Divisible Distributions Research Articles

Local Closure under Infinitely Divisible Distribution Roots and Esscher Transform

In this paper, we show that the local distribution class Lloc∩OSloc is not closed under infinitely divisible distribution roots, i.e., there is an infinitely divisible distribution which belongs to the class, while the corresponding Lévy distribution does not. Conversely, we give a condition, under which, if an infinitely divisible distribution belongs to the class Lloc∩OSloc, then so does the Lévy distribution. Furthermore, we find some sufficient conditions that are more concise and intuitive. Using different methods, we also give a corresponding result for another local distribution class, which is larger than the above class. To prove the above results, we study the local closure under random convolution roots. In particular, we obtain a result on the local closure under the convolution root. In these studies, the Esscher transform of distribution plays a key role, which clarifies the relationship between these local distribution classes and related global distribution classes.

Mutant Number Laws and Infinite Divisibility

Concepts of infinitely divisible distributions are reviewed and applied to mutant number distributions derived from the Lea-Coulson and other models which describe the Luria-Delbrück fluctuation test. A key finding is that mutant number distributions arising from a generalised Lea-Coulson model for which normal cell growth is non-decreasing are unimodal. An integral criterion is given which separates the cases of a mode at the origin, or not.

Domain of existence of the Laplace transform of negative multinomial distributions and simulations

This article provides the domain of existence of the Laplace transform of infinitely divisible negative multinomial distributions, defined by their probability generating functions. It makes it possible to construct all of these distributions. We give examples in dimensions 2 and 3. We give a construction and a simulation of negative multinomial distributions in dimension 2.

Taylor’s Law for Some Infinitely Divisible Probability Distributions from Population Models

In a family of random variables, Taylor’s law or Taylor’s power law of fluctuation scaling is a variance function that gives the variance $$\sigma ^{2}>0$$ of a random variable (rv) X with expectation $$\mu >0$$ as a power of $$\mu $$ : $$\sigma ^{2}=A\mu ^{b}$$ for finite real $$A>0,\ b$$ that are the same for all rvs in the family. Equivalently, TL holds when $$\log \sigma ^{2}=a+b\log \mu ,\ a=\log A$$ , for all rvs in some set. Here we analyze the possible values of the TL exponent b in five families of infinitely divisible two-parameter distributions and show how the values of b depend on the parameters of these distributions. The five families are Tweedie–Bar-Lev–Enis, negative binomial, compound Poisson-geometric, compound geometric-Poisson (or Pólya-Aeppli), and gamma distributions. These families arise frequently in empirical data and population models, and they are limit laws of Markov processes that we exhibit in each case.

A New Bound in the Littlewood–Offord Problem

The paper deals with studying a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the concentration function of a weighted sum of independent identically distributed random variables is estimated in terms of the concentration function of a symmetric infinitely divisible distribution whose spectral measure is concentrated on the set of plus-minus weights.

On Alternative Approximating Distributions in the Multivariate Version of Kolmogorov's Second Uniform Limit Theorem

The aim of the present work is to show that recent results of the authors on the approximation of distributions of sums of independent summands by the infinitely divisible laws on convex polyhedra can be shown via an alternative class of approximating infinitely divisible distributions. We will also generalize the results to the infinite-dimensional case.

Generalizing the Sharpe Ratio and Infinite Divisibility in Financial Markets

Investments with high Sharpe ratios tend to be fixed income investments, particularly short-term bonds. Such securities dominate income investment, but not growth investment. We generalize the Sharpe ratio to interpolate smoothly between risk-averse and risk-seeking behaviors, with the traditional Sharpe ratio as a special case. The generalized ratios may be used to optimize portfolios within a lifestyle strategy or a lifecycle strategy. We demonstrate that the Sharpe ratio has an interpretation connected to value at risk (VaR), if returns are normally distributed. We study broad market indices and indicators for US stocks, US corporate bonds, gold, and Japanese stocks. Using monthly data obtained from FRED, we show that these datasets are neither normal, nor log-normal, nor stable. We argue that none of these distributions is appropriate for market data. We argue instead for using infinitely divisible distributions, and fit Pearson distributions to the data.

