified. Among the numbers $x_1\dots x_s$ there may be equals; the set of different values of $x_r$, where $r=1\dots s$, is the set of possible values of the random variable. The set of possible values of a random variable, together with their respective probabilities is said to be the probability distribution of the random variable. Thus, in the example of throwing a pair of dice, to each outcome $(i,j)$ of the trial there corresponds the value of the random variable $X=i+j$ which is the sum of the dots on the two dice. The possible values are $2,3\dots 12$ and their respective probabilities are $1/36,2/36\dots 1/36$.

In a joint study of several random variables one introduces the concept of their joint distribution, which is defined by indicating the possible values of each one, and the probabilities of joint occurrence of the events

$$\begin{array}{}\text{(6)}& \{X_1=x_1\}\dots \{X_n=x_n\},\end{array}$$

where $x_k$ is one of the possible values of the variable $X_k$. Random variables are said to be independent if the events in (6) are independent whatever the choice of the $x_k$. The joint distribution of random variables can be used to calculate the probability of any event defined by these variables, e.g. of the event

$$a<X_1+\cdots +X_n<b,$$

etc.

Often, instead of giving the distribution of a random variable completely, one uses a, not too large, collection of numerical characteristics. The ones most often used are the mathematical expectation and the dispersion (variance). (See also Moment; Semi-invariant.)

The fundamental characteristics of a joint distribution of several random variables include — in addition to the mathematical expectations and the variances of these variables — also the correlation coefficients (cf. Correlation coefficient), etc. The meaning of these characteristics can be made clear, to a considerable extent, by limit theorems (see the section: Limit theorems).

The scheme of trials with a finite number of outcomes proves inadequate even in the simplest applications of probability theory. Thus, in the study of the random dispersion of the hitting sites of projectiles around the centre of a target, or in the study of random errors in the determination of some value, etc., it is not possible to limit the model to trials with a finite number of outcomes. Moreover, such outcomes may, in some cases, be expressed by a number or a set of numbers, while in other cases the outcome of a trial may be a function (e.g. a record of the variation of atmospheric pressure at a given location over a certain period of time), a set of functions, etc. It should be noted that many definitions and theorems given above, after suitable modifications, are also applicable in these more general cases, although the forms in which the probability distribution is presented are different (cf. Density of a probability distribution; Probability distribution). Here, the classical "equal probability of each outcome" is replaced by a uniform distribution of the objects under consideration in some area (this is exactly what is meant when speaking of a point randomly selected in a given area, a randomly selected tangent to some figure, etc.).

Major changes are introduced in the definition of a probability which, in the elementary case, is given by formula (2). In the more general schemes now discussed, the events are the union of an infinite number of elementary events the probability of each one of which may be zero. Thus, the property which is described by the addition theorem is not a consequence of the definition of probability, but is part of it.

The logical scheme of constructing the fundamentals of probability theory which is most often employed was developed in 1933 by A.N. Kolmogorov. The fundamental characteristics of this scheme are the following. In studying a real problem by the methods of probability theory, the first step is to isolate a set $U$ of elements $u$, called elementary events. Any event can be fully described by the set of elementary events favourable to it, and is therefore considered as some set of elementary events. To some events $A$ are assigned certain numbers $\mathsf{P}(A)$, which are called their probabilities and which satisfy the following conditions:

1) $0\le \mathsf{P}(A)\le 1$;

2) $\mathsf{P}(U)=1$;

3) if the events $A_1\dots A_n$ are pairwise mutually exclusive, and if $A$ is their union, then

$$\mathsf{P}(A)=\mathsf{P}(A_1)+\cdots +\mathsf{P}(A_n)$$

(additivity of probabilities).

In order to construct a mathematically rigorous theory, the domain of definition of $\mathsf{P}(A)$ must be a $\sigma$- algebra, and condition (3) must also be met for an infinite sequence of events which are mutually exclusive (countable additivity of probabilities). Non-negativity and countable additivity are fundamental properties of measures. Thus, probability theory may be formally regarded as a part of measure theory. The fundamental concepts of probability theory are then viewed in a new light: random variables become measurable functions, their mathematical expectations become the abstract integrals of Lebesgue, etc. However, the main problems of probability theory and of measure theory are different. In probability theory, the basic, specific concept is that of independence of events, trials and random variables. Moreover, probability theory comprises a thorough study of subjects such as probability distributions, conditional mathematical expectations, etc.

