Geodesic
Singles in a Markov Chain
Publications de l'Institut Mathématique, _N_S_83 (2008) no. 97, p. 27 .

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Let $\{X_i,i\geq 1\}$ denote a sequence of variables that take values in $\{0,1\}$ and suppose that the sequence forms a Markov chain with transition matrix $P$ and with initial distribution $(q,p)=(P(X_1=0),P(X_1=1))$. Several authors have studied the quantities $S_n$, $Y(r)$ and $AR(n)$, where $S_n=\sum_{i=1}^nX_i$ denotes the number of successes, where $Y(r)$ denotes the number of experiments up to the $r$-th success and where $AR(n)$ denotes the number of runs. In the present paper we study the number of singles $AS(n)$ in the vector $(X_1,X_2,\dots,X_n)$. A single in a sequence is an isolated value of $0$ or $1$, i.e., a run of length $1$. Among others we prove a central limit theorem for $AS(n)$.
DOI : 10.2298/PIM0897027O
Classification : 60J10 60F05, 60K99, 60J20 
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Edward Omey; Stefan Van Gulck. Singles in a Markov Chain. Publications de l'Institut Mathématique, _N_S_83 (2008) no. 97, p. 27 . doi : 10.2298/PIM0897027O. http://geodesic.mathdoc.fr/articles/10.2298/PIM0897027O/

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