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

Summary Consider partial sumsS _n of an i.i.d. sequenceX ₁ X ₂, ..., of centered random variables having a finite moment generating function in a neighborhood of zero. The asymptotic behaviour of is investigated, where 1b _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 n} =1 +0 (1) is determined, where _n depends uponb _n and the distribution ofX ₁. In addition, a weak limit law forU _n is derived. Finally, it is shown how strong invariance takes over if b _n (loglogn)²/log³ n=.