On the Time and the Excess of Linear Boundary Crossings of Sample Sums
The Annals of Statistics · 1978 · Volume 6 · Issue 1
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., $|S_n| \geqq n\varepsilon)$.
Keywords
60F10, 60F15, 60G50, largest excess, last time, Limiting distribution, limiting moments, linear boundary crossings, moment generating functions, sample sums, total number
Enable JavaScript for citation tools and additional paper interactions.