Convergence of martingales with a directed index set

Uniform Convergence of Reversed Martingales

A necessary and sufficient condition for the uniform convergence of a family of reversed martingales converging to a degenerated limiting process is given. The condition is expressed by means of regular convergence (in Hardy’s sense) of corresponding means. It is shown that the given regular convergence is equivalent to Hoffmann-Jørgensen’s eventually total boundedness in the mean which is nece...

Convergence Theorems for Set Valued and Fuzzy Valued Martingales and Smartingales

The purpose of this paper is to give convergence theorems both for closed convex set valued and relative fuzzy valued martingales, and suband supermartingales. These kinds of martingales, suband super-martingales are the extension of classical real valued martingales, suband super-martingales. Here we compare two kinds of convergences, in the Hausdor metric and in the Kuratowski-Mosco sense. We...

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...

Local Times and Almost Sure Convergence of Semi-Martingales

Uniform convergence rates for a class of martingales with application in non-linear co-integrating regression

For a class of martingales, this paper provides a framework on the uniform consistency with broad applicability. The main condition imposed is only related to the conditional variance of the martingale, which holds true for stationary mixing time series, stationary iterated random function, Harris recurrent Markov chain and I(1) processes with innovations being a linear process. Using the estab...