Balram S. Rajput

Spectral representations of infinitely divisible processes

SummaryThe spectral representations for arbitrary discrete parameter infinitely divisible processes as well as for (centered) continuous parameter infinitely divisible processes, which are separable in probability, are obtained. The main tools used for the proofs are (i) a “polar-factorization” of an arbitrary Lévy measure on a separable Hilbert space, and (ii) the Wiener-type stochastic integrals of non-random functions relative to arbitrary “infinitely divisible noise”.

Calculation of multidimensional stable densities

Stable random variables are used in economics, engineering, hydrology and physics to model situations where the underlying distributions are heavy tailed. Stable densities do not generally have explicit formula, even in one variable. This paper describes the steps necessary to calculate multivariate stable densities by numerically inverting the characteristic function. We give a program tocalculate two dimensional stable densites that uses a recent two dimensional adaptive quadratureroutine. Graphs of families of such densities are given for a range of values of a and various spectral measures

Spectral representation of semistable processes, and semistable laws on banach spaces

We introduce the notion of semistable processes and semistable random measures; and give a characterization of semistable laws on Banach spaces. Using this charcterization, we discuss the existence of semistable random measures, define the stochastic integrals with respect to these measures, and obtain the spectral representations of arbitrary (not necessairly symmetric) semistable and stable processes. In addition, we give a criterion of independence for stochastic integrals.

Semistable Laws on Topological Vector Spaces.

SummaryIn this paper we discuss three types of results: Firstly, we present two Lévy-Hinčin type representations of Poisson type infinitely divisible (i.d.) laws on certain topological vector (TV) spaces; one of these complements a known representation due to Dettweiler. Secondly, we define and characterize r-semistable laws on locally convex TV spaces and also obtain good representation of their characteristic functions. Finally, we discuss the existence and the continuity of the semigroup {μt∶t>0} of i.d. laws μ on locally convex TV spaces. These complement similar known results of Siebert.

On some extremal problems in H p and the prediction ofL p -harmonizable stochastic processes

SummaryWe providesimple andsuccinct solutions to two dual extremal problems in the Hardy spacesHp, and to an aspect of the linear prediction problem for a certain class of discrete and continuous parameter “Lp-harmonizable” stochastic processes, for all 1≦p<∞. Two of the results presented appear new. The methods of proof of the rest of the results provide alternatesimpler andshorter proofs for some earlier known theorems.

A representation of the characteristic function of a stable probability measure on certain TV spaces

In this paper, a Levy-Khintchine type representation of the characteristic function of a K-regular stable probability measure on real locally convex topological vector spaces, satisfying certain conditions, is presented.

On the spectral representations of complex semistable and other infinitely divisible stochastic processes

In this paper we extend the definition of stochastic integrals relative to the larger class of complex semistable and other infinitely divisible measures that was given in a previous paper. We obtain spectral representations of complex semistable and other i.d. stochastic processes.

Zeros of the densities of infinitely divisible measures on R n

Let μ be an infinitely divisible probability measure onRn without Gaussian component and let ν be its Lévy measure. Suppose that μ is absolutely continuous with respect to the Lebesgue measure λ. We investigate the structure of the set ℝn of admissible translates of μ. This yields a unified presentation of previously known results. We also show that ifλ(S)>0 then μ is equivalent to λ, under the assumption that supp μ=Rn, whereS is the closure of the semigroup generated by the support of ν.

Supports of semi-stable probability measures on locally convex spaces

LetE be a locally convex space. Let μ be an absolutely convexly tight Radom semi-stable probability measure onE with index 1≤α<2 and Lévy measureM. The main result of this paper shows that the closed semigroup generated by the support ofM and the negative of the barycenter ofM restricted to a suitable compact subset ofE is a (closed) linear space ofE, and that the support of μ is a suitable translate of this linear space. This result complements a few known results concerning the supports of stable and semi-stable probability measures. In particular, it extends an analogous result proved recently for the support of α-stable probability measures 1≤α<2 (Ref. 4). Related results concerning the support of Radon semi-stable probability measures onE of index 0

Tail Behavior for Semigroups of Measures on Groups

Let (μt)t>0 be a convolution semigroup of probability measures on a measurable group (G, ℬ). In this paper, we provide precise information about the asymptotic behavior of μt{q>s, whereq is a measurable seminorm and (μt)t>0 isq-continuous.