Limit laws for the modulus of continuity of the partial sum process and for the Shepp statistic (Q1114998)

Let \(S_ n\) denote the partial sum of an i.i.d. sequence of centered r.v.-s, having a finite moment generating function \(\phi\) in a neighborhood of zero. The authors prove strong and weak limit laws for \[ W_ n=\max_{1\leq i\leq n-k}\max_{1\leq j\leq k}(S_{i+j}-S_ i),\quad V_ n=\max_{1\leq i\leq n-k}\min_{1\leq j\leq k}(k/j)(S_{i+j}-S_ i) \] \[ and\quad T_ n=\max_{1\leq i\leq n-k}(S_{i+k(i)}-S_ i), \] where \(1\leq k=k(n)\leq n\) is an integer sequence such that \(k(n)/n\to 0,\) and \(\lim_{n\to \infty} \inf k(n)/\log n>0.\) These questions cannot be handled by invariance principles. The ``critical choice'' \(k=k(n)=[c \log n]\), where \(0<c<\infty\) is fixed, has been studied extensively starting with \textit{P. Erdős} and \textit{A. Rényi} [J. Analyse Math. 23, 103-111 (1970; Zbl 0225.60015)].