[Paper Review] Scaling limits of general population processes - Wright-Fisher and branching processes in random environment

Vincent Bansaye, Maria-Emilia Caballero+1 more|arXiv (Cornell University)|Feb 7, 2018

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper develops a unified framework for scaling limits of discrete population processes in random environments, using convergence of characteristics to derive diffusions with jumps. It extends classical Wright-Fisher and Galton-Watson processes to random environments, proving convergence in law to SDEs with jumps via novel functional analytic techniques and Laplace exponent methods.

ABSTRACT

Our motivation comes from the large population approximation of individual based models in population dynamics and population genetics. We propose a general method to investigate scaling limits of finite dimensional population size Markov chains to diffusion with jumps. The statements of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the transition of the chain and strongly exploit the structure of the population processes defined recursively as sums of independent random variables. These results allow to reduce the convergence of characteristics of semimartingales to analytically tractable functional spaces. We develop two main applications. First, we extend the classical Wright-Fisher diffusion approximation to independent and identically distributed random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interactions and random environment to the solution of stochastic differential equations with jumps.

Motivation & Objective

• To develop a general method for deriving scaling limits of finite-dimensional Markov chains in population dynamics with random environments.
• To extend classical Wright-Fisher and Galton-Watson processes to settings with i.i.d. random environments.
• To establish convergence in law of generalized population processes to jump-diffusion SDEs using characteristic convergence.
• To overcome limitations of traditional martingale problem and tightness methods by introducing a new functional space for test functions.
• To leverage the independence structure of individual events and Laplace exponent techniques for sum of i.i.d. random variables.

Proposed method

• Proposes a new convergence criterion based on the convergence of characteristics of semimartingales associated with the Markov chains.
• Uses a functional space H, dense in regular functions vanishing at zero, with norm equivalent to ||H|| = sup_u ||H(u)/(1∧|u|²)||_∞.
• Applies Laplace exponent characterization to sums of independent, non-negative random variables in the process increments.
• Introduces a truncation and smoothing strategy for test functions to handle jump components and ensure convergence.
• Employs time scaling v_N → ∞ and renormalization Z_{[v_N t]}^N / N to derive diffusion limits.
• Relies on tightness and weak convergence arguments grounded in asymptotic behavior of triplet characteristics (drift, diffusion, jump components).

Experimental results

Research questions

• RQ1 Can the classical Wright-Fisher diffusion approximation be extended to independent and identically distributed random environments?
• RQ2 Under what conditions does a generalized Galton-Watson process in a random environment converge to a jump-diffusion SDE?
• RQ3 How can the convergence of population processes with dependent or non-Markovian structure be characterized via characteristics of their transition kernels?
• RQ4 What functional space of test functions ensures convergence of the associated martingale problem in the presence of jumps and random environments?
• RQ5 Can the branching property be relaxed while still obtaining convergence to a continuous-state process with jumps?

Key findings

• The sequence of rescaled population processes (Z_{[v_N t]}^N / N : t ≥ 0) converges in law to a diffusion with jumps as N → ∞.
• The limiting SDE is characterized by the limiting triplet of characteristics derived from (v_N, L^N), generalizing Lamperti’s framework to non-branching, non-stable processes.
• Convergence holds under mild moment and regularity conditions on the environment and individual event distributions.
• The method avoids the need for strong moment assumptions by using a tailored functional space H with norm ||H|| = sup_u ||H(u)/(1∧|u|²)||_∞.
• The approach successfully handles explosive and non-stable limiting processes, extending beyond classical CSBP limits.
• Technical lemmas establish uniform convergence of Laplace exponent differences and boundedness of jump components, ensuring tightness and identification of the limit.