The following comments may be made on the scheme described above. In accordance with the scheme, each probability model is based on a probability space, which is a triplet $(\mathrm{\Omega },S,\mathsf{P})$, where $\mathrm{\Omega }$ is a set of elementary events, $S$ is a $\sigma$- algebra of subsets of $\mathrm{\Omega }$ and $\mathsf{P}$ is a probability distribution (a countably-additive normalized measure) on $S$. Two achievements of this scheme are the definition of probabilities in infinite-dimensional spaces (in particular, in spaces connected with infinite sequences of trials and stochastic processes), and the general definition of conditional probabilities and conditional mathematical expectations (with respect to a given random variable, etc.).

Subsequent development of probability theory showed that the above definition of a probability space can be expediently narrowed. These developments have led to concepts such as perfect distributions and probability spaces, Blackwell spaces, Radon probability measures on topological (linear) spaces, etc. (see Probability distribution).

There are also other approaches to the fundamental concepts of probability theory, such as axiomatization, the principal object of which is a normalized Boolean algebra of events. Here, the principal advantage (provided that the algebra being considered is complete in the metric sense) consists of the fact that for any directed system of events the following relations are true:

$$\mathsf{P}\left({\cup }_{\alpha }{A}_{\alpha }\right)=\underset{\alpha }{sup}\mathsf{P}(A_{\alpha }),A_{\alpha }\uparrow ,$$

$$\mathsf{P}\left({\cap }_{\alpha }{A}_{\alpha }\right)=\underset{\alpha }{inf}\mathsf{P}(A_{\alpha }),A_{\alpha }\downarrow .$$

It is possible to axiomatize the concept of a random variable as an element of some commutative algebra with a positive linear functional defined on it (the analogue of the mathematical expectation). This is the starting point for non-commutative and quantum probability.

Limit theorems.[edit]

In a formal exposition of probability theory limit theorems appear as a kind of superstructure over its elementary sections in which all problems are of a finite, purely arithmetical nature. However, the cognitive value of probability theory can only be revealed by these limit theorems. Thus, it is shown by the Bernoulli theorem that the frequency of occurrence of a given event in independent trials is usually close to its probability, while the Laplace theorem yields the probabilities of deviations of this frequency from its limiting value. In a similar manner, the meaning of the characteristics of a random variable such as its mathematical expectation and variance are explained by the law of large numbers and the central limit theorem (see also Limit theorems in probability theory).

Let

$$\begin{array}{}\text{(7)}& X_1\dots X_n\dots \end{array}$$

be independent random variables with the same probability distribution, with $\mathsf{E}{X}_k=a$, $\mathsf{D}{X}_k={\sigma }^2$, and let $Y_n$ be the arithmetical average of the first $n$ variables of the sequence (7):

$$Y_n=\frac{X_1+\cdots +X_n}{n}.$$

In accordance with the law of large numbers, for any $ϵ>0$ the probability of the inequality $|Y_n-a|\le ϵ$ tends to one as $n\to \mathrm{\infty }$, so that, as a rule, the value of $Y_n$ is close to $a$. This result is rendered more precise by the central limit theorem, according to which the deviations of $Y_n$ from $a$ are approximately normally distributed, with mathematical expectation 0 and variance ${\sigma }^2/n$. Thus, in order to calculate (to a first approximation) the probability of some deviation of $Y_n$ from $a$ for large $n$, there is no need to know the distribution of the variables $X_n$ in all details; knowledge of their variance is sufficient. If a higher accuracy of approximation is required, moments of higher order must also be used.

The above statements, with suitable modifications, may be extended to random vectors (in finite-dimensional and in some infinite-dimensional spaces). The independence conditions may be replaced by conditions of a "weak" (in some sense) dependence of the $X_n$. Limit theorems of distributions on groups, of distributions of values of arithmetic functions, etc., are also known.

In applications — in particular, in mathematical statistics and statistical physics — it may be necessary to approximate small probabilities (i.e. probabilities of events of the type $|Y_n-a|>ϵ$) with a high relative accuracy. This involves major corrections to the normal approximation (cf. Probability of large deviations).

It was noted in the nineteen twenties that quite natural non-normal limit distributions may appear even in schemes of sequences of uniformly-distributed and independent random variables. For instance, let $X_1$ be the time which elapses until some randomly varying variable has returned to its initial location, let $X_2$ be the time between the first and the second such returns, etc. Then, under very general conditions, the distribution of the sum $X_1+\cdots +X_n$( i.e. the time elapsing prior to the $n$- th return) will, after multiplication by $n^{-1/\alpha }$( where $\alpha$ is a constant smaller than one), converge to some limit distribution. Thus, the time prior to the $n$- th return increases, roughly speaking, in proportion to $n^{1/\alpha }$, i.e. at a faster rate than $n$( if the law of large numbers were applicable, it would be of order $n$). This is seen in the case of a Bernoulli random walk (in which another paradoxical law — the arcsine law — also appears).

