The strong law of large numbers: a weak-𝑙₂ view

Heinkel, Bernard

doi:10.1090/S0002-9939-1995-1213860-X

The strong law of large numbers: a weak-$l_ 2$ view
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Proc. Amer. Math. Soc. 123 (1995), 273-280 Request permission

Abstract:

Let $({X_k})$ be a sequence of independent, centered, and square integrable real-valued random variables. To that sequence one associates \[ \forall n \in \mathbb {N},\quad {\xi _n} = {\left \| {({2^{ - n}}{X_k}),\;{2^n} + 1 \leq k \leq {2^{n + 1}}} \right \|_{2,\infty }}.\] When there exists $K \geq 1$ such that \[ \sum \limits _{n \geq 1} {{P^K}({\xi _n} > {c_n}) < + \infty ,} \] where $({c_n})$ is a suitable sequence of positive constants, then the strong law of large numbers holds if and only if $({X_k}/k)$ converges almost surely to 0.

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