The Convex Minorant of the Cauchy Process

We determine the law of the convex minorant (Ms,s�[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� has logarithmic dimension one. DOI: https://doi.org/10.1214/ECP.v5-1017 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-79477 Published Version Originally published at: Bertoin, Jean (2000). The convex minorant of the Cauchy process. Electronic Communications in Probability, 5(5):51-55. DOI: https://doi.org/10.1214/ECP.v5-1017 Elect. Comm. in Probab. 5 (2000) 51–55 ELECTRONIC COMMUNICATIONS in PROBABILITY THE CONVEX MINORANT OF THE CAUCHY PROCESS JEAN BERTOIN Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 05, France, email: [email protected] submitted Nov. 11, 1999. Final version accepted Jan. 20, 2000. AMS 1991 Subject classification: 60J30 Cauchy process, convex minorant Abstract We determine the law of the convex minorant (Ms, s ∈ [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 ′ has logarithmic dimension one.We determine the law of the convex minorant (Ms, s ∈ [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 ′ has logarithmic dimension one. For a given real-valued function f defined 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,∞[ and f is a sample path of Brownian motion has been studied in depth by Groeneboom [11], see also Pitman [13] and Çinlar [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,∞[, 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 Lévy 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; we shall establish in particular a simple connection with the gamma process which yields several interesting consequences. More precisely, we shall see that the derivative of the convex minorant of the Cauchy process is continuous on ]0, 1[, specify its behavior near the boundary points 0 and 1, and determine the exact Hausdorff measure of the set of points on which it increases. Let (Cs, s ∈ [0, 1]) be a standard one-dimensional Cauchy process and (Ms, s ∈ [0, 1]) denote its convex minorant. The right-derivative (M ′ s, s ∈ [0, 1[) is an increasing process with rightcontinuous paths, and we write μx = inf {s ∈ [0, 1[: M ′ s > x} , x ∈ R