Wim Vervaat

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

The present paper considers the stochastic difference equation Y n = A n Y n -1 + B n with i.i.d. random pairs ( A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and ( A, B ) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n -1 B n . A second subject is the series ∑ C n f ( T n ) with ( C n ) a sequence of i.i.d. random variables, ( T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Levy–Hincin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f ( x ) = e − x /α .

Limit theorems for records from discrete distributions

Weak and strong functional limit theorems are obtained for record values and record epochs in a sequence of independent random variables with common distribution F. The emphasis is on the case in which F is concentrated on the non-negative integers. For contrast, the well-known case of continuous F is also considered. Analogues of results obtained earlier by Resnick, de Haan and the author for continuous F are presented here for F concentrated on the non-negative integers. Also is investigated under which circumstances the latter case is so close to the continuous F case that the resulting limit theorems are the same.

Capacities, Large Deviations and Loglog Laws

Spaces of capacities are considered with their natural subspaces and two topologies, the vague and the narrow. Large deviation principles are identified as a class of limit relations of capacities. Narrow large deviation principles occasionally can be tied to loglog laws, and this relationship is studied. Specific narrow large deviation principles and loglog laws are presented (without proof) for the Poisson process on the positive quadrant that is the natural foundation for extremal processes and spectrally positive stable motions. Related loglog laws for extremal processes and stable motions are discussed.

Stationary self-similar extremal processes

SummaryLet (ξk)k∞=−∞ be a stationary sequence of random variables, and, forA⊂ℝ, let

Log-fractional stable processes

M_n (A): = \mathop V\limits_{k/n \in A} \gamma _n (\xi _k )

Ignatov's theorem: a new and short proof

where γn is an affine transformation of ℝ (has the forman·+bn,an>0,bn∈ℝ). ThenMn is a random sup measure, that is,

Functional central limit theorems for processes with positive drift and their inverses

M_n (\mathop U\limits_\alpha G_\alpha ) = \mathop V\limits_\alpha M_n (G_\alpha )