Convergence Of Empirical Measures Research Articles

Empirical approximation to invariant measures for McKean–Vlasov processes: Mean-field interaction vs self-interaction

This paper proves that, under a monotonicity condition, the invariant probability measure of a McKean–Vlasov process can be approximated by weighted empirical measures of some processes including itself. These processes are described by distribution dependent or empirical measure dependent stochastic differential equations constructed from the equation for the McKean–Vlasov process. Convergence of empirical measures is characterized by upper bound estimates for their Wasserstein distances to the invariant measure. Numerical simulations of the mean-field Ornstein–Uhlenbeck process are implemented to demonstrate the theoretical results.

Rate of convergence of empirical measures for exchangeable sequences

Abstract Let S be a finite set, (Xn ) an exchangeable sequence of S-valued random variables, and μn = (1/n) ∑ i = 1 n $\sum\limits_{i=1}^{n} $ δ Xi the empirical measure. Then, μn (B) ⟶ a . s . $\overset{a.s.}{\longrightarrow} $ μ(B) for all B ⊂ S and some (essentially unique) random probability measure μ. Denote by 𝓛(Z) the probability distribution of any random variable Z. Under some assumptions on 𝓛(μ), it is shown that a n ≤ ρ [ L ( μ n ) , L ( μ ) ] ≤ b n and ρ [ L ( μ n ) , L ( a n ) ] ≤ c n u $$\begin{array}{} \displaystyle \frac{a}{n}\le\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(\mu)\bigr]\le\frac{b}{n}\qquad\text{and}\qquad\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(a_n)\bigr]\le\frac{c}{n^u} \end{array} $$ where ρ is the bounded Lipschitz metric and an (⋅) = P(X n+1 ∈ ⋅ | X 1, …, Xn ) is the predictive measure. The constants a, b, c > 0 and u ∈ ( 1 2 $\frac{1}{2} $ , 1] depend on 𝓛(μ) and card (S) only.

Extremes of independent stochastic processes: a point process approach

For each n ≥ 1, let \(\{ X_{in}, \quad i \geqslant 1 \}\) be independent copies of a nonnegative continuous stochastic process Xn = (Xn(s))s∈S indexed by a compact metric space S. We are interested in the process of partial maxima \(\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0, s\in S,\) where the brackets [ ⋅ ] denote the integer part. Under a regular variation condition on the sequence of processes Xn, we prove that the partial maxima process \(\tilde M_{n}\) weakly converges to a superextremal process \(\tilde M\) as \(n\to \infty \). We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

Information Theory in the mathematical Statistics

In the paper we present an outline of the information theory from the probabilistic and statistical point of view. Such a direction of the information theory has been intensively developed in recent decades and significantly influenced a progress in the statistical methodology. The aim of the article is to introduce the reader into these problems, provide some intuitions and acquaint with a specific information-theoretic approach to the mathematical statistics. The first part of the paper is devoted to brief and easy of approach introduction to the main notions of the information theory like entropy, relative entropy (Kullback-Leibler distance), information projection and Fisher information as well as presentation of their most important properties including de Bruijn's identity, Fisher information inequalities and entropy power inequalities. In the short second part we give applications of the notions and results from the first part to limit theorems of the probability theory such as the asymptotic equipartition property, the convergence of empirical measures in the entropy distance, large deviation principle with emphasis to Sanov theorem, the convergence of distributions of homogeneous Markov chains in the entropy distance and the central limit theorem. The main, last part of the article shows some most significant and important applications of the information theory to the mathematical statistics. We discuss connections of the maximum likelihood estimators with the information projections and the notion of sufficient statistic from the information-theoretic point of view. The problems of source coding, channel capacity and an amount of information provided by statistical experiments are presented in a statistical framework. Some attention is paid to the expansion of Clarke and Barron and its corollaries e.g. in density estimation. Next, applications of the information theory to hypothesis testing is discussed. We give the classical Stein's Lemma and its generalization to testing composite hypothesis obtained by Bahadur and show their connections with the asymptotic efficiency of statistical tests. Finally, we briefly mention the problem of information criteria in a model seletion including the most popular two-stage minimal description length criterion of Rissanen. The enclosed literature is limited only to papers and books which are referred to in the paper.

Nonparametric maximum likelihood estimation in a non locally compact setting

Maximum likelihood estimation is considered in the context of infinite dimensional parameter spaces. It is shown that in some locally convex parameter spaces sequential compactness of the bounded sets ensures the existence of minimizers of objective functions and the consistency of maximum likelihood estimators in an appropriate topology. The theory is applied to revisit some classical problems of nonparametric maximum likelihood estimation, to study location parameters in Banach spaces, and finally to obtain Varadarajan’s theorem on the convergence of empirical measures in the form of a consistency result for a sequence of maximum likelihood estimators. Several parameter spaces sharing the crucial compactness property are identified.

Entropy and maximal spacings for random partitions

Asymptotic properties of the partitions of the unit interval induced by some special random sequences of points are studied from two points of view. The partition-entropy and largest spacings for independent identically distributed (i.i.d.) uniform sequences in [0,1] are first compared with the corresponding quantities for a random sequence generated by a scheme of S. Kakutani. A general combinatorial technique exploiting the tree structure of interval splittings is then introduced to prove almost-sure (a.s.) equidistribution of the points of Kakutani's construction, along with a rate of convergence of empirical measures to Lebesgue measure and such other results as a.s. equidistribution of k-tuples in [0,1]k