Random self-decomposability and autoregressive processes

We introduce the notion of random self-decomposability and discuss its relation to the concepts of self-decomposability and geometric infinite divisibility. We present its connection with time series autoregressive schemes with a regression coefficient that randomly turns on and off. In particular, we provide a characterization of random self-decomposability as well as that of marginal distributions of stationary time series that follow this scheme. Our results settle an open question related to the existence of such processes.     

Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ?_λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ?_λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ?_λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ.     

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.     

Random capacities and their distributions

Summary We formalize the notion of an increasing and outer continuous random process, indexed by a class of compact sets, that maps the empty set on zero. Existence and convergence theorems for distributions of such processes are proved. These results generalize or are similar to those known in the special cases of random measures, random (closed) sets and random (upper) semicontinuous functions. For the latter processes infinite divisibility under the maximum is introduced and characterized. Our result generalizes known characterizations of infinite divisibility for random sets and max-infinite divisibility for random vectors. Also discussed is the convergence in distribution of the row-vise maxima of a null-array of random semicontinuous functions.Research supported by the Swedish Natural Science Research Council     

A transformed geometric distribution in stochastic modelling

The analytical concepts of infinite divisibility and (0) unimodality are fundamental to the study of probability distributions in general and to discrete distributions in particular. In this paper, a one-one correspondence is established between these two important properties which will permit any infinitely divisible discrete distribution (with finite mean value) to be transformed into a (0) unimodal discrete distribution. When this transformation is applied specifically to the geometric distribution, the result is a novel distribution, which can be fully and explicitly specified and whose factorial moments can be expressed in closed forms. This transformed geometric distribution is found to apply to underreported geometrically distributed decision processes, embedded renewal processes with logarithmically distributed components, and M/M/1 queues in which the service mechanism has been uniformly improved.     

Negative binomial distributions for point processes

Negative binomial point processes are defined for which all finite-dimensional distributions associated with disjoint bounded Borel sets are negative binomial in the usual sense. For these processes we study classical notions such as infinite divisibility, conditional distributions, Palm probabilities, convergence, etc. Negative binomial point processes appear to be of interest because they are mathematically tractable models which can be used in many situations. The general results throw some new light on some well-known special cases like the Polya process and the Yule process.     

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.     

Tempered stable laws as random walk limits

Stable laws can be tempered by modifying the Lévy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random walk models can provide a basic physical model for the underlying physical phenomena.     

Infinite divisibility II

Khinchin's theorem on the infinite divisibility of the limit of an infinitesimal triangular array of distributions is extended to distributions on a broad class of groups.     

Generalization to the sufficient conditions for a discrete random variable to be infinitely divisible

The convenient sufficient conditions for the infinite divisibility of discrete distributions are generalized. A numerical example is presented. The role of infinitely divisible distributions in applications is discussed.     

First hitting time models for the generalized inverse Gaussian distribution

Any generalized inverse Gaussian distribution with a non-positive power parameter is shown to be the distribution of the first hitting time of level 0 for each of a variety of time-homogeneous diffusions on the interval [0, ∞). The infinite divisibility of the generalized inverse Gaussian distributions is a simple consequence of this and an elementary convolution formula for these distributions.     

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.     

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.     

On permanental processes

Permanental processes can be viewed as a generalization of squared centered Gaussian processes. We analyze the connections of these processes with the local time process of general Markov processes. The obtained results are related to the notion of infinite divisibility.     

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.     

New partial geometric difference sets and partial geometric difference families

Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets (and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns (which were recently coined by Cunsheng Ding in “Codes from Difference Sets” (2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.     

Operator geometric stable laws

Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions, their divisibility and domains of attraction, and present their application to finance. Operator geometric stable laws are useful for modeling financial portfolios where the cumulative price change vectors are sums of a random number of small random shocks with heavy tails, and each component has a different tail index.     

Structural,continuity, and asymptotic properties of a branching particle system

We delineate a connection between the stochastic evolution of the cluster structure of a specific branching–diffusing particle system and a certain previously unknown structure-invariance property of a related class of distributions. Thus, we demonstrate that a Pólya–Aeppli sum of i.i.d.r.v.’s with a common zero-modified geometric distribution also follows a Pólya–Aeppli law. The consideration of these classes is motivated by and applied to studying subtle properties of this branching–diffusing particle system, which belongs to the domain of attraction of a continuous Dawson–Watanabe superprocess. We illustrate this structure-invariance property by considering the Athreya–Ney-type representation of the cluster structure of our particle system. Also, we apply this representation to prove the continuity in mean square of a related real-valued stochastic process. In contrast to other works in this field, we impose the condition that the initial random number of particles follows a Pólya–Aeppli law – a condition that is consistent with stochastic models that emerge in such varied fields as population genetics, ecology, insurance risk, and bacteriophage growth. Our results extend some recent work of Vinogradov. Specifically, we resolve the issue of noninvariance of the initial field and manage to avoid related anomalies that arose in earlier studies. Also, we demonstrate that under natural additional assumptions, our particle system must have evolved from a scaled Poisson field starting at a specified time. In some sense, this result provides a partial justification for assuming that the system had originated at a certain time in the past from a Poisson field of particles. We demonstrate that the corresponding high-density limit of our branching–diffusing particle system inherits an analogous backward-evolution property. Several of our results illustrate a general convergence theorem of Jørgensen et al. to members of the power-variance family of distributions. Finally, combining a Poisson mixture representation for the branching particle system considered with certain sharp analytical methods gives us an explicit representation for the leading error term of the high-density approximation as a linear combination of related Bessel functions. This refines a theorem of Vinogradov on the rate of convergence.     

COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.