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November, 1985 The Robbins-Siegmund Series Criterion for Partial Maxima
Michael J. Klass
Ann. Probab. 13(4): 1369-1370 (November, 1985). DOI: 10.1214/aop/1176992820


Let $X, X_1, X_2,\cdots$ be i.i.d. random variables and let $M_n = \max_{1\leq j \leq n}X_j$. For each nondecreasing real sequence $\{b_n\}$ such that $P(X > b_n) \rightarrow 0$ and $P(M_n \leq b_n) \rightarrow 0$ we show that $P(M_n \leq b_n i.o.) = 1$ if and only if $\sum_nP(X > b_n)\exp\{- nP(X > b_n)\} = \infty$. The restrictions on the $b_n's$ can be removed.


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Michael J. Klass. "The Robbins-Siegmund Series Criterion for Partial Maxima." Ann. Probab. 13 (4) 1369 - 1370, November, 1985.


Published: November, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0576.60023
MathSciNet: MR806233
Digital Object Identifier: 10.1214/aop/1176992820

Primary: 60F15
Secondary: 60F10 , 60F20 , 60G99

Keywords: minimal growth rate , Partial maxima , strong limit theorems , upper and lower class sequences

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 4 • November, 1985
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