For an ergodic Markov chain {X(t)} on N, with a stationary
distribution Ο, let Tnβ>0 denote a hitting time for [n]c, and let
Xnβ=X(Tnβ). Around 2005 Guy Louchard popularized a conjecture that, for nββ, Tnβ is almost Geometric(p), p=Ο([n]c), Xnβ is almost
stationarily distributed on [n]c, and that Xnβ and Tnβ are almost
independent, if p(n):=supiβp(i,[n]c)β0 exponentially fast. For the
chains with p(n)β0 however slowly, and with
supi,jββ₯p(i,β )βp(j,β )β₯TVβ<1, we show that Louchard's
conjecture is indeed true even for the hits of an arbitrary SnββN
with Ο(Snβ)β0. 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 ksupiβp(i,Snβ), by a k-long sequence of independent copies of
(βnβ,tnβ), where βnβ=Geometric(Ο(Snβ)), tnβ is
distributed stationarily on Snβ, and βnβ is independent of tnβ. 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 Ξ½, 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 ΞΌ=o(lnΞ½)
and ΞΌ=o(Ξ½1/2) largest parts of the random cca composition and the
random C-composition, respectively. (Submitted to Annals of Probability in
August, 2009.