Additive Markov chain

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 X_n has a certain value x_n 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,

:$\backslash Pr(X\_n=x\_n\backslash mid\; X\_\{n-1\}=x\_\{n-1\},\; X\_\{n-2\}=x\_\{n-2\},\; \backslash dots,\; X\_\{n-m\}=x\_\{n-m\})\; =\; \backslash sum\_\{r=1\}^\{m\}\; f(x\_n,x\_\{n-r\},r).$

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

: $\backslash Pr(X\_n=1\backslash mid\; X\_\{n-1\}=x\_\{n-1\},\; X\_\{n-2\}=x\_\{n-2\},\; \backslash dots)\; =\; \backslash bar\{X\}\; +\; \backslash sum\_\{r=1\}^\{m\}\; F(r)\; (x\_\{n-r\}-\backslash bar\{X\}),$

: $\backslash Pr(X\_n=0\backslash mid\; X\_\{n-1\}=x\_\{n-1\},\; X\_\{n-2\}=x\_\{n-2\},\; \backslash dots)\; =\; 1\; -\; \backslash Pr(X\_n=1\backslash mid\; X\_\{n-1\}\; =\; x\_\{n-1\},\; X\_\{n-2\}\; =\; x\_\{n-2\},\; \backslash dots).$

Here $\backslash bar\{X\}$ is the probability to find X_n = 1 in the sequence and

F(r) is referred to as the memory function. The value of $\backslash bar\{X\}$ and the function F(r) contain all the information about correlation properties of the Markov chain.

In the binary case, the correlation function between the variables $X\_n$ and $X\_k$ of the chain depends on the distance $n\; -\; k$ only. It is defined as follows:

:$K(r)\; =\; \backslash langle\; (X\_n-\backslash bar\{X\})(X\_\{n+r\}-\backslash bar\{X\})\; \backslash rangle\; =\; \backslash langle\; X\_n\; X\_\{n+r\}\; \backslash rangle\; -\{\backslash bar\{X\}\}^2,$

where the symbol $\backslash langle\; \backslash cdots\; \backslash rangle$ denotes averaging over all n. By definition,

:$K(-r)=K(r),\; K(0)=\backslash bar\{X\}(1-\backslash bar\{X\}).$

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

:$K(r)=\backslash sum\_\{s=1\}^m\; K(r-s)F(s),\; \backslash ,\; \backslash ,\; \backslash ,\; \backslash ,\; r=1,\; 2,\; \backslash dots\backslash ,.$

• Examples of Markov chains

{{More footnotes|date=September 2010}}

{{Reflist|60em}}

• 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
• {{cite journal|author1=S. Hod |author2=U. Keshet |year=2004|title=Phase transition in random walks with long-range correlations|journal=Phys. Rev. E|volume=70|issue=1 Pt 2 |pages=015104|doi=10.1103/PhysRevE.70.015104|pmid=15324113 |arxiv=cond-mat/0311483|bibcode=2004PhRvE..70a5104H|s2cid=18169687 }}
• {{cite journal|author1=S.L. Narasimhan |author2=J.A. Nathan |author3=K.P.N. Murthy |year=2005|title=Can coarse-graining introduce long-range correlations in a symbolic sequence?|journal=Europhys. Lett.|volume=69|issue=1|pages=22|arxiv=cond-mat/0409042|doi=10.1209/epl/i2004-10307-2|bibcode=2005EL.....69...22N|s2cid=250845691 }}
• Ramakrishnan, S. (1981) "Finitely Additive Markov Chains", Transactions of the American Mathematical Society, 265 (1), 247–272 {{JSTOR|1998493}}

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Category:Markov processes