Hereditary Hsu-Robbins-Erdös Law of Large Numbers

We show that every sequence $f_1,f_2,\cdots$ of real-valued random variables with ${\sup }_{n\in \mathbb{N}}\text{E}(f_n^2)<\mathrm{\infty }$ contains a subsequence $f_{k_1},f_{k_2},\cdots$ converging in $\text{textsc}Ces\grave{a}ro$ mean to some $f_{\mathrm{\infty }}\in {\mathbb{L}}^2$ {\it completely,} to wit, ${\sum }_{N\in \mathbb{N}}\mathrm{\P }(\mid \frac{1}{N}{\sum }_{n=1}^N{f}_{k_n}-f_{\mathrm{\infty }}\mid >\text{eps})<\mathrm{\infty },\mathrm{\forall }\text{eps}>0;$ and {\it hereditarily,} i.e., along all further subsequences as well. We also identify a condition, slightly weaker than boundedness in ${\mathbb{L}}^2,$ which turns out to be not only sufficient for the above hereditary complete convergence in $\text{textsc}Ces\grave{a}ro$ mean, but necessary as well.