On the Time and the Excess of Linear Boundary Crossings of Sample Sums

Abstract

For an infinite sequence of independent and identically distributed random variables $X, X_1, X_2,\cdots; X_n,\cdots$ for which $EX = 0$ and $\operatorname{Var} X = 1$, the behavior of crossings that $S_n = \sum_{1\leqq k\leqq n} X_k \geqq n\varepsilon (\varepsilon > 0)$ for some $n \geqq 1$ has recently been under intensive investigation, of which the subject matters are the largest excess $Z = \sup_{n\geqq 1} (S_n - n\varepsilon)^+$, the last time $M = \sup \{n; S_n \geqq n\varepsilon, 1 \leqq n < \infty\}$, or $= 0$ if no such $\sup$ exists, and the number of crossings $N = \sum_{1\leqq n<\infty} I\{S_n \geqq n\varepsilon\}$ ($I$ means the indicator function). This paper describes a striking distributional similarity between $\varepsilon^2N$ and $\varepsilon^2(M - N)$ in the limiting sense as $\varepsilon \rightarrow 0$. Moreover, a new and systematic treatment for the moments problem unifies the previously published results as well as giving some new results. Existence of the limiting moments as $\varepsilon \rightarrow 0$ and of the moment generating function is also considered in detail. Most of the results for the one-sided crossings (i.e., $S_n \geqq n\varepsilon$) are then extended to cover their analogues in two-sided crossings (i.e., $|S_n| \geqq n\varepsilon)$.