Summary
Consider partial sumsS n of an i.i.d. sequenceX 1 X 2, ..., 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(logp 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 ofX 1. 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/log3 n=∞.
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Deheuvels, P., Steinebach, J. Exact convergence rates in strong approximation laws for large increments of partial sums. Probab. Th. Rel. Fields 76, 369–393 (1987). https://doi.org/10.1007/BF01297492
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DOI: https://doi.org/10.1007/BF01297492