Showing papers on "Infinite divisibility published in 2016"
TL;DR: In this paper, the authors investigated conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has a different scaling factor.
Abstract: We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has a different scaling factor. We give necessary and sufficient conditions in terms of non-negativity of some elementary functions and some more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein’s theorem for both equal and non-equal scaling factors. This leads to conditions on parameters under which Meijer’s G-function or Fox’s H-function represents an infinitely divisible probability distribution on the positive half-line. Moreover, we present new integral equations for both G-function and H-function. The results of the paper generalize those due to Ismail (with Bustoz, Muldoon and Grinshpan) and Alzer who considered previously the case of unit scaling factors.
29 citations
TL;DR: It is proved that finite divisibility of stochastic matrices is an NP-complete problem, and this result is extended to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochy matrices.
20 citations
TL;DR: In this paper, it was shown that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in the limit theorem.
Abstract: We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.
10 citations
TL;DR: This work considers a continuous, infinitely divisible random field in Rd given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent LÉvy measure and finds that the asymptotic probability is equivalent to the right tail of the underlying Lévey measure.
Abstract: We consider a continuous, infinitely divisible random field in Rd given as an integral of a kernel function with respect to a Levy basis with convolution equivalent Levy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level x as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Levy measure.
7 citations
TL;DR: In this article, it was shown that if a standard semicircular element is a standard random variable, then it is freely infinitely divisible for Ω(n) √ √ n/ √ log n/ log n) for every n √ N.
Abstract: We prove that $X^r$ follows an FID distribution if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r\in(-\infty,0]\cup[1,\infty)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r\geq1$; (3) $X$ follows a mixture of some HCM distributions and $|r|\geq1$; (4) $X$ follows some beta distributions and $r$ is taken from some interval. In particular, if $S$ is a standard semicircular element then $|S|^r$ is freely infinitely divisible for $r\in(-\infty,0]\cup[2,\infty)$. Also we consider the symmetrization of the above probability measures, and in particular show that $|S|^r \,\text{sign}(S)$ is freely infinitely divisible for $r\geq2$. Therefore $S^n$ is freely infinitely divisible for every $n\in\mathbb N$. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables.
5 citations
TL;DR: In this paper, the most recently introduced continuous univariate distributions are characterized based on a simple relationship between two truncated moments and the truncated moment of the sequence of the two moments.
Abstract: We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of ce...
4 citations
01 Jan 2016
TL;DR: In this article, necessary and sufficient conditions for existence and infinite divisibility of α-determinantal processes were given for negative binomial and ordinary binomial multivariate distributions.
Abstract: We give necessary and sufficient conditions for existence and infinite divisibility of α-determinantal processes. For that purpose we use results on negative binomial and ordinary binomial multivariate distributions.
4 citations
TL;DR: In this article, the authors deduce tight Turan type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature.
Abstract: In this paper we deduce some tight Turan type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turan type inequalities. Moreover, by using these Turan type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turan type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution.
4 citations
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TL;DR: The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying theory of regular variation, without appeal to the general L\'evy-Khintchine integral representation of infinitely divisible distributions.
Abstract: The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying theory of regular variation, without appeal to the general L\'evy-Khintchine integral representation of infinitely divisible distributions.
3 citations
01 Jan 2016
TL;DR: In this paper, the authors characterize exchangeability of infinitely divisible distributions in terms of the characteristic triplet and make a connection to Levy copulas, and further study general mappings between classes of measures that preserve exchangeability and give various examples which arise from discrete time settings, such as stationary distributions of AR(1) processes.
Abstract: We characterize exchangeability of infinitely divisible distributions in terms of the characteristic triplet. This is applied to stable distributions and self-decomposable distributions, and a connection to Levy copulas is made. We further study general mappings between classes of measures that preserve exchangeability and give various examples which arise from discrete time settings, such as stationary distributions of AR(1) processes, or from continuous time settings, such as Ornstein–Uhlenbeck processes or Upsilon-transforms.
1 citations
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TL;DR: In this article, it was shown that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible, and for each fixed n ≥ 2, it is shown that for any fixed n > 0, the smallest eigen value is not infinite divisible.
Abstract: The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the $\beta$-Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of a $\beta$-Hermite ensemble, is not infinitely divisible. Furthermore, for each fixed $N \ge 2$ it is proved that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible.
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TL;DR: In this article, the authors decompose random populations with a genealogy in subfamilies of a given degree of kinship and obtain a notion of infinitely divisible genealogies.
