We consider strictly stationary infinitely divisible processes and first extend the mixing conditions given in Maruyama [Theory Probab. Appl. 15 (1970) 1–22] and Rosiński and Żak [Stoc. Proc. Appl. 61 (1996) 277–288] from the univariate to the d-dimensional case. Thereafter, we show that multivariate Lévy-driven mixed moving average processes satisfy these conditions and hence a wide range of well-known processes such as superpositions of Ornstein − Uhlenbeck (supOU) processes or (fractionally integrated)...

We study the problem of finding the smallest $m$ such that every element of an exponential family can be written as a mixture of $m$ elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that $m=q^{N-1}$ is the smallest number for which any distribution of $N$$q...$

The introduced three parameter (position μ, scale ∑ and shape γ) multivariate generalized Normal distribution (γ-GND) is based on a strong theoretical background and emerged from Logarithmic Sobolev Inequalities. It includes a number of well known distributions such as the multivariate Uniform, Normal, Laplace and the degenerated Dirac distributions. In this paper, the cumulative distribution, the truncated distribution and the hazard rate of the γ-GND are presented. In addition, the Maximum Likelihood...

We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type inequality. As a consequence we obtain dimension-free two-level concentration results for convex functions of independent random variables with sufficiently regular tail decay. We also provide a link between modified log-Sobolev inequalities...

Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S=\sum a_{i₁,...,i_d}{X}_{i₁}^{\left(1\right)}\cdots X_{i_d}^{\left(d\right)}$ generated by symmetric random variables $X_{i₁}^{\left(1\right)},...,X_{i_d}^{\left(d\right)}$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order d. Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of S and moments of some...

This paper gives upper and lower bounds for moments of sums of independent random variables $\left(X_k\right)$ which satisfy the condition ${P\left(\right|X|}_k\ge t)=exp(-N_k\left(t\right))$, where $N_k$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N\left(t\right)={\left|t\right|}^r$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.

Consider the sequence ${\left(Cₙ\right)}_{n\ge 1}$ of positive numbers defined by C₁ = 1 and $C_{n+1}=1+Cₙ²/4$, n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

We calculate the moments $m_{k,0}$ of the measure orthogonalizing the 2-dimensional Chebyshev polynomials introduced by Koornwinder.

Let X=(X₁,..., Xₙ) be a sample from a distribution with density f(x;θ), θ ∈ Θ ⊂ ℝ. In this article the Bayesian estimation of the parameter θ is considered. We examine whether the Bayes estimators of θ are pointwise ordered when the prior distributions are partially ordered. Various cases of loss function are studied. A lower bound for the survival function of the normal distribution is obtained.