# Weak convergence of random processes with immigration at random times

## TL;DR

This paper investigates the weak convergence of a broad class of shot noise processes with randomly timed responses, extending previous models and establishing conditions for convergence to Gaussian processes.

## Contribution

It introduces generalized conditions for weak convergence of shot noise processes with random response functions and times, broadening the scope of previous models.

## Key findings

- Established sufficient conditions for weak convergence to Gaussian processes.
- Extended previous models to more general random times and response functions.
- Provided specialized results for particular instances of random times and responses.

## Abstract

By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. The so defined random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in [Iksanov et al. (2017). Bernoulli, 23, 1233--1278] and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.