Fluctuations for mean-field interacting age-dependent Hawkes processes
نویسنده
چکیده
The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes n goes to +∞) being granted by the study performed in [9], the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale n) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead of Poisson (which occurs for the law of large numbers limit).
منابع مشابه
Mean-field inference of Hawkes point processes
We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of t...
متن کاملA Hierarchical Kinetic Theory of Birth, Death and Fission in Age-Structured Interacting Populations
We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results unde...
متن کاملStimulus sensitivity of a spiking neural network model
Some recent papers relate the criticality of complex systems to their maximal capacity of information processing. In the present paper, we consider high dimensional point processes, known as age-dependent Hawkes processes, which have been used to model spiking neural networks. Using mean-field approximation, the response of the network to a stimulus is computed and we provide a notion of stimul...
متن کاملKinetic theory of age-structured stochastic birth-death processes.
Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quant...
متن کاملFinite-size fluctuations in interacting particle systems.
Fluctuations may govern the fate of an interacting particle system even on the mean-field level. This is demonstrated via a three species cyclic trapping reaction with a large, yet finite number of particles, where the final number of particles N(f) scales logarithmically with the system size N, N(f) approximately ln N. Statistical fluctuations, that become significant as the number of particle...
متن کامل