Poisson Approximations for $r$-Scan Processes

Abstract

Let $X_i$ be positive i.i.d. random variables (or more generally a uniformly mixing positive-valued ergodic stationary process). The $r$-scan process induced by $\{X_i\}$ is $R_i = \sum^{i+r-1}_{k=i} X_k, i = 1, 2, \ldots, n - r + 1$. Limiting distributions for the extremal order statistics among $\{R_i\}$ suitably normalized (and appropriate threshold values $a = a_n$ and $b = b_n$) are derived as a consequence of Poisson approximations to the Bernoulli sums $N^-(a) = \sum^{n+r-1}_{i=1} W^-_i(a)$ and $N^+(b) = \sum^{n-r+1}_{i=1}W^+_i(b)$, where $W^-_i(a) \lbrack W^+_i(b) \rbrack = 1$ or 0 according as $R_i \leq a (R_i > b)$ occurs or not. Applications include limit theorems for $r$-spacings based on i.i.d. uniform $\lbrack 0, 1 \rbrack$ r.v.'s, for extremal $r$-spacings based on i.i.d. samples from a general density and for the $r$-scan process with a variable time horizon.