Abstract: The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space U which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of collections of subfamilies of fixed kinship h (described as ultrametric measure spaces), for every depth h as a measurable functional of the genealogy. Random elements in the semigroup are studied, in particular infinitely divisible random variables. Here we define infinite divisibility of random genealogies as the property that the h-tops can be represented as concatenation of independent identically distributed h-forests for every h and obtain a Levy-Khintchine representation of this object and a corresponding representation via a concatenation of points of a Poisson point process of h-forests. Finally the case of discrete and marked um-spaces is treated allowing to apply the results to both the individual based and most important spatial populations. The results have various applications. In particular the case of the genealogical (U-valued) Feller diffusion and genealogical (U V -valued) super random walk is treated based on the present work in [DG18b] and [GRG]. In the part II of this paper we go in a different direction and refine the study in the case of continuum branching populations, give a refined analysis of the Laplace functional and give a representation in terms of a Cox process on h-trees, rather than forests.
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TL;DR: In this paper, the reciprocal of the exit time from a cone of planar Brownian motion was studied and the exponential functional of an associated Brownian Motion was shown to correspond to the exponential function of a Levy Process.
Abstract: Motivated by a common Mathematical Finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of an associated Brownian motion. We prove a conjecture by Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and we present a novel simple proof of De Blassie's result (1987-1988) about the asymptotic behaviour of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, for t large. Similar issues for the exponential functional of a Levy process are also discussed. We finally use the findings obtained by the windings approach in order to get results for quantities associated to the pricing of Asian options.
01 Jan 2016
TL;DR: In this paper, the authors considered multivariate gamma distributions of third and higher dimensions and showed that these distributions are infinitely divisible in certain special cases, and extended a result of one of the authors for the corresponding bivariate distribution.
Abstract: SUMMARY. Multivariate gamma distributions of third and higher dimension are considered and shown to be infinitely divisible in certain special cases. This extends, for these special cases, a result of one of the authors for the corresponding bivariate distribution.
01 Jan 2016
TL;DR: In this paper, necessary and sufficient conditions for the existence of upper or lower bounds for an infinitely divisible random variable are given for the mono tonicity of the sample functions of certain stochastic processes.
Abstract: SUMMARY. Necessary and sufficient conditions for the existence of upper or lower bounds for an infinitely divisible random variable are found. The results are applied to give conditions for the mono tonicity of the sample functions of certain stochastic processes.
01 Jan 2016
TL;DR: In this paper, it was shown that for any d-dimensional centered Gaussian vector with a nonsingular covariance, the vector is not self-decomposable unless its components are independent.
Abstract: Exponential variables, gamma variables or squared centered Gaussian variables, are always selfdecomposable. Does this property extend to multivariate gamma distributions? We show here that for any d-dimensional centered Gaussian vector (η1, …, η d ) with a nonsingular covariance, the vector (η12, …, η d 2) is not selfdecomposable unless its components are independent. More generally, permanental vectors with nonsingular kernels are not selfdecomposable unless their components are independent.
TL;DR: In this paper, a study of free analysis in the quaternionic setting was conducted, and Boolean convolution for quaternion-valued measures was studied. And an integral representation for Caratheodory and Herglotz functions was presented.
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TL;DR: In this article, an infinite dimensional compound Poisson limit theorem in free probability has been proved and characterized in terms of its free cumulants in a non-commutative probability space.
Abstract: Inspired by R. Speicher's multidimensional free central limit theorem and semicircle families, we prove an infinite dimensional compound Poisson limit theorem in free probability, and define infinite dimensional compound free Poisson distributions in a non-commutative probability space. Infinite dimensional free infinitely divisible distributions are defined and characterized in terms of its free cumulants. It is proved that for a distribution of a sequence of random variables, the following statements are equivalent. (1) The distribution is multidimensional free infinitely divisible. (2) The distribution is the limit distribution of triangular trays of families of random variables. (3) The distribution is the distribution of $\{a_1^{(i)}: i=1, 2, \cdots\}$ of a multidimensional free Levy process $\{\{a_t^{(i)}:i=1, 2, \cdots\}: t\ge 0\}$. (4) The distribution is the limit distribution of a sequence of multidimensional compound free Poisson distributions.
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TL;DR: In this article, the authors considered the problem of infinite divisibility of R-diagonal *-distributions with respect to the operation of free additive convolution and proved that the set of infinitely divisible R-Diagonal distributions is closed under free multiplicative convolution.
Abstract: The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distributions to infinitely divisible R-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible R-diagonal distribution that can arise in a C*-probability space, by a pair of compactly supported Borel probability measures on $[ 0, \infty )$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible R-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution.
01 Jan 2016
TL;DR: In this article, a necessary and sufficient condition for the n-divisibility of a random variable (r.v.) with support contained in {0, 1, 2,...} is given.
Abstract: SUMMARY. In this note, a necessary and sufficient condition for the n-divisibility of a random variable (r.v.) with support contained in {0, 1, 2, ...} is given. Next a characteri zation of the geometric distribution is obtained. Also bounds for P(X = k-\-1) are given when the distribution of an infinitely divisible r.v. X is known to coincide with the geometric distribution at the points 0, 1, ..., k.