We present covariance identities and inequalities for functionals of the Wiener Poisson processes. Using Malliavin calculus techniques, an expansion with a remainder term is obtained such functionals. Our results extend known functions multivariate random vectors.

A class of upsilon transformations Lévy measures for matrix subordinators is introduced. Some regularizing properties these are derived, such as absolute continuity and complete monotonicity. The with completely monotone densities characterized. Examples infinitely divisible nonnegative definite random matrices constructed using an transformation.

For testing the fit of a discrete distribution, use probability generating function and its empirical counterpart has been suggested in Koeherlakota Kocherlakota (1986). In present paper, particular functional corresponding process is proposed as measure to test discrepancy between evidence hypothesis. The asymptotic behavior when parameter estimated obtained, study exemplified for Poisson case only but procedure can be extended other distributions.

The Voiculescu $S$-transform is an analytic tool for computing free multiplicative convolutions of probability measures. It has been studied measures with non-negative support and having all moments zero mean. We extend the to symmetric unbounded without moments. As application, a representation stable derived as convolution semicircle measure positive measure.

The so-called Bercovici-Pata bijection maps the set of classical infinitely divisible laws to free laws. purpose this work is study corresponding Generalized Gamma Convolutions (GGC). Characterizations their cumulant transforms are derived as well integral representations with respect process. A random matrix model for GGC built consisting integrals a Nested subclasses shown converge stable class distributions.

The family of power semicircle distributions defined as normalized real powers the density is considered.The marginals uniform on spheres in high-dimensional Euclidean spaces are included this and a boundary case classical Gaussian distribution.A review some results including genesis so-called Poincaré's theorem presented.The moments these related to super Catalan numbers their Cauchy transforms terms hypergeometric functions derived.Some members class play role distribution with respect...

There is a one-to-one correspondence between classical one-dimensional infinitely divisible distributions and free distributions. In this work we study the corresponding to type G A new characterization of given first class introduced. The are studied role special symmetric beta distribution shown as building block for It proved that multiplicative convolution an arcsine with Marchenko–Pastur distribution.

Two transformations $\mathcal{A}_1$ and $\mathcal{A}_2$ of L\'{e}vy measures on $\mathbb{R}^d$ based the arcsine density are studied their relation to general Upsilon is considered. The domains definition determined it shown that they have same range. class infinitely divisible distributions with being in common range called $A$ any distribution expressed as law a stochastic integral $\int_0^1\cos(2^{-1}\uppi t)\,\mathrm{d}X_t$ respect process $\{X_t\}$. This new includes proper subclass...

In this paper we study functional asymptotic behavior of p-trace processes n × Hermitian matrix valued Brownian motions, when goes to infinity.For each p ≥ 1 establish uniform a.s. and L q laws large numbers the a.s.convergence supremum (respectively infimum) over a compact interval largest smallest) eigenvalue process.We also prove that fluctuations around limiting process, converge weakly one-dimensional centered Gaussian process Zp, given as Wiener integral with deterministic Volterra...

Almost sure and $L^k$-convergence of the traces Laguerre processes to family dilations standard free Poisson distribution are established. We also prove that fluctuations around limiting process, converge weakly a continuous centered Gaussian process. The almost convergence on compact time intervals largest smallest eigenvalues is established

We say that a random variate on Euclidean space is marginal infinitely divisible with respect to class of linear mappings if each these results in an variate. Special cases are applied multivariate extension the concept type G probability laws. Random nonnegative matrices play central role.