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On a functional central limit theorem for random walks conditioned to stay positive


Bolthausen, E (1976). On a functional central limit theorem for random walks conditioned to stay positive. The Annals of Probability, 4(3):480-485.

Abstract

Let $\{X_k: k \geqq 1\}$ be a sequence of i.i.d.rv with $E(X_i) = 0$ and $E(X_i^2) = \sigma^2, 0 < \sigma^2 < \infty$. Set $S_n = X_1 + \cdots + X_n$. Let $Y_n(t)$ be $S_k/\sigma n^\frac{1}{2}$ for $t = k/n$ and suitably interpolated elsewhere. This paper gives a generalization of a theorem of Iglehart which states weak convergence of $Y_n(t)$, conditioned to stay positive, to a suitable limiting process.

Abstract

Let $\{X_k: k \geqq 1\}$ be a sequence of i.i.d.rv with $E(X_i) = 0$ and $E(X_i^2) = \sigma^2, 0 < \sigma^2 < \infty$. Set $S_n = X_1 + \cdots + X_n$. Let $Y_n(t)$ be $S_k/\sigma n^\frac{1}{2}$ for $t = k/n$ and suitably interpolated elsewhere. This paper gives a generalization of a theorem of Iglehart which states weak convergence of $Y_n(t)$, conditioned to stay positive, to a suitable limiting process.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Conditioned limit theorem; functional central limit theorem; random walks; weak convergence
Language:English
Date:1976
Deposited On:04 Nov 2009 13:15
Last Modified:05 Apr 2016 13:30
Publisher:Institute of Mathematical Statistics
ISSN:0091-1798
Publisher DOI:https://doi.org/10.1214/aop/1176996098
Related URLs:http://www.zentralblatt-math.org/zmath/en/search/?q=an:0336.60024
http://www.ams.org/mathscinet-getitem?mr=415702

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