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Short Communications

On the probability of the non-appearence of a given number of $s$-tuples in compound Markov chains

P. F. Belyaev

Moscow

Abstract: Let $\{j_r\}$, $r=\overline{1,n}$, $j_r=\overline{1,k}$ be a sequence obtained by realizations of $n$ trials which are bound into a compound Markov chain of order $s$ with $k$ outcomes.
Let $s$-tuple denote a subsequence of $\{j_r\}$ consisting of $s$ consecutive symbols and let $P(n,k;m)$ be the probability that in the sequence $\{j_r\}$ of all possible $k^s$ $s$-tuples exactly $m$ $s$-tuples are missing.
The asymptotic behaviour of the probability $P(n,k;m)$ as $n\to\infty$; $k\to\infty$; $k^re^{-n/k^s}<c<\infty$ is considered.

Received: 24.12.1964

 English version: Theory of Probability and its Applications, 1965, 10:3, 496–499

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