On the closure under infinitely divisible distribution roots

On the closure under infinitely divisible distribution roots

On directional convolution equivalent densities

We propose a definition of directional multivariate subexponential and convolution equivalent densities and find a useful characterization of these notions for a class of integrable and almost radial decreasing functions. We apply this result to show that the density of the absolutely continuous part of the compound Poisson measure built on a given density f is directionally convolution equivalent and inherits its asymptotic behaviour from f if and only if f is directionally convolution equivalent. We also extend this characterization to the densities of more general infinitely divisible distributions on ℝ d, d≥1, which are not pure compound Poisson.

A unified approach to Stein’s method for stable distributions

In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and discuss the classification of infinitely divisible distributions. Next, we establish a Stein identity for an infinitely divisible random variable via the Lévy-Khintchine representation of the characteristic function. In particular, we establish and unify the Stein identities for an α-stable random variable available in the existing literature. Next, we derive the solutions of the Stein equations and its regularity estimates. Further, we derive error bounds for α-stable approximations, and also obtain rates of convergence results in Wasserstein-δ,δ<α for α∈(0,1) and Wasserstein-type distances for α∈(1,2). Finally, we compare these results with existing literature.

Some Positive Conclusions Related to the Embrechts–Goldie Conjecture

We give some conditions under which if an infinitely divisible distribution supported on \( [0,\infty) \) belongs to the intersection of the distribution class \( {\mathfrak{L}}(\gamma) \) for some \( \gamma\geq 0 \) and the distribution class \( {\mathfrak{OS}} \), then so does the corresponding Lévy distribution or its convolution with itself. To this end, we discuss the closure under compound convolution roots for the class and provide some types of distributions satisfying the above conditions. Therefore, this leads to some positive conclusions related to the Embrechts–Goldie conjecture in contrast to the fact that all corresponding previous results for the distribution class \( {\mathfrak{L}}(\gamma)\cap{\mathfrak{OS}} \) were negative.

The fluctuations of the giant cluster for percolation on random split trees

A split tree of cardinality $n$ is constructed by distributing $n$ balls in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as $n \rightarrow \infty$ are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random $m$-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.

On infinitely divisible multivariate gamma distributions

We provide an explicit construction of variables from a bivariate gamma distribution which is infinitely divisible. These distributions are currently only known through their (complicated) density functions or moment generating functions. The sampling construction uses independent gamma and Poisson random variables and sheds further light on these distributions and their infinitely divisible property. The extension to multivariate gamma infinitely divisible variables is also provided, as well as a negative binomial case.

Estimation and simulation for multivariate tempered stable distributions

We introduce a methodology for the simulation and parameter estimation of multivariate tempered stable distributions with an emphasis on the bivariate case. Our approach is based on an approximation due to a discretization of the spectral measure. It is then applied to two bivariate financial datasets related to exchange rates. The first is comprised of exchange rates between standard currencies, while the second is based on exchange rates related to cryptocurrencies. Our approximation results hold for a wide class of multivariate infinitely divisible distributions.

Non-Debye relaxations: The characteristic exponent in the excess wings model

The characteristic (Laplace or Lévy) exponents uniquely characterize infinitely divisible probability distributions. Although of purely mathematical origin they appear to be uniquely associated with the memory functions present in evolution equations which govern the course of such physical phenomena like non-Debye relaxations or anomalous diffusion. Commonly accepted procedure to mimic memory effects is to make basic equations time smeared, i.e., nonlocal in time. This is modeled either through the convolution of memory functions with those describing relaxation/diffusion or, alternatively, through the time smearing of time derivatives. Intuitive expectations say that such introduced time smearings should be physically equivalent. This leads to the conclusion that both kinds of so far introduced memory functions form a “twin” structure familiar to mathematicians for a long time and known as the Sonine pair. As an illustration of the proposed scheme we consider the excess wings model of non-Debye relaxations, determine its evolution equations and discuss properties of the solutions.

