Iid finite variance strong law

Identical distribution and IID families: Let $(X_i{)}_{i\in I}$ be random elements with the same measurable target $(E,\mathcal{E})$. They are identically distributed if $\mathbb{P}(X_i\in B)=\mathbb{P}(X_j\in B)$ for all $i,j\in I$ and $B\in \mathcal{E}$, that is, their laws in def-law-or-distribution-of-a-random-element agree. They are independent and identically distributed (IID) if, in addition, the whole family is independent in def-independent-random-elements. Independence means mutual independence, not merely pairwise independence. No moment assumption is part of either definition. The empty family satisfies these universal conditions vacuously.

Kolmogorov strong law for independent uniformly bounded variances: Independent square-integrable real $(X_n{)}_{n\ge 1}$ with $C={\sup }_n\mathrm{Var}(X_n)<\mathrm{\infty }$ satisfy $(S_n-\mathbb{E}{S}_n)\mathrm{/}n\to 0$ almost surely.