## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 48, Number 3 (2016), 792-811.

### Analysis of a stochastic approximation algorithm for computing quasi-stationary distributions

J. Blanchet, P. Glynn, and S. Zheng

#### Abstract

We study the convergence properties of a Monte Carlo estimator proposed in the physics literature to compute the quasi-stationary distribution on a transient set of a Markov chain (see De Oliveira and Dickman (2005), (2006), and Dickman and Vidigal (2002)). Using the theory of stochastic approximations we verify the consistency of the estimator and obtain an associated central limit theorem. We provide an example showing that convergence might occur very slowly if a certain eigenvalue condition is violated. We alleviate this problem using an easy-to-implement projection step combined with averaging.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 792-811.

**Dates**

First available in Project Euclid: 19 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1474296315

**Mathematical Reviews number (MathSciNet)**

MR3568892

**Zentralblatt MATH identifier**

1352.60106

**Subjects**

Primary: 60J22: Computational methods in Markov chains [See also 65C40]

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Quasi-stationary distribution stochastic approximation Markov chain central limit theorem

#### Citation

Blanchet, J.; Glynn, P.; Zheng, S. Analysis of a stochastic approximation algorithm for computing quasi-stationary distributions. Adv. in Appl. Probab. 48 (2016), no. 3, 792--811. https://projecteuclid.org/euclid.aap/1474296315