48 research outputs found
Convergence of U-statistics for interacting particle systems
The convergence of U-statistics has been intensively studied for estimators
based on families of i.i.d. random variables and variants of them. In most
cases, the independence assumption is crucial [Lee90, de99]. When dealing with
Feynman-Kac and other interacting particle systems of Monte Carlo type, one
faces a new type of problem. Namely, in a sample of N particles obtained
through the corresponding algorithms, the distributions of the particles are
correlated -although any finite number of them is asymptotically independent
with respect to the total number N of particles. In the present article,
exploiting the fine asymptotics of particle systems, we prove convergence
theorems for U-statistics in this framework
Particle systems with a singular mean-field self-excitation. Application to neuronal networks
We discuss the construction and approximation of solutions to a nonlinear
McKean-Vlasov equation driven by a singular self-excitatory interaction of the
mean-field type. Such an equation is intended to describe an infinite
population of neurons which interact with one another. Each time a proportion
of neurons 'spike', the whole network instantaneously receives an excitatory
kick. The instantaneous nature of the excitation makes the system singular and
prevents the application of standard results from the literature. Making use of
the Skorohod M1 topology, we prove that, for the right notion of a 'physical'
solution, the nonlinear equation can be approximated either by a finite
particle system or by a delayed equation. As a by-product, we obtain the
existence of 'synchronized' solutions, for which a macroscopic proportion of
neurons may spike at the same time
First hitting times for general non-homogeneous 1d diffusion processes: density estimates in small time
Motivated by some applications in neurosciences, we here collect several estimates for the density of the first hitting time of a threshold by a non-homogeneous one-dimensional diffusion process and for the density of the associated process stopped at the threshold. We first remind the reader of the connection between both. We then provide some Gaussian type bounds for the density of the stopped process. We also discuss the stability of the density with respect to the drift. Proofs mainly rely on the parametrix expansion