# Convergence of the age structure of general schemes of population processes

## TL;DR

This paper establishes a Central Limit Theorem for a family of age-structured population processes, showing that fluctuations around the law of large numbers limit are governed by a stochastic PDE, advancing understanding of population dynamics models.

## Contribution

It proves the CLT for age-structured population processes with parameters depending on age and population structure, extending previous LLN results to include fluctuations.

## Key findings

- Fluctuation processes converge weakly in Skorokhod space.
- Limit is characterized by a stochastic partial differential equation.
- Provides a rigorous probabilistic framework for population process fluctuations.

## Abstract

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.