An extension of a convergence theorem for Markov chains arising in population genetics. (September 2016) - Möhle, Martin; Notohara, Morihiro
Abstract: 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 Π N = I + d N B N, where I is the unit matrix, Q a generator matrix, ( B N ) N a sequence of matrices, lim N ℩∞ c N = lim N →∞ d N =0 and lim N →∞ c N ∕ d N =0. Suppose that the limits P ≔lim m →∞ ( I + d N Q ) m and G ≔lim N →∞ P B N P exist. If the sequence of initial distributions P X N (0) converges weakly to some probability measure μ, then the finite-dimensional distributions of ( X N ([ t ∕ c N )) t ≥0 converge to those of the Markov process ( X t ) t ≥0 with initial distribution μ, transition matrix P e t G and lim N →∞ ( I + d N Q + c N B N ) [ t ∕ c N ]