By Raphael Cerf

The objective of this paintings is to suggest a finite inhabitants counterpart to Eigen's version, which includes stochastic results. the writer considers a Moran version describing the evolution of a inhabitants of dimension m of chromosomes of size over an alphabet of cardinality ?. The mutation likelihood in keeping with locus is q. He offers in simple terms with the pointy height panorama: the replication cost is s>1 for the grasp series and 1 for the opposite sequences. He experiences the equilibrium distribution of the method within the regime the place ? 8,m? 8,q?0, q?a?]0, 8[,m?a?[0, eight]

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M We deﬁne further a vector function H : A → { 0, . . , ⎛ ⎛ ⎞ x(1) H m ⎜ .. ⎟ ⎜ ∀x = ⎝ . ⎠ ∈ A H(x) = ⎝ x(m) The partition of A m }m by setting ⎞ x(1) ⎟ .. ⎠. H x(m) induced by the map H is H−1 ({ d }) , d ∈ { 0, . . , }m . We deﬁne ﬁnally the distance process (Dt )t≥0 by ∀t ≥ 0 Dt = H Xt . Our next goal is to prove that the process (Xt )t≥0 is lumpable with respect to the m induced by the map H, so that the distance process (Dt )t≥0 is partition of A a genuine Markov process. 1. 2 (H Lumpability).

M } and c ∈ { 0, . . , } such that e = d(j ← c) and d(j) = c. Naturally, d(j ← c) is the vector d in which the j–th component d(j) has been replaced by c: ⎞ d(1) ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎜d(j − 1)⎟ ⎟ ⎜ ⎟ c d(j ← c) = ⎜ ⎟ ⎜ ⎜d(j + 1)⎟ ⎟ ⎜ ⎟ ⎜ ... ⎠ ⎝ ⎛ d(m) We have then p x, x(j ← w) . 1, we have p x, x(j ← w) = w∈A H(w)=c w∈A H(w)=c = 1 m 1 m 1≤i≤m A(x(i))M (x(i), w) A(x(1)) + · · · + A(x(m)) AH H(x(i)) MH (H(x(i)), c) . A H(x(1)) + · · · + AH H(x(m)) 1≤i≤m H ¨ CERF RAPHAEL 28 This sum is a function of H(x) and c only.

For i ∈ { 0, . . , m − 1 }, θ θ P Zt+1 = i + 1 | Ztθ = i = P Ot+1 (0) = i + 1 | Otθ (0) = i = σi(m − i)MH (0, 0) + (m − i)2 MH (θ, 0) . 3. A renewal argument We prove here a formula for the invariant measure involving two stopping times. The proof is based on a standard renewal argument and the formula is a variant of other well known formulas. Because this formula is a key of the whole analysis, we include its proof here. Let (Xt )t≥0 be a discrete time Markov chain with values in a ﬁnite state space E which is irreducible and aperiodic.