## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 2, Number 2 (1992), 329-357.

### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 2, Number 2 (1992), 329-357.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005707

**Digital Object Identifier**

doi:10.1214/aoap/1177005707

**Mathematical Reviews number (MathSciNet)**

MR1161058

**Zentralblatt MATH identifier**

0761.60018

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60E20 60G50: Sums of independent random variables; random walks

**Keywords**

$r$-scans $r$-spacings extremal distributions Poisson approximation

#### Citation

Dembo, Amir; Karlin, Samuel. Poisson Approximations for $r$-Scan Processes. Ann. Appl. Probab. 2 (1992), no. 2, 329--357. doi:10.1214/aoap/1177005707. https://projecteuclid.org/euclid.aoap/1177005707