The maximal variation of a bounded martingale and the central limit theorem
(1998) l’Institut Henri Poincare. Annales (B). Probabilites et Statistiques — Vol. 34, n° 1, p. 49-59 (1998)
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- Mertens and Zamir's paper [3] is concerned with the asymptotic behavior of the maximal L-1-variation xi(n)(1)(p) of a [0,1]-valued martingale of length n starting at p. They prove the convergence of xi(n)(1)(p)/root n to the normal density evaluated at its p-quantile. This paper generalizes this result to the conditional L-q-variation for q is an element of [1,2). The appearance of the normal density remained unexplained in Mertens and Zamir's proof: it appeared as the solution of a differential equation. Our proof however justifies this normal density as a consequence of a generalization of the central limit theorem discussed in the second part of this paper. (C) Elsevier, Paris.
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De Meyer, B. (1998). The maximal variation of a bounded martingale and the central limit theorem. l’Institut Henri Poincare. Annales (B). Probabilites et Statistiques, 34(1), 49-59. https://doi.org/10.1016/S0246-0203(98)80017-4 (Original work published 1998)