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

Abstract

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