  We propose two numerical schemes for approximating quasi-stationary
distributions (QSD) of finite state Markov chains with absorbing states. Both
schemes are described in terms of certain interacting chains in which the
interaction is given in terms of the total time occupation measure of all
particles in the system and has the impact of reinforcing transitions, in an
appropriate fashion, to states where the collection of particles has spent more
time. The schemes can be viewed as combining the key features of the two basic
simulation-based methods for approximating QSD originating from the works of
Fleming and Viot (1979) and Aldous, Flannery and Palacios (1998), respectively.
The key difference between the two schemes studied here is that in the first
method one starts with $a(n)$ particles at time $0$ and number of particles
stays constant over time whereas in the second method we start with one
particle and at most one particle is added at each time instant in such a
manner that there are $a(n)$ particles at time $n$. We prove almost sure
convergence to the unique QSD and establish Central Limit Theorems for the two
schemes under the key assumption that $a(n)=o(n)$. When $a(n)\sim n$, the
fluctuation behavior is expected to be non-standard. Some exploratory numerical
results are presented to illustrate the performance of the two approximation
schemes.
