The Annals of Probability

The Robbins-Siegmund Series Criterion for Partial Maxima

Michael J. Klass

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab., Volume 13, Number 4 (1985), 1369-1370.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992820

Digital Object Identifier
doi:10.1214/aop/1176992820

Mathematical Reviews number (MathSciNet)
MR806233

Zentralblatt MATH identifier
0576.60023

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60F20: Zero-one laws 60F10: Large deviations 60G99: None of the above, but in this section

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

Citation

Klass, Michael J. The Robbins-Siegmund Series Criterion for Partial Maxima. Ann. Probab. 13 (1985), no. 4, 1369--1370. doi:10.1214/aop/1176992820. https://projecteuclid.org/euclid.aop/1176992820


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