arXiv subjects

Jie Yen Fan

Publications and source records attributed to Jie Yen Fan.

4 recordsLinked to original sources

arXiv2025

Estimation of rates in population-age-dependent processes by means of test functions

This paper aims to develop practical applications of the model for the highly technical measure-valued populations developed by the authors in \cite{FanEtal20}. We consider the problem of estimation of parameters in the general age and population-dependent model, in which the individual birth and death rates depend not only on the age of the individual but also on the whole population composition. We derive new estimators of the rates based on the use of test functions in the functional Law of Large Numbers and Central Limit Theorem for populations with a large carrying capacity. We consider the rates to be simple functions, that take finitely many values both in age $x$ and measure $A$, which leads to systems of linear equations. The proposed method of using test functions for estimation is a radically new approach which can be applied to a wide range of models of dynamical systems.

math.ST↗

arXiv2019

Limit theorems for multi-type general branching processes with population dependence

A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population density as a measure-valued process and obtain its asymptotics, as the population grows with the environmental carrying capacity. "Density" in this paper generally refers to the population size as compared to the carrying capacity. Thus, a deterministic approximation is given, in the form of a Law of Large Numbers, as well as a Central Limit Theorem. Migration can also be incorporated. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

math.PR↗

arXiv2019

Convergence of the age structure of general schemes of population processes

We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter $K$, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper the Law of Large Numbers as $K\to\infty$ was derived. Here we prove the Central Limit Theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.

math.PR↗

arXiv2015

Mimicking self-similar processes

We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable Lévy processes as well as an example of an artificial process having the marginals of $t^κV$ for some symmetric random variable $V$. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.

math.ST↗