Kolmogorov iid l1 strong law

Measurable coordinatewise functions preserve independence: Let $(X_i{)}_{i\in I}$ be an independent family of random elements $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$. For each $i$, let $g_i:(S_i,{\mathrm{\Sigma }}_i)\to (T_i,{\mathcal{T}}_i)$ be measurable. Then the family $(g_i\circ X_i{)}_{i\in I}$ is independent.

Summability of truncated normalized variances: For identically distributed integrable real $(X_n)$ and $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$, ${\sum }_{n\ge 1}\mathrm{Var}(Y_n)\mathrm{/}{n}^2\le 2\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$. No independence is required.

Strong law under summable normalized variances: Let $(X_n{)}_{n\ge 1}$ be independent square-integrable real random variables. Let $0<b_n$ be deterministic and nondecreasing with $b_n\to \mathrm{\infty }$. If ${\sum }_{n\ge 1}\frac{\mathrm{Var}(X_n)}{b_n^2}<\mathrm{\infty },$ then $\frac{1}{b_n}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)⟶0\text{almost surely}\mathrm{.}$ In particular, for IID centered square-integrable variables and any $\epsilon >0$, $S_n\mathrm{/}[\sqrt{n}(\log n{)}^{1\mathrm{/}2+\epsilon }]\to 0$ almost surely (the displayed normalization is used for $n\ge 2$).

Cesaro limit of truncated means: For identically distributed integrable real $(X_n)$, with $Y_k=X_k{\mathbf{1}}_{\{\mathrm{\mid }{X}_k\mathrm{\mid }\le k\}}$, one has $n^{-1}{\sum }_{k=1}^n\mathbb{E}{Y}_k\to \mathbb{E}{X}_1$.

Iid linear truncation occurs only finitely often: For identically distributed integrable real $(X_n)$, put $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$. Almost surely $Y_n=X_n$ for all sufficiently large $n$. Consequently $n^{-1}{\sum }_{k=1}^n(X_k-Y_k)\to 0$. Independence is unnecessary.