Additive Markov chain

In probability theory, an additive Markov chain is a Markov chain with an additive conditional probability function. Here the process is a discrete-time Markov chain of order m and the transition probability to a state at the next time is a sum of functions, each depending on the next state and one of the m previous states.

## Contents

* 1 Definition
* 2 Binary case
* 2.1 Relation between the memory function and the correlation function
* 3 See also
* 4 Notes
* 5 References

## Definition

An additive Markov chain of order m is a sequence of random variables X1, X2, X3, ..., possessing the following property: the probability that a random variable Xn has a certain value xn under the condition that the values of all previous variables are fixed depends on the values of m previous variables only (Markov chain of order m), and the influence of previous variables on a generated one is additive,

Pr⁡(Xn=xn∣Xn−1=xn−1,Xn−2=xn−2,…,Xn−m=xn−m)=∑r=1mf(xn,xn−r,r).

## Binary case

A binary additive Markov chain is where the state space of the chain consists on two values only, Xn ∈ { x1, x2 }. For example, Xn ∈ { 0, 1 }. The conditional probability function of a binary additive Markov chain can be represented as

Pr⁡(Xn=1∣Xn−1=xn−1,Xn−2=xn−2,…)=X¯+∑r=1mF(r)(xn−r−X¯),

Pr⁡(Xn=0∣Xn−1=xn−1,Xn−2=xn−2,…)=1−Pr⁡(Xn=1∣Xn−1=xn−1,Xn−2=xn−2,…).

Here X¯ is the probability to find Xn = 1 in the sequence and F(r) is referred to as the memory function. The value of X¯ and the function F(r) contain all the information about correlation properties of the Markov chain.

### Relation between the memory function and the correlation function

In the binary case, the correlation function between the variables Xn and Xk of the chain depends on the distance n−k only. It is defined as follows:

K(r)=⟨(Xn−X¯)(Xn+r−X¯)⟩=⟨XnXn+r⟩−X¯2,

where the symbol ⟨⋯⟩ denotes averaging over all n. By definition,

K(−r)=K(r),K(0)=X¯(1−X¯).

There is a relation between the memory function and the correlation function of the binary additive Markov chain:[1]

K(r)=∑s=1mK(r−s)F(s),r=1,2,….

## See also

* Examples of Markov chains

## Notes

1. ↑ S.S. Melnyk, O.V. Usatenko, and V.A. Yampol’skii. (2006) "Memory functions of the additive Markov chains: applications to complex dynamic systems", Physica A, 361 (2), 405–415 doi:10.1016/j.physa.2005.06.083

## 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, 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
* S. Hod; U. Keshet (2004). "Phase transition in random walks with long-range correlations". Phys. Rev. E 70 (1 Pt 2): 015104. doi:10.1103/PhysRevE.70.015104. PMID 15324113. Bibcode: 2004PhRvE..70a5104H.
* S.L. Narasimhan; J.A. Nathan; K.P.N. Murthy (2005). "Can coarse-graining introduce long-range correlations in a symbolic sequence?". Europhys. Lett. 69 (1): 22. doi:10.1209/epl/i2004-10307-2. Bibcode: 2005EL.....69...22N.
* Ramakrishnan, S. (1981) "Finitely Additive Markov Chains", Transactions of the American Mathematical Society, 265 (1), 247–272 JSTOR 1998493

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