A Poisson process in space{time is used to generate a linear Kolmogorov's birth{growth model. Points start to form on 0; L] at time zero. Each newly formed point initiates two bidirectional moving frontiers of constant speed. New points continue to form on not-yet passed over parts of 0; L]. The whole interval will eventually be passed over by moving frontiers. Let N L be the total number of po...

Let us say that we want to analyze the total sum of a certain kind of result in a series of repeated independent random experiments each of which has a well-defined expected value and finite variance. In other words, a certain kind of result (e.g. whether the experiment is a “success”) has some probability to be produced in each experiment. We would like to repeat the experiment many times inde...

In this lecture, we describe two proofs of a central theorem of mathematics , namely the central limit theorem. One will be using cumulants, and the other using moments. Actually, our proofs won't be entirely formal, but we will explain how to make them formal. What it the central limit theorem? The theorem says that under rather general circumstances, if you sum independent random variables an...

We explore the relevance of the celebrated Central Limit Theorem (CLT) of statistics to citation averages, namely, Impact Factors (IF). The CLT predicts that, first, due to random fluctuations, the range of IF values that are statistically available to a journal of size n, follows a 1/ √ n behavior. Large journals (high-n), whose IF’s vary within a narrower range, are thus penalized in IF ranki...

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be cons...

Under left truncation, data (X(i), Y(i)) are observed only when Y(i) ≤ X(i). Usually, the distribution function F of the X(i) is the target of interest. In this paper, we study linear functionals ∫ φ dF(n) of the nonparametric maximum likelihood estimator (MLE) of F, the Lynden-Bell estimator F(n). A useful representation of ∫ φ dF(n) is derived which yields asymptotic normality under optimal m...

A central limit theorem is proved for α-mixing random fields. The sets of locations where the random field is observed become more and more dense in an increasing sequence of domains. The central limit theorem concerns these observations. The limit theorem is applied to obtain asymptotic normality of kernel type density estimators. It turns out that in our setting the covariance structure of th...

The goal of this paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss th...

For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence of probability measures in the transportation distance when the cost function is a continuous and non-decreasing function of the ...