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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  \[    ∑  n ≥  1 P K ( ξ  n &gt; c n ) &gt; + ∞  , \sum \limits _{n \geq 1} {{P^K}({\xi _n} &gt; {c_n}) &gt; + \infty ,}     \]  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.

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