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.
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\midXn-1=xn-1,Xn-2=xn-2,...,Xn-m=xn-m)=
| m | |
| \sum | |
| r=1 |
f(xn,xn-r,r).
A binary additive Markov chain is where the state space of the chain consists on two values only, Xn ∈ . For example, Xn ∈ . The conditional probability function of a binary additive Markov chain can be represented as
\Pr(Xn=1\midXn-1=xn-1,Xn-2=xn-2,...)=\bar{X}+
| m | |
| \sum | |
| r=1 |
F(r)(xn-r-\bar{X}),
\Pr(Xn=0\midXn-1=xn-1,Xn-2=xn-2,...)=1-\Pr(Xn=1\midXn-1=xn-1,Xn-2=xn-2,...).
Here
\bar{X}
\bar{X}
In the binary case, the correlation function between the variables
Xn
Xk
n-k
K(r)=\langle(Xn-\bar{X})(Xn+r-\bar{X})\rangle=\langleXnXn+r\rangle-{\bar{X}}2,
where the symbol
\langle … \rangle
K(-r)=K(r),K(0)=\bar{X}(1-\bar{X}).
There is a relation between the memory function and the correlation function of the binary additive Markov chain:[1]
| m | |
| K(r)=\sum | |
| s=1 |
K(r-s)F(s),r=1,2,....