Convergence in distribution of random mappings of finite sets to branching processes

A countable set $X=\bigcup_{t=0}^\infty X(t)$ is partitioned into pairwise disjoint finite layers $X(t)$, the cardinalities $|X(t)|$ of the sets $X(t)$, $t=0,1,2,\dots$, are finite. Each layer is partitioned into $r$ disjoint sets $X_i(t)$, $i=1,\ldots,r$, so that $X(t)=\bigcup_{i=1}^rX_i(t)$, $N_i(t)=|X_i(t)|$ and $N_i(t)\sim N\theta_i(t)$ as $N\to\infty$. We set $X'=X\setminus X(0)$.
We consider random mappings $y=f(x)$ of the set $X'$ into the set $X$. We assume that for any pairwise unequal $x_i$, $i=1,\ldots,k$, the random variables $y_i=f(x_i)$, $i=1,\ldots,k$, are independent and $f(X(t))\subseteq X(t-1)$, $t=1,2,\dots$ . Let $Y_i(0)\subseteq X_i(0)$ be some fixed subsets and $Y_i(t)=f^{-1}(Y(t-1))\cap X_i(t)$, $t=1,2,\dots$, be the sequence of preimages of $Y_i(0)$ in these random mappings. It is shown that $\mu_i(t,N)=|Y_i(t)|$, $i=1,\ldots, r$, converges in distribution as $N\to\infty$ to a non-homogeneous in time branching process with $r$ types of particles.
This research was supported by the Russian Foundation for Basic Research, grant 02–01–00266, and by the program of the President of Russian Federation for support of leading scientific schools, grant 1758.2003.1.