Singles in a Markov chain

Edward Omey () and Stefan Van Gulck ()
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Edward Omey: Hogeschool-Universiteit Brussel (HUB), Belgium
Stefan Van Gulck: Hogeschool-Universiteit Brussel (HUB), Belgium

No 2007/17, Working Papers from Hogeschool-Universiteit Brussel, Faculteit Economie en Management

Abstract: Let fXi; i _ 1g denote a sequence of variables that take values in f0; 1g and suppose that the sequence forms a Markov chain with transition matrix P and with initial distribution (q; p) = (P(X1 = 0); P(X1 = 1)). Several authors have studied the quantities Sn, Y (r) and AR(n), where Sn = Pn i=1 Xi 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 (X1;X2; :::;Xn). A single in a sequence is an isolated value of 0 or 1, i.e. a run of lenght 1. Among others we prove a central limit theorem for AS(n).

Pages: 14 pages
Date: 2007-09
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