An extension of a convergence theorem for Markov chains arising in population genetics. (September 2016)

• Record Type: Journal Article
• Authors: Möhle, Martin; Notohara, Morihiro
• Is Part Of: Journal of applied probability. Volume 53:Number 3(2016)
• Page Start: 953
• Page End: 956
• Publication Date: 2016-09
• Subjects: Convergence, -- Markov chain, -- population genetics, -- separation of time-scales; Primary 60F05, -- 92D10, -- Secondary 60J27, -- 92D25; 519.2
• DOI: 10.1017/jpr.2016.54
• Languages: English
• ISSNs: 0021-9002

Abstract

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 ]