## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 1 (1976), 115-121.

### An Invariance Principle for Random Walk Conditioned by a Late Return to Zero

#### Abstract

Let $\{S_n: n \geqq 0\}$ denote the recurrent random walk formed by the partial sums of i.i.d. integer-valued random variables with zero mean and finite variance. Let $T = \min \{n \geqq 1: S_n = 0\}$. Our main result is an invariance principle for the random walk conditioned by the event $\lbrack T = n\rbrack$. The limiting process is identified as a Brownian excursion on [0, 1].

#### Article information

**Source**

Ann. Probab., Volume 4, Number 1 (1976), 115-121.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996189

**Digital Object Identifier**

doi:10.1214/aop/1176996189

**Mathematical Reviews number (MathSciNet)**

MR415706

**Zentralblatt MATH identifier**

0332.60047

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B10: Convergence of probability measures

Secondary: 60G50: Sums of independent random variables; random walks 60J15 60K99: None of the above, but in this section 60F05: Central limit and other weak theorems 60J65: Brownian motion [See also 58J65]

**Keywords**

Conditioned limit theorems hitting time invariance principle random walk weak convergence

#### Citation

Kaigh, W. D. An Invariance Principle for Random Walk Conditioned by a Late Return to Zero. Ann. Probab. 4 (1976), no. 1, 115--121. doi:10.1214/aop/1176996189. https://projecteuclid.org/euclid.aop/1176996189