Nonidentical strong law under summable normalized variances

Variance and covariance identities for random variables: Let $X,Y$ be square-integrable real random variables on one probability space. Then $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2,$ $\mathrm{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\mathrm{.}$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents: For $a,b>0$ and $r,s\in \mathbb{R}$, $a^{r+s}=a^r{a}^s,(ab{)}^r=a^r{b}^r,(a\mathrm{/}b{)}^r=a^r\mathrm{/}{b}^r,(a^r{)}^s=a^{rs}\mathrm{.}$

The p-series for a real exponent p converges exactly when p is greater than one: For every real $p$, ${\sum }_{k\ge 1}\frac{1}{k^p}\text{ converges}⟺p>1.$

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 $n\ge 2$).