Maxima of increments of partial sums for certain subexponential distributions (Q1411891)

The following result is proved for partial sums \(S_n\) of i.i.d. random variables that exhibit subexponential tails: \[ \lim _{n\to \infty }\left \{ \max _{0\leq j <n} \max _ {1\leq k \leq n - j} \frac {S_{j+k} - S_j}{\varphi (k/ \log n)^{2p - 1} (\log n)^p }\right \} = 1 \text{ a.s.}, \] where \(\varphi \) is a function specified in the paper and \(p > 1.\) Instead of the assumption of existence and finiteness of the moment generating function of the underlying distribution some other moment conditions are considered.