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.
Publié le : 1985-11-14
Classification:
Partial maxima,
minimal growth rate,
upper and lower class sequences,
strong limit theorems,
60F15,
60F20,
60F10,
60G99
@article{1176992820,
author = {Klass, Michael J.},
title = {The Robbins-Siegmund Series Criterion for Partial Maxima},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 1369-1370},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992820}
}
Klass, Michael J. The Robbins-Siegmund Series Criterion for Partial Maxima. Ann. Probab., Tome 13 (1985) no. 4, pp. 1369-1370. http://gdmltest.u-ga.fr/item/1176992820/