Exact convergence rates in strong approximation laws for large increments of partial sums

Abstract

Consider partial sumsS n of an i.i.d. sequenceX1X2, ..., of centered random variables having a finite moment generating function ϕ in a neighborhood of zero. The asymptotic behaviour of\(U_n = \mathop {\max }\limits_{0 \leqq k \leqq n - b_n } (S_{k - b_n } - S_k )\) is investigated, where 1≦b n ≦n denotes an integer sequence such thatb n /logn→∞ asn→∞. In particular, ifb n =o(log p n) asn→∞ for somep>1, the exact convergence rate ofU n /b n α n =1 +0 (1) is determined, where α n depends uponb n and the distribution ofX1. In addition, a weak limit law forU n is derived. Finally, it is shown how strong invariance takes over if\(\mathop {\lim }\limits_{n \to \infty }\)b n (loglogn)2/log3n=∞.