On extending the Brunk–Prokhorov strong law of large numbers

We prove that the sequence $\{b_n^{-1}(X_1+\dots+X_n)\}_{n\ge 1}$ converges almost everywhere to zero if $\{X_n\}_{n\ge 1}$ is a martingale difference with respect to some increasing sequence of $\sigma$-algebras of the basic probability space, the series $\sum_{n=1}^{\infty}n^{r-1}b_n^{-2r}E|X_n|^{2r}$ converges for some $r > 1$, the sequence of positive numbers $\{b_n\}_{n\ge 1}$ does not decrease and is unbounded, and there exists a strictly increasing sequence of positive integers $\{k_n\}_{n\ge 1}$ such that $\sup_{n\ge 1}k_{n+1}k_n^{-1}=d < \infty$ and
$$ 0<b=\inf_{n\ge 1}b_{k_n}b_{k_{n+1}}^{-1}\le \sup_{n\ge 1}b_{k_n}b_{k_{n+1}}^{-1}=c<1. $$

For $b_n=n$, all hypotheses are satisfied and the theorem reduces to the well-known theorem due to Brunk and Prokhorov for independent random variables.