On the Minimum Number of Fixed Length Sequences with Fixed Total Probability

Abstract

Let $X_1, X_2,\cdots$ be a stationary sequence of $B$-valued random variables, where $B$ is a finite set. For each positive integer $n$, and number $\lambda$ such that $0 < \lambda < 1$, let $N(n, \lambda)$ be the cardinality of the smallest set $E \subset B^n$ such that $P\lbrack(X_1, X_2,\cdots, X_n) \in E\rbrack > 1 - \lambda$. An example is given to show that $\lim_{n\rightarrow\infty}n^{-1} \log N(n, \lambda)$ may not exist for some $\lambda$, thereby settling in the negative a conjecture of Parthasarathy.