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Contents of PMS, Vol. 11, Fasc. 2,
pages 193 - 222
 

MÉTHODES DE CHANGEMENT DE TEMPS POUR LA CONVERGENCE EN LOI DES MARTINGALES

A. Touati

Abstract: Let M = (M (t);t (- T ) be a centred, square integrable martingale, indexed by T = N or T = R+, whose predictible quadratic variation is denoted by (<M >(t);t (- T ). The main problem we investigate is the study of the joint convergence in law, when c --> oo , of the processes

( ) 1 1 V~ ---M o t(ct), v(c)<M >o t(ct);t > 0 , v(c)
where v and t are two increasing functions. To solve this problem we use three technical tools (each of them having its one interest):
  1. a limit theorem for composed processes;
  2. a limit theorem for random change of time;
  3. a method of enlarging the probability space on which M is defined.

This approach looks to be efficient as far as the asymptotic behaviour of functionals of recurrent Markov or semi-Markov processes is concerned. Several examples illustrate the developed theory.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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