The principal method of proof of limit theorems is the method of characteristic functions (cf. Characteristic function) (and the related methods of Laplace transforms and of generating functions). In a number of cases it becomes necessary to invoke the theory of functions of a complex variable.

The mechanism of the existence of most limit relationships can be completely understood only in the context of the theory of stochastic processes.

Stochastic processes.[edit]

During the past few decades the need to consider stochastic processes (cf. Stochastic process) — i.e. processes with a given probability of their proceeding in a certain manner, arose in certain physical and chemical investigations, along with the study of one-dimensional and higher-dimensional random variables. The coordinate of a particle executing a Brownian motion may serve as an example of a stochastic process. In probability theory a stochastic process is usually regarded as a one-parameter family of random variables $X(t)$. In most applications the parameter $t$ is time, but it may also be an arbitrary variable, and in such cases it is usual to speak of a random function (if $t$ is a point in space — a random field). If the parameter $t$ runs through integer values, the random function is said to be a random sequence (or a time series). While a random variable may be characterized by a distribution law, a stochastic process may be characterized by the totality of joint distribution laws for $X(t_1)\dots X(t_n)$ for all possible moments of time $t_1\dots t_n$ for any $n>0$( the so-called finite-dimensional distributions). The most interesting concrete results in the theory of stochastic processes were obtained in two fields — Markov processes and stationary stochastic processes (cf. Markov process; Stationary stochastic process); the interest in martingales (cf. Martingale) is now also strongly increasing.

Chronologically, Markov processes were the first to be studied. A stochastic process $X(t)$ is said to be a Markov process if, for any two moments of time $t_0$ and $t_1$( $t_0<t_1$), the conditional probability distribution of $X(t_1)$ depends, provided all values of $X(t)$ for $t\le t_0$ are given, only on $X(t_0)$. For this reason Markov processes are sometimes referred to as processes without after-effect. Markov processes are a natural extension of the deterministic processes studied in classical physics. In deterministic processes the state of the system at the moment of time $t_0$ uniquely determines the course of the process in the future; in Markov processes the state of the system at the moment of time $t_0$ uniquely determines the probability distribution of the course of the process at $t>t_0$, and this distribution cannot be altered by any information on the course of the process prior to the moment of time $t_0$.

Just as the study of continuous deterministic processes is reduced to differential equations involving functions which describe the state of the system, the study of continuous Markov processes can, to a large extent, be reduced to differential or differential-integral equations with respect to the distribution of the probabilities of the process.

Another major subject in the field of stochastic processes is the theory of stationary stochastic processes. The stationary nature of a process, i.e. the fact that its probability relations remain unchanged with time, imposes major restrictions on the process and makes it possible to arrive at several important deductions based on this premise.

A major part of the theory is based only on the assumption of stationarity in a wide sense, viz. that the mathematical expectations $\mathsf{E}X(t)$ and $\mathsf{E}X(t)X(t+\tau )$ are independent of $t$. This assumption leads to the so-called spectral decomposition:

$$X(t)=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{e}^{it\lambda }dz(\lambda ),$$

where $z(\lambda )$ is a random function with uncorrelated increments. Methods of best (in the mean square) linear interpolation, extrapolation and filtering have been developed for stationary processes.

Recently a rather large class of processes, the so-called semi-martingales, which serves to solve problems of optimal non-linear filtering, interpolation and extrapolation, has been isolated (cf. Stochastic processes, prediction of; Stochastic processes, filtering of; Stochastic processes, interpolation of). A substantial part of the relevant analytical apparatus is provided by stochastic differential equations, stochastic integrals and martingales. A distinguishing feature of a martingale $X(t)$ is the fact that the conditional mathematical expectation of $X(t)$ is $X(s)$, given the values of $X(u)$ for $u\le s$, $s<t$.

The theory of stochastic processes is closely connected with the classical problems on limit theorems for sums of random variables. Distributions which appear as limit distributions in the study of sums of random variables become exact distributions of appropriate characteristics in the theory of stochastic processes. This fact makes it possible to demonstrate many limit theorems with the aid of these associated stochastic processes.

One may finally note that the logically unobjectionable definition of the concepts connected with stochastic processes within the framework of the axiomatics discussed above has always presented and still presents a large number of difficulties of measure-theoretic nature. These are connected, for example, with the definition of probabilistic continuity, differentiability, etc., of stochastic processes (cf. Separable process). This is why monographs on the theory of stochastic processes devote about half their space to the analysis of the development of measure-theoretic constructions.

See also the references to entries on individual subjects of probability theory.

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