Abstract
We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.
Highlights
This work is a contribution to the study of scaling limits of discrete population models.The parameter N ∈ N scales the population sizes
Galton-Watson processes correspond to the case when FN (z) = z and LNi,n = LN, i.e it does not depend on (z, e), while Wright-Fisher processes are obtained when FN (z) = N and LNi,n(z, e) are Bernoulli random variables with parameter z/N
Our results extend the criterion for the convergence of a sequence of Galton-Watson processes as well as the results we know in random environment [21, 4] or with interactions [11, 32]
Summary
This work is a contribution to the study of scaling limits of discrete population models. Galton-Watson processes correspond to the case when FN (z) = z and LNi,n = LN , i.e it does not depend on (z, e), while Wright-Fisher processes are obtained when FN (z) = N and LNi,n(z, e) are Bernoulli random variables with parameter z/N These population models can take into account the effect of random environment and include many additional ecological forces such as competition, cooperation and sexual reproduction. Applying for instance this method to the classical Galton-Watson framework seems to lead to moment assumptions It is well known from the works of Lamperti [24, 25] and Grimvall [16] that the finite dimensional convergence of the renormalized processes (Z[NvN t]/N : t ≥ 0) with a time scale vN → ∞ is equivalent to the convergence of a characteristic triplet associated with (vN , LN ) when N tends to infinity. For any U subset of Rd containing a neighborhood of 0, we define U ∗ as U \ {0}
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