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The Robbins-Siegmund series criterion for partial maxima. (English) Zbl 0576.60023

Summary: Let \(X,X_ 1,X_ 2,..\). 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 \(P(X>b_ n)\to 0\) and \(P(M_ n\leq b_ n)\to 0\) we show that \[ P(M_ n\leq b_ n\quad i.o.)=1\quad if\quad and\quad only\quad if\quad \sum_{n}P(X>b_ n)\exp \{-nP(X>b_ n)\}=\infty. \] The restrictions on the \(b_ n's\) can be removed.

MSC:

60F15 Strong limit theorems
60F10 Large deviations
60F20 Zero-one laws
60G99 Stochastic processes
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