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May, 1992 Poisson Approximations for $r$-Scan Processes
Amir Dembo, Samuel Karlin
Ann. Appl. Probab. 2(2): 329-357 (May, 1992). DOI: 10.1214/aoap/1177005707


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.


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Amir Dembo. Samuel Karlin. "Poisson Approximations for $r$-Scan Processes." Ann. Appl. Probab. 2 (2) 329 - 357, May, 1992.


Published: May, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0761.60018
MathSciNet: MR1161058
Digital Object Identifier: 10.1214/aoap/1177005707

Primary: 60F05
Secondary: 60E20 , 60G50

Keywords: $r$-scans , $r$-spacings , extremal distributions , Poisson approximation

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.2 • No. 2 • May, 1992
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