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Exact convergence rate of an Erdös-Rényi strong law for moving quantiles

Published online by Cambridge University Press:  14 July 2016

Paul Deheuvels  and
Josef Steinebach
Paul Deheuvels*
Affiliation:
Université de Paris
Josef Steinebach*
Affiliation:
Universität Marburg
*
∗Postal address: Institut de Statistique, Université de Paris VI, 4 Place Jussieu, F-75230 Paris Cedex 05, France.
∗∗Postal address: Fachbereich Mathematik, Philipps-Universität, Hans-Meerwein-Straße, D-3550 Marburg, W. Germany.
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Abstract

Consider a sequence U1, U2, · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un+1, · ··, Un+K, and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α < u < 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X1, X2, · ·· with d.f. F satisfying a strict monotonicity condition.

Keywords

BEST STRONG CONVERGENCE RATEMAXIMUM MOVING QUANTILESLARGE DEVIATIONSSTOCHASTIC GEYSER PROBLEM
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 355 - 369
Copyright
Copyright © Applied Probability Trust 1986 

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