Almost Sure Convergence of the Quadratic Variation of Martingales: A Counterexample

Local Times and Almost Sure Convergence of Semi-Martingales

Examples and counterexamples to almost-sure convergence of bilateral martingales

Given a stationary process (Xp)p∈Z and an event B ∈ σ(Xp, p ∈ Z), we study the almost sure convergence as n and m go to infinity of the “bilateral” martingale E [1B |X−n, X−n+1, . . . , Xm−1, Xm ] . We show that almost sure convergence holds in some classical examples such as i.i.d. or Markov processes, as well as for the natural generator of Chacon’s transformation. However, we also prove that...

Almost Sure Convergence Rates for the Estimation of a Covariance Operator for Negatively Associated Samples

Let {Xn, n >= 1} be a strictly stationary sequence of negatively associated random variables, with common continuous and bounded distribution function F. In this paper, we consider the estimation of the two-dimensional distribution function of (X1,Xk+1) based on histogram type estimators as well as the estimation of the covariance function of the limit empirical process induced by the se...

On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formul...

Almost sure convergence of the Bartlett estimator

We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a. s. behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate nor...