Decomposition of infinitely divisible characteristic functions with absolutely continuous poisson spectral measure

We shall consider the problem of characterizing infinitely divisible characteristic functions which have only infinitely divisible factors. Infinitely divisible characteristic functions treated in this paper are those which have absolutely continuous Poisson spectral measures and have no Gaussian component in their Lévy canonical representations. It will be shown that Ostrovskii's sufficient condition is also necessary in this case.     

Decomposition of multivariate infinitely divisible characteristic functions with absolutely continuous Poisson spectral measures

We shall consider the decomposition problem of multivariate infinitely divisible characteristic functions which have no Gaussian component and have absolutely continuous Poisson spectral measures. Under the condition that A = {x;f(x) > 0} is open, where f is the density of spectral measure, we shall show that a known sufficient condition for the membership of the class I_0m (i.e., infinitely divisible characteristic functions having only infinitely divisible factors) is also necessary.     

On the decomposition of infinitely divisible characteristic functions of several variables

Summary We prove here some theorems describing infinitely divisible characteristic functions defined on R ⁿ which have positive Poisson spectrum and belong to the class I ₀ of characteristic functions without indecomposable factor. These theorems are generalizations to the case of several variables of results due to I.V. Ostrovskiy in the case of one variable.This work was supported by the National Science Foundation under grant NSF-GP-6175.     

Stability bounds of decompositions of infinitely divisible distribution functions with Gaussian component of linnik's class\mathfrak{L}

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 131–142, 1987.     

On infinitely divisible matrices,kernels, and functions

Summary Generalizations and applications of a recent theorem of C. Loewner on nonnegative quadratic forms have led to interesting new results and to some especially simple derivations of well-known theorems from a unified point of view. In this note we discuss two applications to the study of characteristic functions and completely monotonic functions, and show how the classical representation theorems for infinitely divisible laws may be obtained.Research partially sponsored by National Science Foundation under Grants GP 5855 and GP 5358     

A condition for absolute continuity of infinitely divisible distribution functions

Summary Throughout this paper the symbols r.v., d.f., ch.f., and i.d. will stand, respectively, for random variable, distribution function, characteristic function, and infinitely divisible.Let F(x) be an i.d.d.f. Hartman and Wintner [5] and Blum and Rosenblatt [1] have given a condition, necessary and sufficient, for F(x) to be a continuous d.f. In this note a sufficient condition for F(x) to be an absolutely continuous d.f. is given.Research supported by ONR Contract No. NONR-285(46).Research supported in part by a National Science Foundation fellowship.     

On infinitely divisible semimartingales

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.     

Stability estimates for the expansions of infinitely divisible distribution functions whose Gaussian components are of Linnik class ℭ. 2

Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 142–161.     

Explicit estimates for the asymptotics of subexponential infinitely divisible distribution functions

In this paper, order-sharp explicit estimates are first obtained for the asymptotics at infinity of a wide class of subexponential infinitely divisible distributions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 295–301, February, 2000.     

On a characterization of infinitely divisible characteristic functionals on a Hubert space

Summary A characterization of infinitely divisible characteristic functional on a Hilbert space, analogous to that of Johansen [1], is given.     

On multivariate infinitely divisible distributions

Simple conditions are given which characterize the generating function of a nonnegative multivariate infinitely divisible random vector. Necessary conditions on marginals, linear combinations, tail behavior, and zeroes are discussed, and a sufficient condition is given. The latter condition, which is a multivariate generalization of ordinary log-convexity, is shown to characterize only certain products of univariate infinitely divisible distributions.