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

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
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