The Annals of Probability

The Minimal Growth Rate of Partial Maxima

Michael J. Klass

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Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random variables and let $M_n = \max_{i \leq j \leq n} X_j$. For each real sequence $\{b_n\}$, a sequence $\{b^\ast_n\}$ and a sub-sequence of integers $\{n_k\}$ is explicitly constructed such that $P(M_n \leq b_n \text{i.o.}) = 1 \operatorname{iff} \sum_k P(M_{n_k} \leq b^\ast_{n_k}) = \infty$. This result gives a complete characterization of the upper and lower-class sequences (as introduced by Paul Levy) for the a.s. minimal growth rate of $\{M_n\}$.

Article information

Source
Ann. Probab., Volume 12, Number 2 (1984), 380-389.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993296

Digital Object Identifier
doi:10.1214/aop/1176993296

Mathematical Reviews number (MathSciNet)
MR735844

Zentralblatt MATH identifier
0536.60038

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 Minimal Growth Rate of Partial Maxima. Ann. Probab. 12 (1984), no. 2, 380--389. doi:10.1214/aop/1176993296. https://projecteuclid.org/euclid.aop/1176993296


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