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May, 1984 The Minimal Growth Rate of Partial Maxima
Michael J. Klass
Ann. Probab. 12(2): 380-389 (May, 1984). DOI: 10.1214/aop/1176993296


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 $\{M_n\}$.


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Michael J. Klass. "The Minimal Growth Rate of Partial Maxima." Ann. Probab. 12 (2) 380 - 389, May, 1984.


Published: May, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0536.60038
MathSciNet: MR735844
Digital Object Identifier: 10.1214/aop/1176993296

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

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

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 2 • May, 1984
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