Strong law under summable normalized variances

Kolmogorov convergence criterion: For independent centered square-integrable real random variables $(X_n{)}_{n\ge 1}$, if ${\sum }_{n\ge 1}\mathrm{Var}(X_n)<\mathrm{\infty }$, then ${\sum }_{n\ge 1}{X}_n$ converges almost surely and in $L^2$ to the same finite real random variable.

Kronecker summation lemma: Let $(x_n{)}_{n\ge 1}$ be real and let $0<b_n$ be deterministic, nondecreasing, and tend to infinity. If ${\sum }_{n\ge 1}{x}_n\mathrm{/}{b}_n$ converges in $\mathbb{R}$, then $\frac{1}{b_n}{\sum }_{k=1}^n{x}_k⟶0.$ Repeated values of $b_n$ are allowed.

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.

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 integral test: for $f\ge 0$ nonincreasing on $[0,\mathrm{\infty })$, ${\sum }_{k}f(k)$ converges if and only if the sequence $({\int }_0^{N}f{)}_N$ is bounded, with ${\int }_0^{N}f\le {\sum }_{k<N}f(k)\le f(0)+{\int }_0^{N}f$: For nonnegative nonincreasing $f$ on $[0,\mathrm{\infty })$, the series ${\sum }_{k\ge 0}f(k)$ converges if and only if the proper integrals ${\int }_0^{N}f$ are bounded above as integers $N$ vary.