The Convex Minorant of the Cauchy Process

We determine the law of the convex minorant $(M_s, s\in [0,1])$ 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 Ç 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, ∞[, 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 (C s , s ∈ [0, 1]) be a standard one-dimensional Cauchy process and (M s , 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 for its right-continuous inverse. Recall that the standard gamma process, (γ s , s ∈ [0, 1]) is a subordinator with marginal distributions Our analysis relies on the following.

Lemma 1
The process (µ x , x ∈ R) has the same law as On the one hand, µ x is the (a.s. unique) instant in [0, 1] at which the Cauchy process with drift C (x) reaches its overall infimum on [0, 1], so On the other hand, it is immediately seen that L (x) , x ∈ R is a nested family of random sets, in the sense that L (x) ⊆ L (x ) for x ≤ x . Moreover, the strong Markov property of the Cauchy process easily entails the following regenerative property. Consider an arbitrary finite sequence x 1 ≤ x 2 ≤ . . . ≤ x n of increasing real numbers, and T a stopping time in the filtration (F s ) s≥0 such that T ∈ L (x1) a.s. Then the shifted ladder time sets are jointly independent of F T and have the same (joint) distribution as L (x1) , . . . , L (xn) . Note also that the probability P C (xi) s ≤ 0 is the same for all s > 0, and if we denote this quantity by ρ(x i ), then Hence each regenerative set L (xi) is stable with index ρ(x i ), i.e. it can be identified with the closed range of some stable subordinator with index ρ(x i ), see Lemma VIII.1 in [2]. We have checked that the general framework of [3] applies to the present setting, and Proposition 9 there entails that the law of the n-tuple , that is the same as that of It follows that the two processes in the statement have the same finite-dimensional distributions, and as both are increasing and right-continuous, they have the same law.

Remark.
Two different nested families of stable regenerative sets have been considered in [5].
In this direction, we point out that the argument in section 4 there can be adapted to give an alternative proof of Lemma 1.
Our main result derives immediately from Lemma 1 and the fact that the gamma process has strictly increasing paths with probability one.

Theorem 2
The process (M s , 0 < s < 1) is continuous and has the same law as where γ is a standard gamma process and denotes its inverse process.
Proof: It follows from Lemma 1 that the process (µ x , x ∈ R) has strictly increasing paths with probability one; as a consequence, (M s , 0 < s < 1) can be recovered from (µ x , x ∈ R) by the identity This entails the continuity of M and the stated identity in law (again by Lemma 1).
As a check, let us calculate the distribution of the variable C 1 , which coincides with M 1 = 1 0 M s ds. It follows from Theorem 2 that the latter has the same law as Next, recall that γ 1 has an exponential distribution with parameter 1 and is independent of the Dirichlet process (γ s /γ 1 , s ∈ [0, 1]), and a fortiori ofM 1 . It follows that for every λ ∈ R, the Stieltjes transform (i.e. the iterated Laplace transform) of M 1 is given by Using the identity E(exp {iλγ s }) = (1 − iλ) −s , we see that the right-hand side equals When λ > 0 (respectively, λ < 0), the function z → (1 + z 2 ) −1 log(1 + iλz) is meromorphic on the lower (respectively, upper) complex half-plane with a single pole at z = −i (respectively, at z = i). Using a contour integral, one gets Putting the pieces together, we conclude that E (1/(1 + iλM 1 )) = 1/(1 + |λ|) , and hence that M 1 follows the standard Cauchy distribution. The same method applies for instance to calculate the distribution of the length of the graph of the convex minorant. One gets after a few lines of elementary calculation Next, we deduce from known properties about the regularity of L the following results on the rate of growth of M near the boundary points 0 and 1. (the latter can be seen for instance from Theorem 1 in [4] and the fact that the Laplace exponent of the gamma process is log(1 + ·)). The proof of the lim sup result is similar, using Theorem 2 on page 321 in [10]. In particular, Supp(dM ) is a random closed set with logarithmic Hausdorff dimension 1. Proof: It follows from Theorem 2 that Supp(dM ) = {µ x , x ∈ R} cl is distributed as the normalized closed range of a gamma process on the unit time interval. It is a consequence of a general result by Fristedt and Pruitt [9] that there is some constant c > 0 such that for every t > 0 the h-Hausdorff measure of {γ s , 0 ≤ s ≤ t} cl is ct. This entails our claim.
Remark. Theorem 2 and Corollaries 3 and 4 bear the same flavor as the results of Cranston et al. [7] and Evans [8] on the convex hull of planar Brownian path on the unit time interval (recall e.g. that the convex hull is a C 1 curve and that the set of times in [0, 1] at which the Brownian path touches the convex hull has Hausdorff dimension zero). These similarities are certainly not surprising, considering the close connections between the planar Brownian motion and the linear Cauchy process.