Decomposition of infinitely divisible characteristic functions without Gaussian component

Summary The problem of characterizing the infinitely divisible characteristic functions which have only infinitely divisible factors is considered. Under the assumption that both the absolutely continuous and the singular (or the discrete) components exist in Poisson spectral measures, several necessary conditions for this problem are obtained. These conditions admit partial converses and new examples of infinitely divisible characteristic functions which have only infinitely divisible factors are given.     

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.     

Harmonic analysis for the bi-free partial S-transform

We develop an analytic machinery to study Voiculescu's bi-free partial S-transform and then use the results to characterize the multiplicative bi-free infinite divisibility. It is shown that the class of infinitely divisible distributions coincides with the class of limit distributions for products of bi-free pairs of left and right infinitesimal unitaries, where the pairs are not required to be identically distributed but all left variables are assumed to commute with all right variables. Furthermore, necessary and sufficient conditions for convergence to a given infinitely divisible distribution are found.     

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.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

Stable convergence of the log-likelihood ratio to a mixture of infinitely divisible distributions

Some results concerning the asymptotic behavior of the log-likelihood ratio (LLR) and also of certain other random variables closely associated with the likelihood ratio are presented. More specifically, in the present paper we formulate the conditions for the stable convergence in distribution of the LLR for two sequences of the probability measures to a mixture of infinitely divisible distributions with finite variance. Moreover, the notion of a locally asymptotically mixed infinitely divisible (LAMID) sequence of parametric families of the probability measures is introduced, and it is shown that when a certain kind of differentiability-type regularity condition is satisfied, the given sequence of families satisfies the LAMID condition. These results extend and supplement the previous investigations of the author concerning non-Gaussian asymptotic distributions in statistics. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

Some recent results in infinite divisibility

In this paper, a survey is given of some recent developments in infinite divisibility. There are three main topics: (i) the occurrence of infinitely divisible distributions in applied stochastic processes such as queueing processes and birth-death processes, (ii) the construction of infinitely divisible distributions, mainly by mixing, and (iii) conditions for infinite divisibility in terms of distribution functions and densities.     

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.     

Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

Characterization of palm distributions and infinitely divisible random measures

The purpose of the present paper is to characterize Campbell measures and Palm distributions of random measures and to apply these results in a new approach to the characterizations of infinitely divisible random measures by their Laplace functionals and their Palm distributions. The results on infinitely divisible random measures are well known. They can be found together with a detailed list of references in Kallenberg's monograph [2], which also contains proofs of almost all statements in Section 1 of this paper (see his note on page 9 concerning the Polish space setting).     

The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes

The equivalence of ergodicity and weak mixing for general infinitely divisible processes is proven. Using this result and [9], simple conditions for ergodicity of infinitely divisible processes are derived. The notion of codifference for infinitely divisible processes is investigated, it plays the crucial role in the proofs but it may be also of independent interest.     

A new look at Bergström’s theorem on convergence in distribution for sums of dependent random variables

In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ?-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.     

A differential version of the Efron-Stein inequality: bounding the variance of a function of an infinitely divisible variable

Upon a suitable passage to the limit, the Efron-Stein inequality produces a general variance bound for an absolutely continuous function of an infinitely divisible variable. A necessary and sufficient condition for attainment of the bound is also given.     

Convergence of approximate martingale arrays to mixtures of infinitely divisible distributions in locally compact abelian groups

Sufficient conditions are given for the stable weak convergence of the row sums of an approximate martingale triangular array to a mixture of infinitely divisible distributions on a locally compact abelian group.     

ON THE CONVERGENCE FOR NULL-ARRAYS OF THE POINT PROCESSES

The purpose of this paper is first to establish a representation of the Laplace transformfor the regular infinitely divisible point processes,and then to give a sufficient and neccesarycondition for convergence of the null-arrays toward a regular infinitely divisible pointprocess.     

On Infinitely Divisible Distributions on Locally Compact Abelian Groups

Our aim in this paper is to characterize some classes of infinitely divisible distributions on locally compact abelian groups. Firstly infinitely divisible distributions with no idempotent factor on locally compact abelian groups are characterized by means of limit distributions of sums of independent random variables. We introduce semi-selfdecomposable distributions on topological fields, and in case of totally disconnected fields we give a limit theorem for them. We also give a characterization of semistable laws on p-adic field and show that semistable processes are constructed as scaling limits of sums of i.i.d.     

Infinitely divisible probabilities on discrete linear groups

We investigate the structure of infinitely divisible probability measures on a discrete linear group. It is shown that for any such measure there is an infinitely divisible elementz in the centralizer of the support of the measure, such that the translate of the measure byz is embeddable over the subgroup generated by the support of the measure. Examples are given to show that this reult is best possible.     

Infinitely divisible stochastic processes

Summary It often happens that a stochastic process may be approximated by a sum of a large number of independent components no one of which contributes a significant proportion of the whole. For example the depth of water in a lake with many small rivers flowing into it from distant sources, or the point process of calls entering a telephone exchange (considered as the sum of a number of point processes of calls made by individual subscribers) may approximately fulfil these hypotheses. In the present work we formulate and solve the problem of characterizing stochastic processes all of whose finite-dimensional distributions are infinitely divisible. Although some of our results could be derived from known theorems on probabilities on general algebraic structures, many could not and it seems preferable to take the vector-valued infinitely divisible laws as our starting point. We emphasize that an infinitely divisible process (in our sense) on the real line is not necessarily a decomposable process in the sense of Lévy (cf. § 4) though decomposable processes are particular cases.In § 1 a representation theorem for non-negative (and possibly infinite) stochastic processes is derived, while an analogous theorem in the real-valued case is to be found in § 2. § 3 is devoted to the statement of a central limit theorem and the investigation of some of the properties of infinitely divisible processes. The investigation is continued in § 4 by an examination of processes on the real line giving, for example, a representation theorem under weak conditions for infinitely divisible processes which are a.s. sample continuous. Finally in § 5 a study is made of infinitely divisible point processes and random measures.The author is indebted to Professor J. F. C. Kingman for advice and encouragement.     

The normal distribution is ?-infinitely divisible

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a sub-family of Askey–Wimp–Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.     

Association and random measures

Summary Our point of departure is the result, due to Burton and Waymire, that every infinitely divisible random measure has the property variously known as association, positive correlations, or the FKG property. This leads into a study of stationary, associated random measures onR ^d . We establish simple necessary and sufficient conditions for ergodicity and mixing when second moments are present. We also study the second moment condition that is usually referrent to as finite susceptibility. As one consequence of these results, we can easily rederive some central limit theorems of Burton and Waymire. Using association techniques, we obtain a law of the iterated logarithm for infinitely divisible random measures under simple moment hypotheses. Finally, we apply these results to a class of stationary random measures related to measure-valued Markov branching processes.Research supported in part by NSF Grant DMS-8701212 at the University of Virginia