On Convergence and Compactness in Variation with a Shift of Discrete Probability Laws

We consider a class of discrete distribution functions whose characteristic functions are separated from zero, i.e., their absolute values are larger than a certain positive constant on the entire real line. This class is rather wide: it contains discrete infinitely divisible distribution functions, functions of lattice distributions whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently, we demonstrated in our study that characteristic functions corresponding to elements of this class admit the Levy–Khinchin type representation with a nonmonotonic spectral function. Thus, our class is included in the set of quasi-infinitely divisible distribution functions. Using these representations, we also obtained limit theorems and theorems of compactness with convergence in variation for the sequences from this class. This paper is devoted to analogous results concerning convergence and compactness, but with somewhat weakened convergence in variation. Changing the type of convergence notably expands the applicability of the results.

The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Exponential Distribution

The sum of random variables that are identical and independent from an exponential distribution creates the compound distribution. It is called compound negative binomial distribution as the sum of exponential distribution when the number of random variables added follows the negative binomial distribution. This compound distribution’s characteristic function is established by using mathematical analysis methods, included its uniform continuity property. The characteristic function's parametric curves never disappear from the complex plane, which means it is a positively defined function. Another characteristic function's property shows that this compound distribution is one of infinitely divisible distribution.

On a denseness result for quasi-infinitely divisible distributions

A probability distribution μ on Rd is quasi-infinitely divisible if its characteristic function has the representation μ̂=μ1̂∕μ2̂ with infinitely divisible distributions μ1 and μ2. In Lindner et al. (2018, Thm. 4.1) it was shown that the class of quasi-infinitely divisible distributions on R is dense in the class of distributions on R with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on Rd is not dense in the class of distributions on Rd with respect to weak convergence if d≥2.

Risk Neutral Jump Arrival Rates Implied in Option Prices and Their Models

ABSTRACT Characteristic functions of risk neutral densities are constructed from the prices of options at a fixed maturity using well-known procedures. The logarithm of these characteristic functions are shown to synthesize the Fourier transform of jump arrival tails. The formal arrival rate tails are actual arrival rates if their derivatives have an appropriate sign. The derivatives of formal arrival rate tails embedded in option prices are observed on occasion to be negative, reflecting signed jump arrival rates. Although puzzling at first, we further observe that simple analytical cosine perturbations of the symmetric variance gamma Lévy density provides theoretical examples of such signed arrival rates consistent with a probability density. Additionally signed arrival rates also arise when models of signals perturbed by independent noise yield examples of characteristic functions for signal densities that are ratios of pure jump infinitely divisible characteristic functions. Such ratio characteristic functions can reflect signed arrival rates. Specific models using ratios of bilateral gamma and CGMY models are developed and calibrated to short maturity option prices. The ratio models provide significant improvements over their non-ratio counterparts. The models fall in the class of what have recently been termed to be quasi-infinitely divisible distributions.

Stochastic decomposition for [formula omitted]-norm symmetric survival functions on the positive orthant

We derive a stochastic representation for the probability distribution on the positive orthant (0,∞)d whose association between components is minimal among all probability laws with ℓp-norm symmetric survival functions. It is given by a transformation of a uniform distribution on the standard unit simplex that is multiplied with an independent finite mixture of certain beta distributions and an additional atom at unity. On the one hand, this implies an efficient simulation algorithm for arbitrary probability laws with ℓp-norm symmetric survival function. On the other hand, this result is leveraged to construct an exact simulation algorithm for max-infinitely divisible probability distributions on the positive orthant whose exponent measure has ℓp-norm symmetric survival function. Both applications generalize existing results for the case p=1 to the case of arbitrary p≥1.