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Exact convergence rate in the limit theorems of Erdős-Rényi and Shepp. (English) Zbl 0595.60033
Let \(X_ 1,X_ 2,..\). be a nondegenerate i.i.d. sequence satisfying \(EX_ 1=0\) and \(\phi (t)<\infty\) for \(0\leq t<t_ 0\). Set \(S_ 0=0\), \(S_ n=X_ 1+...+X_ n\) and \(U_ n=\max_{0\leq i\leq n-k}(S_{i+k}- S_ i),\) where \(k=k_ n=[c \log n],\) \(c>0\). The authors establish the exact a.s. convergence rate (including the best constants) for the (so- called) Erdős-Rényi theorem [P. Erdős and A. Rényi, J. Anal. Math. 23, 103-111 (1970; Zbl 0225.60015)] which says that, for \(\alpha\in (0,A)\), \(A=\sup \{\phi '(t)/\phi (t):\quad 0<t<t_ 0\},\) and \(c=c(\alpha)\) such that \(\exp (-1/c)=\inf_{t}\phi (t)\exp (-t\alpha),\) one has \(U_ n/(k\alpha)=1+o(1).\)
By presenting precise lim inf and lim sup results it is shown that the o- term in the latter relation can be replaced by \(O(k^{-1}\log k)\). The same limits are obtained for the statistic \(T_ n=\max_{0\leq i\leq n}(S_{i+k_ i}-S_ i),\) which has been investigated by L. A. Shepp [Ann. Math. Stat. 35, 424-428 (1964; Zbl 0146.391)]. Moreover, some useful assertions concerning moment-generating functions and an earlier large deviation result of V. V. Petrov [Teor. Veroyatn. Primen. 10, 310-322 (1965; Zbl 0235.60028)] are also derived.
Reviewer: J.Steinebach

MSC:
60F15 Strong limit theorems
60F10 Large deviations
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