Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

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...

Limiting laws of supercritical branching random walks

In this note, we make explicit the limit law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in [5] for a continuous analogue of the branching random walk. Also, in the case of a branching random walk on a homogeneous tree, we express the law of the corresponding limiting renormalized Gibbs measures, confirming, in this discrete model, conjectur...

The Almost Sure Convergence for Weighted Sums of Linear Negatively Dependent Random Variables

In this paper, we generalize a theorem of Shao [12] by assuming that is a sequence of linear negatively dependent random variables. Also, we extend some theorems of Chao [6] and Thrum [14]. It is shown by an elementary method that for linear negatively dependent identically random variables with finite -th absolute moment the weighted sums converge to zero as where and is an array of...

THE ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF NEGATIVELY DEPENDENT RANDOM VARIABLES

In this paper we study the almost universal convergence of weighted sums for sequence {x ,n } of negatively dependent (ND) uniformly bounded random variables, where a, k21 is an may of nonnegative real numbers such that 0(k ) for every ?> 0 and E|x | F | =0 , F = ?(X ,…, X ) for every n>l.