On the almost sure minimal growth rate of partial sum maxima (Q1904485)

Let \(\{X_j\}_j\) be a sequence of i.i.d. random variables, \(S_n=\sum^n_{j=1} X_j\), and let \(\{a_n\}\) be an increasing sequence of positive constants tending to \(\infty\). This paper studies the almost lower limit of \(\max_{1\leq j \leq n}S_j/a_n\). It is shown that the lower limit is either 0 or \(\infty\) under mild conditions, and some integral tests to determine which is the case are given. Let \(\tau=\inf \{n\geq 1:S_n>0\}\) and \(\tau_-=\inf \{n\geq 1:S_n \leq 0\}\). Several inequalities are gotten that determine up to scale constants various quantities, which involve truncated moments of the ladder variables \(S_\tau\) and \(\tau\) under three different conditions: \(ES_\tau < \infty\), \(E|S_{\tau_-} |<\infty\) and \(X_j\) symmetric. Moments of ladder variables are also discussed. The problems discussed here have their origin in the desire on the part of many authors to refine, extend and achieve a deeper understanding of the strong law of large numbers and the law of the iterated logarithm. In 1965, Hirsch considered a specific question of concern here.