Limit laws of Erdős-Rényi-Shepp type
Let \(S_ n\) denote the n-th partial sum of an i.i.d. sequence of centered random variables \(X_ i\) possessing a finite moment-generating function in a right neighborhood of zero. Consider the Shepp- resp. Erdős-Rényi-type statistics \[ T_ n=\max_{1\leq i\leq n}(S_{i+\kappa (i)}-S_ i),\quad U_ n=\max_{0\leq i\leq n- k}(S_{i+k}-S_ i), \] \[ W_ n=\max_{0\leq i\leq n-k}\max_{1\leq j\leq k}(S_{i+j}-S_ i),\quad V_ n=\max_{0\leq i\leq n-k}\min_{i\leq j\leq k}(k/j)(S_{i+j}-S_ i), \] where \(k=\kappa (n)=[c \log n]\) with a given constant \(c>0\). Extending their earlier results for \(U_ n\), the authors provide almost sure and convergence in probability statements for the random variables \(T_ n\), \(V_ n\) and \(W_ n\), including also the precise convergence rate results. Moreover, extended versions of the \textit{P. Erdős} and \textit{A. Rényi} [J. Anal. Math. 23, 103-111 (1970; Zbl 0225.60015)] and \textit{L. A. Shepp} [Ann. Math. Stat. 35, 424-428 (1964; Zbl 0146.391)] theorems are presented, covering the full range of constants \(c>0\) appearing in \(k=[c \log n]\). Last but not least, a limiting extreme value distribution is derived for \(U_ n\) (after suitable normalization), if the underlying distribution of \(X_ i\) is non-lattice. Some possible applications are briefly outlined.
scientific article; zbMATH DE number 4038915