Strong law under summable normalized variances

The variables $Y_n=(X_n-\mathbb{E}{X}_n)\mathrm{/}{b}_n$ are independent, centered, and square-integrable, with variances $\mathrm{Var}(X_n)\mathrm{/}{b}_n^2$. Measurable transformations give independence, and covariance bilinearity gives the variance identity. The convergence criterion makes ${\sum }_n{Y}_n$ converge on one probability-one event.

On each path in that event apply Kronecker to $x_n=X_n-\mathbb{E}{X}_n$ and the given $b_n$. The deterministic conclusion is precisely the asserted normalized convergence. Zero variances and repeated positive normalizers present no exception.

To apply the general assertion of step 2.1 to the rate, put $q=1\mathrm{/}2+\epsilon >0$, $p=2q=1+2\epsilon >1$, and $b_n=n^{1\mathrm{/}2}(\log n{)}^q$ for $n\ge 2$, with $b_1=b_2$. By [F5], $\log n\ge \log 2>0$, both factors are positive and nondecreasing, and $b_n\ge (\log 2{)}^q\sqrt{n}\to \mathrm{\infty }$. Here the positive real power $n^{1\mathrm{/}2}$ is the positive square root because its square is $n$ by the power laws; its unboundedness follows directly from $n>A^2$ implying $\sqrt{n}>A$ for $A>0$. Also $b_n^2=n(\log n{)}^p$. Thus the normalizers satisfy every hypothesis of the general assertion, once variance summability is checked.

For each integer $j\ge 1$, the block $2^j\le n<2^{j+1}$ has exactly $2^j$ terms. The logarithm product law gives $\log (2^j)=j\log 2$ by finite induction. Consequently every term in that block is at most $1\mathrm{/}[2^j(j\log 2{)}^p]$, and the whole block satisfies ${\sum }_{n=2^j}^{2^{j+1}-1}1\mathrm{/}[n(\log n{)}^p]\le (\log 2{)}^{-p}{j}^{-p}$. Every finite partial sum starting at $n=2$ is bounded by a sum of finitely many complete blocks: the elementary induction $2^J\ge J+1$ ensures such an upper endpoint exists. By [F6] these block bounds have uniformly bounded partial sums, at most $(\log 2{)}^{-p}{\sum }_{j\ge 1}{j}^{-p}<\mathrm{\infty }$. Applying the nonnegative bounded-partial-sum criterion, rather than assuming an infinite regrouping identity, proves ${\sum }_{n\ge 2}1\mathrm{/}[n(\log n{)}^p]<\mathrm{\infty }$.

The common variance is a finite ${\sigma }^2\ge 0$. By steps 3.1–4.1, ${\sum }_n\mathrm{Var}(X_n)\mathrm{/}{b}_n^2={\sigma }^2\mathrm{/}{b}_1^2+{\sigma }^2{\sum }_{n\ge 2}1\mathrm{/}[n(\log n{)}^p]<\mathrm{\infty }$. The general assertion proved in steps 1.1–2.1 now gives the stated rate, including ${\sigma }^2=0$. All normalizers, blocks and measurable transforms were explicitly defined; no Choice assumption or later strong-law/rate theorem is used.