The Maximal Variation of a Bounded Martingale and the Central Limit Theorem

Bernard De Meyer

No 1996035, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: Mertens and Zamir's (1977) paper is concerned with the asymptotic behaviour of the maximal L[exp.1]-variation [xi1.n(p)] of a [0,1]-valued martingale of length n starting at p. They prove the convergence of [ [xi1.n(p)] / [square root.n]]. to the normal density evaluated at its p-quantile. This paper generalises this result to the conditional L[exp.q]-variation for q [belong] [1,2). The appearance of the normal density remained unexplained in Mertens and Zamir's proof: it appeared there as the solution of a differential equation. Our proof however justifies this normal density as a consequence of a generalisation of the CLT discussed in the second part of this paper.

Date: 1996-08-01
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