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August, 1974 Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive
Donald L. Iglehart
Ann. Probab. 2(4): 608-619 (August, 1974). DOI: 10.1214/aop/1176996607


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.


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Donald L. Iglehart. "Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive." Ann. Probab. 2 (4) 608 - 619, August, 1974.


Published: August, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0299.60053
MathSciNet: MR362499
Digital Object Identifier: 10.1214/aop/1176996607

Primary: 60F05
Secondary: 60G65 , 60K25

Keywords: conditioned limit theorems , functional central limit theorem , invariance principle , Random walks , renewal theory , weak convergence

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 4 • August, 1974
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