Tight Markov chains and random compositions

For an ergodic Markov chain $\{X(t)\}$ on $\Bbb N$, with a stationary distribution $\pi$, let $T_n>0$ denote a hitting time for $[n]^c$, and let $X_n=X(T_n)$. Around 2005 Guy Louchard popularized a conjecture that, for $n\to \infty$, $T_n$ is almost Geometric($p$), $p=\pi([n]^c)$, $X_n$ is almost stationarily distributed on $[n]^c$, and that $X_n$ and $T_n$ are almost independent, if $p(n):=\sup_ip(i,[n]^c)\to 0$ exponentially fast. For the chains with $p(n) \to 0$ however slowly, and with $\sup_{i,j}\,\|p(i,\cdot)-p(j,\cdot)\|_{TV}<1$, we show that Louchard's conjecture is indeed true even for the hits of an arbitrary $S_n\subset\Bbb N$ with $\pi(S_n)\to 0$. More precisely, a sequence of $k$ consecutive hit locations paired with the time elapsed since a previous hit (for the first hit, since the starting moment) is approximated, within a total variation distance of order $k\,\sup_ip(i,S_n)$, by a $k$-long sequence of independent copies of $(\ell_n,t_n)$, where $\ell_n= \text{Geometric}\,(\pi(S_n))$, $t_n$ is distributed stationarily on $S_n$, and $\ell_n$ is independent of $t_n$. The two conditions are easily met by the Markov chains that arose in Louchard's studies as likely sharp approximations of two random compositions of a large integer $\nu$, a column-convex animal (cca) composition and a Carlitz (C) composition. We show that this approximation is indeed very sharp for most of the parts of the random compositions. Combining the two approximations in a tandem, we are able to determine the limiting distributions of $\mu=o(\ln\nu)$ and $\mu=o(\nu^{1/2})$ largest parts of the random cca composition and the random C-composition, respectively. (Submitted to Annals of Probability in August, 2009.