EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 83(97), pp. 27–35 (2008)

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Edward Omey and Stefan Van Gulck

Department of Mathematics and Statistics, HUB, Stormstraat 2, 1000 Brussels, Belgium

Abstract: 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 $AS(n)$.

Classification (MSC2000): 60J10; 60F05; 60K99; 60J20

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Electronic fulltext finalized on: 21 Oct 2008. This page was last modified: 10 Dec 2008.

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