On the rate of convergence in the strong law of large numbers

Let $X_1,X_2,\dots$ be independent random variables, $S_n=X_1+\dots+X_n$, $\{b_n\}_{n=1}^\infty$ be a positive nondecreasing sequence, $\{n_i\}_{i=1}^\infty$ be an increasing sequence of integers satisfying some conditions. We obtain relations between $\displaystyle\mathbf P\{\sup_{k\ge n_m}S_k/b_k\ge\varepsilon\}$ and
$$ Q_m(\varepsilon)=\mathbf P\{S_{n_m}\ge \varepsilon b_{n_m}\}+\sum_{k=m}^\infty\mathbf P\{S_{n_{k+1}}-S_{n_k}\ge\varepsilon b_{n_{k+1}}\},\qquad\varepsilon>0,m\ge 1. $$