An extension of a convergence theorem for Markov chains arising in population genetics. (English) Zbl 1351.60025

Möhle, Martin; Notohara, Morihiro

J. Appl. Probab. 53, No. 3, 953-956 (2016).

Summary: An extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer \(N\) let \((X_N(r))_r\) be a Markov chain with the same finite state space \(S\) and transition matrix \(\Pi_N=I+d_NQ+c_NB_N\), where \(I\) is the unit matrix, \(Q\) a generator matrix, \((B_N)_N\) a sequence of matrices, \(\lim_{N\to\infty}c_N=\lim_{N\to\infty}d_N=0\) and \(\lim_{N\to\infty}c_N/d_N=0\). Suppose that the limits \(P:=\lim_{m\to\infty}(I+d_NQ)^m\) and \(G:=\lim_{N\to\infty}PB_NP\) exist. If the sequence of initial distributions \(P_{X_N(0)}\) converges weakly to some probability measure \(\mu\), then the finite-dimensional distributions of \((X_N([t/c_N]))_{t\geq0}\) converge to those of the Markov process \((X_t)_{t\geq0}\) with initial distribution \(\mu\), transition matrix \(P\mathrm{e}^{tG}\) and \(\lim_{N\to\infty}(I+d_NQ+c_NB_N)^{[t/c_N]}=P-I+e^{tG}=P\mathrm{e}^{tG}=\mathrm{e}^{tG}P\) for all \(t>0\).

MSC:

60F05	Central limit and other weak theorems
92D10	Genetics and epigenetics
60J27	Continuous-time Markov processes on discrete state spaces
92D25	Population dynamics (general)
17D92	Genetic algebras

Keywords:

convergence; Markov chain; population genetics; separation of time-scales