  We study a discrete-time stochastic process that can also be interpreted as a
model for a viral evolution. A distinguishing feature of our process is
power-law tails due to dynamics that resembles preferential attachment models.
In the model we study, a population is partitioned into sites, with each site
labeled by a uniquely-assigned real number in the interval $[0,1]$ known as
fitness. The population size is a discrete-time transient birth-and-death
process with probability $p$ of birth and $1-p$ of death. The fitness is
assigned at birth according to the following rule: the new member of the
population either "mutates" with probability $r$, creating a new site uniformly
distributed on $[0,1]$ or "inherits" with probability $1-r$, joining an
existing site with probability proportional to the site's size. At each death
event, a member from the site with the lowest fitness is killed. The number of
sites eventually tends to infinity if and only if $pr>1-p$. Under this
assumption, we show that as time tends to infinity, the joint empirical measure
of site size and fitness (proportion of population in sites of size and fitness
in given ranges) converges a.s. to the product of a modified Yule distribution
and the uniform distribution on $[(1-p)/(pr),1]$. Our approach is based on the
method developed in \cite{similar-but-different}. The model and the results
were independently obtained by Roy and Tanemura in [RT].
