On the smallest maximal increment of partial sums of i.i.d. random variables (Q1355715)

The authors study the almost sure limiting behaviour of the minimum \(m_n(k_n)\) of maximal increments of partial sums (suitably centered and normalized), taken over subintervals of size \(k_n\) in a large interval of size \(n\). The limiting behaviour is different for the cases (i) \(k_n/\log n\to\infty\), (ii) \(k_n=O(\log n)\), or (iii) \(k_n=o(\log n)\) as \(n\to\infty\). Lim inf and lim sup results are obtained under almost no assumptions on the underlying distribution. Relations to an earlier result of Csörgö and Révész (1981) are also discussed.