Jerzy Szulga

Multiple integration with respect to Poisson and Lévy processes

SummaryNecessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form ∫...∫fdX1...dXd, where X1, ..., Xd are components of a positive or symmetric pure jump type Lévy process in ℝd. Conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity, or to be tight. All arguments proceed via reduction to the special case of Poisson integrals.

Hypercontraction principle and random multilinear forms

SummaryWe study a Banach space valued random multilinear forms in independent real random variables extensively using the concept of hypercontractive maps between Lq-spaces. We show that multilinear forms share with linear forms a lot of properties, like comparability of Lq-,L0-and almost sure convergence.

The Weierstrass–Mandelbrot Process Revisited

We derive a functional central limit theorem for quasi-Gaussian processes. In particular, we prove that the limit of the Mandelbrot–Weierstrass process is a complex fractional Brownian motion.

A Comment on “On the Rotation Matrix in Minkowski Space-Time” by Özdemir and Erdoğdu

We comment on the article by M. Ozdemir and M. Erdogdu. We indicate that the exponential map onto the Lorentz group can be obtained in two elementary ways. The first way utilizes a commutative algebra involving a conjugate of a semi-skew-symmetric matrix, and the second way is based on the classical epimorphism from SL(2,C) onto SO_0(3,1)

On Simulating Fractional Brownian Motion

We discuss how a computer simulation affects the properties of random trajectories, like stationarity or self-similarity, focusing on the Weierstrass-Mandelbrot approximation of the fractional Brownian motion.

Limit distributions of U-statistics resampled by symmetric stable laws

SummaryIf (Yi) and (Vi) are independent random sequences such thatYi are i.i.d. random variables belonging to the normal domain of attraction of a symmetric α-stable law, 0<α<2, andVi are i.i.d. random variables, then the limit distributions of U-statistics

Series expansions of multiple Lévy integrals

n^{ - 1/\alpha } \sum\limits_{1 \leqq i_t , \ldots ,i_d \leqq n} {Y_{i_1 } \ldots Y_{i_d } f(V_{i_1 } , \ldots ,V_{i_d } )}

( p, r )-convex functions on vector lattices

, coincide with the probability laws of multiple stochastic integralsXdf =∫ ...∫f (t1, ... ,td)dX(td) with respect to a symmetric α-stable processX(t).