Limit behavior of the convolution iterates of a probability measure on a semigroup of matrices

Halfplane Trimming for Bivariate Distributions

We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site ('local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to infinity. Two limit procedures are considered, according to the order in which M and N go to infinity; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x(*), the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts.

Limit Behaviour of Convolution Products of Probability Measures

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an α-stable law, α ∈ (0, 2). When 1 ≤ α < 2, we show that for a suitable regularly varying sequence κn of index 1 − 1/α, the limiting spectral distribution μα of κnK coincides with the one of the random symmetric matrix of the un-normalized weights (Lévy matrix with i.i.d. entries). In contrast, when 0 < α < 1, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law μ̃α supported on [−1, 1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of μα and μ̃α. We also study the limiting behavior of the invariant probability measure of K.

Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/resampling schemes. We develop a stability analysis by funtional inequalities for a nonlinear flow of probability measures describing the limit behavior of the methods as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures.