COMMUNICATIONS in PROBABILITY THE CONVEX MINORANT OF THE CAUCHY PROCESS

We determine the law of the convex minorant (Ms; s 2 [0; 1]) of a real-valued Cauchy process on the unit time interval, in terms of the gamma process. In particular, this enables us to deduce that the paths of M have a continuous derivative, and that the support of the Stieltjes measure dM 0 has logarithmic dimension one. For a given real-valued function f dened on some interval I R, one calls the convex minorant of f the largest convex function on I which is bounded from above by f. The case when I = [0;1 [ and f is a sample path of Brownian motion has been studied in depth by Groeneboom [11], see also Pitman [13] and C inlar [6]. In particular, it has been shown in these works that the convex minorant of Brownian motion is almost surely a piecewise linear function on the open interval]0;1[, and that the distribution of its derivative can be characterized in terms of a certain process with independent (non-stationary) increments. The more general case when f is a sample path of a Markov process (respectively, a Levy process) has been considered by Bass [1] (respectively, by Nagasawa and Tanaka [12]). In this note, we carry out a similar study for the Cauchy process on the unit time interval