The maximal variation of a bounded martingale and the central limit theorem

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.