## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 2 (1992), 801-824.

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

#### Abstract

Let $\{S_n\}$ be a random walk on the integers with negative drift, and let $A_n = \{S_k \geq 0, 1 \leq k \leq n\}$ and $A = A_\infty$. Conditioning on $A$ is troublesome because $P(A) = 0$ and there is no natural sigma-field of events "like" $A. A$ natural definition of $P(B\mid A)$ is $\lim_{n\rightarrow\infty}P(B\mid A_n)$. The main result here shows that this definition makes sense, at least for a large class of events $B$: The finite-dimensional conditional distributions for the process $\{S_k\}_{k\geq 0}$ given $A_n$ converge strongly to the finite-dimensional distributions for a measure $\mathbf{Q}$. This distribution $\mathbf{Q}$ is identified as the distribution for a stationary Markov chain on $\{0,1,\ldots\}$.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 2 (1992), 801-824.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989807

**Digital Object Identifier**

doi:10.1214/aop/1176989807

**Mathematical Reviews number (MathSciNet)**

MR1159575

**Zentralblatt MATH identifier**

0756.60062

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J15

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

Large deviations Markov chains conditional limit theorems quasistationary distributions

#### Citation

Keener, Robert W. Limit Theorems for Random Walks Conditioned to Stay Positive. Ann. Probab. 20 (1992), no. 2, 801--824. doi:10.1214/aop/1176989807. https://projecteuclid.org/euclid.aop/1176989807