Every sequence of real-valued random variables with sup E(f_n²) < ∞ admits a subsequence that converges completely and hereditarily in Cesàro mean to some f_∞ in L², with a marginally weaker condition being necessary and sufficient.
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Hereditary Hsu-Robbins-Erd\"os Law of Large Numbers
Every sequence of real-valued random variables with sup E(f_n²) < ∞ admits a subsequence that converges completely and hereditarily in Cesàro mean to some f_∞ in L², with a marginally weaker condition being necessary and sufficient.