Consider a sequence (ηN(t):t≥0) of continuous-time, irreducible
Markov chains evolving on a fixed finite set E, indexed by a parameter N.
Denote by RN(η,ξ) the jump rates of the Markov chain ηtN, and
assume that for any pair of bonds (η,ξ), (η′,ξ′)arctan{RN(η,ξ)/RN(η′,ξ′)} converges as N↑∞. Under a
hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a
recursive procedure which provides a sequence of increasing time-scales
\theta^1_N, \dots, \theta^{\mf p}_N, θNj≪θNj+1, and of
coarsening partitions \{\ms E^j_1, \dots, \ms E^j_{\mf n_j}, \Delta^j\},
1\le j\le \mf p, of the set E. Let \phi_j: E \to \{0,1, \dots, \mf n_j\}
be the projection defined by \phi_j(\eta) = \sum_{x=1}^{\mf n_j} x \, \mb
1\{\eta \in \ms E^j_x\}. For each 1\le j\le \mf p, we prove that the hidden
Markov chain XNj(t)=ϕj(ηN(tθNj)) converges to a Markov
chain on \{1, \dots, \mf n_j\}