Finite variance logarithmic rate for iid sums

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: The function $\log :(0,\mathrm{\infty })\to \mathbb{R}$ is continuous and strictly increasing, is onto $\mathbb{R}$, and satisfies, for $x,y>0$, $\log (xy)=\log x+\log y,\log (x\mathrm{/}y)=\log x-\log y,\log (1\mathrm{/}x)=-\log x\mathrm{.}$ Also $\log 1=0$.

Continuity and derivatives of positive-base real powers: For $a>0$, the function $x\mapsto a^x$ is continuous on $\mathbb{R}$ and $(a^x{)}^{\mathrm{\prime }}=a^x\log a\mathrm{.}$ For $\alpha \in \mathbb{R}$, the function $x\mapsto x^{\alpha }$ is continuous and differentiable on $(0,\mathrm{\infty })$, with $(x^{\alpha }{)}^{\mathrm{\prime }}=\alpha x^{\alpha -1}\mathrm{.}$

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 $p$, ${\sum }_{k\ge 1}\frac{1}{k^p}\text{ converges}⟺p>1.$

If $0\le a_k\le b_k$ eventually, convergence of $\sum b_k$ gives convergence of $\sum a_k$, and divergence of $\sum a_k$ gives divergence of $\sum b_k$: Let $(a_k)$ and $(b_k)$ be sequences of reals and suppose there is $K\in \mathbb{N}$ with

$0\le a_k\le b_k\text{for all }k\ge K\mathrm{.}$

Then:

1. if $\sum b_k$ converges then $\sum a_k$ converges (def-series);
2. if $\sum a_k$ diverges then $\sum b_k$ diverges.

The same statement holds verbatim for series with a general starting index $m$, applied to the shifted sequences of def-series.

The hypothesis is on the terms from some index on, not on all of them: finitely many terms of either sequence may violate it, or be negative, without affecting the conclusion. What may not be dropped is nonnegativity of $(a_k)$ from that index on.

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 $n\ge 2$).