Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums

Let $X_1, X_2,\cdots$ be an i.i.d. sequence with $EX_1 = 0, EX^2_1 = 1, Ee^{tX_1} < \infty (|t| < t_0)$, and partial sums $S_n = X_1 + \cdots + X_n$. Starting from some analogous results for the Wiener process, this paper studies the almost sure limiting behaviour of $\max_{0 \leq n \leq N - a_N} a^{-1/2}_N (S_{n + a_N} - S_n)$ as $N \rightarrow \infty$ under various conditions on the integer sequence $a_N$. Improvements of the Erdos-Renyi law of large numbers for partial sums are obtained as well as strong invariance principle-type versions via the Komlos-Major-Tusnady approximation. An appearing gap between these two results is also going to be closed.