Abstract
In this paper, it is shown that the iterated Lévy transforms (β n) of a standard Brownian motion β, so defined:
$$\beta ^0 = \beta ,and:\beta _t^{n + 1} = \int\limits_0^t {\operatorname{sgn} (\beta _s^n )} d\beta _s^n (n \geqq 0)$$
satisfy the following property: a.s.,β n andβ m have common zeros, as soon asm>n+1. This property bears some relation with the conjectured ergodicity of the Lévy transform.
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Référence
Dubins, L., Smorodinsky, M.: The Modified, discrete, Lévy transformation is Bernoulli. In: Azéma, J., Meyer, P.A. (eds.) Sém. Probas.XXVI. (Lect. Notes Math. vol., 1526 pp. 157–161) Berlin Heidelberg New York: Springer 1992