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October, 1987 Limit Laws of Erdos-Renyi-Shepp Type
Paul Deheuvels, Luc Devroye
Ann. Probab. 15(4): 1363-1386 (October, 1987). DOI: 10.1214/aop/1176991982


Let $S_n = X_1 + \cdots + X_n$ be the $n$th partial sum of an i.i.d. sequence of random variables. We describe the limiting behavior of \begin{equation*}\begin{split}T_n = \max_{1\leq i\leq n}(S_{i+\kappa(i)} - S_i), \\ U_n = \max_{0\leq i\leq n-k}(S_{i+k} - S_i), \\ W_n = \max_{0\leq i\leq n-k} \max_{1\leq j\leq k}(S_{i+j} - S_i) \\ \end{split}\end{equation*} and $V_n = \max_{0\leq i\leq n-k} \min_{1\leq j\leq k}(k/j)(S_{i+j} - S_i),$ for $k = \kappa(n) = \lbrack c \log n\rbrack$, and where $c > 0$ is a given constant. We assume that the random variables $X_i$ are centered and have a finite moment generating function in a right neighborhood of zero, and obtain among other results the full form of the Erdos-Renyi (1970) and Shepp (1964) theorems. Our conditions extend those of Deheuvels, Devroye and Lynch (1986) to cover a larger class of distributions.


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Paul Deheuvels. Luc Devroye. "Limit Laws of Erdos-Renyi-Shepp Type." Ann. Probab. 15 (4) 1363 - 1386, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0637.60039
MathSciNet: MR905337
Digital Object Identifier: 10.1214/aop/1176991982

Primary: 60F15
Secondary: 60F10

Keywords: Erdos-Renyi-Shepp laws , large deviations , Law of the iterated logarithm , laws of large numbers , moving averages

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 4 • October, 1987
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