Description
We consider a sequence of Markov processes $X^1,X^2,\ldots$ with state space $E$, where $X^N$ is subject to a strong drift towards a subset $D\subseteq E$, and where $\Phi(X^N)$ evolves on a slower timescale for a suitable map $\Phi:E\to D$. Using the method of martingale problems, we prove a limit theorem showing that, as $N\to\infty$, $\Phi(X^N)\Rightarrow Z$ in the space of càdlàg paths, while $X^N$ converges to $X$ in measure. We apply this general limit result to models of copy number variation of genetic elements in a diploid Moran population of size $N$. At time $t$, the population is described by $X_t^N\in\mathcal P(\mathbb N_0)$, where $X_t^N(k)$ denotes the frequency of individuals with copy number $k$, and $\Phi:\mathcal P(\mathbb N_0)\to\mathbb R$ is the first-moment map.