The Annals of Probability

Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive

Donald L. Iglehart

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Abstract

Let $\{\xi_k: k \geqq 1\}$ be a sequence of independent, identically distributed random variables with $E\{\xi_1\} = 0$ and $E\{\xi_1^2\} = \sigma^2, 0 < \sigma^2 < \infty$. Form the random walk $\{S_n: n \geqq 0\}$ by setting $S_0 = 0, S_n = \xi_1 + \cdots + \xi_n, n \geqq 1$. Let $T$ denote the hitting time of the set $(-\infty, 0\rbrack$ by the random walk. The main result in this paper is a functional central limit theorem for the random functions $S_{\lbrack nt \rbrack}/\sigma n^{\frac{1}{2}}, 0 \leqq t \leqq 1$, conditional on $T > n$. The limit process, $W^+$, is identified in terms of standard Brownian motion. Similar results are obtained for random partial sums and renewal processes. Finally, in the case where $E\{\xi_1\} = \mu > 0$, it is shown that the conditional (on $T > n$) and unconditional weak limit for $(S_{\lbrack nt\rbrack} - \mu nt)/\sigma n^{\frac{1}{2}}$ is the same, namely, Brownian motion.

Article information

Source
Ann. Probab., Volume 2, Number 4 (1974), 608-619.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996607

Digital Object Identifier
doi:10.1214/aop/1176996607

Mathematical Reviews number (MathSciNet)
MR362499

Zentralblatt MATH identifier
0299.60053

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G65 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Conditioned limit theorems functional central limit theorem invariance principle random walks renewal theory weak convergence

Citation

Iglehart, Donald L. Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive. Ann. Probab. 2 (1974), no. 4, 608--619. doi:10.1214/aop/1176996607. https://projecteuclid.org/euclid.aop/1176996607


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