Additive Markov Chain

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An Additive Markov Chain is a Markov Chain with an additive conditional probability function.



References

2015

  • Definition
An additive Markov chain of order m is a sequence of random variables X1X2X3, ..., 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,
[math]\displaystyle{ \Pr(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots, X_{n-m}=x_{n-m}) = \sum_{r=1}^{m} f(x_n,x_{n-r},r) }[/math].
  • Binary case
A binary additive Markov chain is where the state space of the chain consists on two values only, Xn ∈ { x1x2 }. For example, Xn ∈ { 0, 1 }. The conditional probability function of a binary additive Markov chain can be represented as
[math]\displaystyle{ \Pr(X_n=1|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots) = \bar{X} + \sum_{r=1}^{m} F(r) (x_{n-r}-\bar{X}), }[/math]
[math]\displaystyle{ \Pr(X_n=0|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots) = 1 - \Pr(X_n=1|X_{n-1} = x_{n-1}, X_{n-2} = x_{n-2}, \dots). }[/math]
Here [math]\displaystyle{ \bar{X} }[/math] is the probability to find Xn = 1 in the sequence and
F(r) is referred to as the memory function. The value of [math]\displaystyle{ \bar{X} }[/math] 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 [math]\displaystyle{ X_n }[/math] and [math]\displaystyle{ X_k }[/math] of the chain depends on the distance [math]\displaystyle{ n - k }[/math] only. It is defined as follows:
[math]\displaystyle{ K(r) = \langle (X_n-\bar{X})(X_{n+r}-\bar{X}) \rangle = \langle X_n X_{n+r} \rangle -{\bar{X}}^2, }[/math]
where the symbol [math]\displaystyle{ \langle \cdots \rangle }[/math] denotes averaging over all n. By definition,
[math]\displaystyle{ K(-r)=K(r), K(0)=\bar{X}(1-\bar{X}). }[/math]
There is a relation between the memory function and the correlation function of the binary additive Markov chain:
[math]\displaystyle{ K(r)=\sum_{s=1}^m K(r-s)F(s), \, \, \, \, r=1, 2, \dots\,. }[/math]

2006

2006

2003