The Annals of Probability

Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive

Donald L. Iglehart

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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


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

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


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.

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