Abstract
We present some recent developments for limit theorems in probability theory, illustrating the variety of this field of activity. The recent results we discuss range from Stein’s method, as well as for infinitely divisible distributions as applications of this method in stochastic geometry, to asymptotics for some discrete models. They deal with rates of convergence, functional convergences for correlated random walks and shape theorems for growth models.
Highlights
Limit theorems form a branch of probability theory whose activity still has a great influence on other probabilistic fields since most probabilistic analyses are usually completed by a limit theorem
Another kind of limit theorems deals with functional counterparts of (1) where stochastic processes are considered instead of random variables. The most famous such limit theorem is Donsker’s invariance principle: this functional counterpart of the central limit theorem (CLT) states that the piecewise constant process (S nt )t∈[0,1] associated to the partial sum Sn of iid square integrable random variables (Xi)i≥1, say centered with unit variance, satisfies
A measurable function R : X × N → R+ is a radius of stabilization for the score function ξ if the contribution ξ(x, ηs) of a point x ∈ X to the random variable F essentially depends on the points of ηs falling inside Bd x, R(x, ηs) : for all ζ ⊂ X with #ζ ≤ 7, ξ x, ηs ∪ ζ = ξ x, ∩ Bd(x, R(x, ηs)), x ∈ X, s ≥ 1
Summary
Limit theorems form a branch of probability theory whose activity still has a great influence on other probabilistic fields since most probabilistic analyses are usually completed by a limit theorem. They have numerous applications starting from statistical properties of related estimators to stochastic approximation in various models The aim of this survey article is to highlight some various recent developments in some selected directions: sharp rates of convergence in limit theorems with Stein’s method, functional limit theorems under dependence, and shape theorems for discrete growth models. Another kind of limit theorems deals with functional counterparts of (1) where stochastic processes are considered instead of random variables. The most famous such limit theorem is Donsker’s invariance principle: this functional counterpart of the CLT states that the piecewise constant process (S nt )t∈[0,1] associated to the partial sum Sn of iid square integrable random variables (Xi)i≥1, say centered with unit variance, satisfies.
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