Long memory and self-similar processes
This paper is a survey of both classical and new results and ideas on long memory, scaling and self-similarity, both in the light-tailed and heavy-tailed cases.
Measure concentration for compound Poisson distributions.
Measure concentration for stable laws with index close to 2.
Metastable behaviour of small noise Lévy-Driven diffusions
We consider a dynamical system in driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature...
Metrics for multivariate stable distributions
Metrics are proposed for the distance between two multivariate stable distributions. The first set of metrics are defined in terms of the closeness of the parameter functions of one dimensional projections of the laws. Convergence in these metrics is equivalent to convergence in distribution and an explicit bound on the uniform closeness of two stable densities is given. Another metric based on the Prokhorov metric between the spectral measures is related to the first metric. Consequences for approximation,...
Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model
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)...
Multilinear forms in Pareto-like random variables and product random measures
Multiply self-decomposable measures in generalized convolution algebras
Multiply self-decomposable probability measures on Banach spaces
Non-commutative probability limit theorems
On a characterization of symmetric stable processes.
On a characterization of symmetric stable processes. (Short Communication).
On a conditional Cauchy functional equation of several variables and a characterization of multivariate stable distributions.
On a functional equation for characteristic functions
On a theorem of Kłosowska about generalised convolutions
On a Transformation for Probability Generating Functions of some Infinitely Divisible Distributions
On a Transformation of Analytic Characteristic Functions
On a two-variable zeta function for number fields
This paper studies a two-variable zeta function attached to an algebraic number field , introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When this function becomes the completed Dedekind zeta function of the field . The function is a meromorphic function of two complex variables with polar divisor , and it satisfies the functional equation . We consider the special case , where for this function...
On asymptotic behaviors and convergence rates related to weak limiting distributions of geometric random sums
Geometric random sums arise in various applied problems like physics, biology, economics, risk processes, stochastic finance, queuing theory, reliability models, regenerative models, etc. Their asymptotic behaviors with convergence rates become a big subject of interest. The main purpose of this paper is to study the asymptotic behaviors of normalized geometric random sums of independent and identically distributed random variables via Gnedenko's Transfer Theorem. Moreover, using the Zolotarev probability...
On certain classes of distributions