Compound Poisson approximation for multiple runs in a Markov chain
Let \((X_i)\) be a stationary Markov chain with \(r\)-valued terms. Consider the runs of \(k_i\) \((i = 1,\dots,r)\) consecutive random variables having value \(i\), that end in the finite sequence \(X_1,\dots,X_n\). Then the random variable \(W=\sum^n_{a=1} \sum^r_{i=1} I[(X_{a-k+1},\dots,X_a)=(i,\dots,i)]\) enumerates the overlapping run. Here \(I\) denotes the indicator. The aim of the paper is to estimate the approximation of \(W\)-distribution by compound Poisson; the so called waiting time \(T_k\), i.e. the number of trials until the \(k\)th run appearance \(P(T_k>n)=P(W<k)\), is shown for large \(k\).
consecutive \(k_1,\dots,k_r\) out of \(n\):MFM system
scientific article; zbMATH DE number 1901376