Tomasz Żak

The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes

The equivalence of ergodicity and weak mixing for general infinitely divisible processes is proven. Using this result and [9], simple conditions for ergodicity of infinitely divisible processes are derived. The notion of codifference for infinitely divisible processes is investigated, it plays the crucial role in the proofs but it may be also of independent interest.

On inversions and Doob h-transforms of linear diffusions

Let X be a regular linear diffusion whose state space is an open interval \(E \subseteq \mathbb{R}\). We consider the dual diffusion X∗ whose probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. We provide a construction of X∗ as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X∗ is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the three-dimensional hyperbolic Bessel process.

Zeros of the densities of infinitely divisible measures on R n

Let μ be an infinitely divisible probability measure onRn without Gaussian component and let ν be its Lévy measure. Suppose that μ is absolutely continuous with respect to the Lebesgue measure λ. We investigate the structure of the set ℝn of admissible translates of μ. This yields a unified presentation of previously known results. We also show that ifλ(S)>0 then μ is equivalent to λ, under the assumption that supp μ=Rn, whereS is the closure of the semigroup generated by the support of ν.

Supports of semi-stable probability measures on locally convex spaces

LetE be a locally convex space. Let μ be an absolutely convexly tight Radom semi-stable probability measure onE with index 1≤α<2 and Lévy measureM. The main result of this paper shows that the closed semigroup generated by the support ofM and the negative of the barycenter ofM restricted to a suitable compact subset ofE is a (closed) linear space ofE, and that the support of μ is a suitable translate of this linear space. This result complements a few known results concerning the supports of stable and semi-stable probability measures. In particular, it extends an analogous result proved recently for the support of α-stable probability measures 1≤α<2 (Ref. 4). Related results concerning the support of Radon semi-stable probability measures onE of index 0

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let μ be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant ϰ such that |μ{‖·‖<t}−μ{‖·+r‖<t}|⩽ϰ‖r‖q, whereq depends on the properties of the norm. We specify it in the case ofLα spaces, α>1.