A note on the two-sided regulated random walkстатья
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Аннотация:In this paper we address the two-sided regulated random walk defined by the relation X-N (t + 1) = min(N, max(O, XN (t) + A (t + 1))) where (A (t); t greater than or equal to 1) is a sequence of i.i.d r.v’s with integer values such that A (t) greater than or equal to -1, E A = 0 and E r(A) < + infinity for an r > 1. Denoting by pi(N) its stationary distribution, F-N (chi) = pi(N) ([0, Nchi]) and G(chi) the d.f of a uniform r.v on [0,1]. It is shown that 0 < lim N\textbackslash\textbackslashF-N - G\textbackslash\textbackslash(p) less than or equal to (lim) over barN\textbackslash\textbackslashF-N - G\textbackslash\textbackslash(p) less than or equal to +infinity for 1 less than or equal to p less than or equal to + infinity, that is: 1/N is the exact convergence rate of F-N to G. This result improves (in the particular case considered) earlier results claiming that lim(N) \textbackslash\textbackslashF-N - G\textbackslash\textbackslashinfinity = 0. (C) 2003 Elsevier SAS. All rights reserved.