## The Annals of Probability

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

### Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive

#### 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