Random Sums Research Articles

Intensity distribution in random lasers: comparison between a stochastic differential model of interacting modes and random phase sum-based models

Random lasers (RLs) are a special type of laser with a feedback mechanism arising from the random photon scattering in a disordered medium. Their emitted intensity is inherently stochastic. Here we compare results for the intensity distribution from two classes of models. The first concerns electromagnetic wave scattering in a random medium with field amplitudes and phases as independent random or locally correlated variables [random phase sum (RPS)-based models]. In the second, stochastic differential equations describe the mode dynamics in a random medium. Whereas RPS-based models imply Rayleigh, exponential, and K distributions, in the second class we extend to any degree f of optical nonlinearity previous results valid only up to the sixth order, introducing a novel family of intensity distributions, the generalized Izrailev distributions of order f . Model predictions are compared to very large experimental datasets from two quite distinct RLs: a N d 3 + -doped nanopowder and a mixture of colloids containing T i O 2 particles and a dye solution. While RPS models do not provide good data fits, excellent agreement is found with the stochastic differential model, indicating that it properly captures the influence of high-order nonlinearities on the intensity distribution of RLs.

On a Multivariate Analog of the Zolotarev Problem

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.

Fractional Poisson random sum and its associated normal variance mixture

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are run to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed.

On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

Stochastic Integrals and Random Sums of Power Contractions in Systemics

Stochastic integrals, random sums, random contractions, and selfdecomposable random variables constitute fundamental concepts of probability theory with significant applications in several areas of systemics. The main results of the paper are a characterization of a selfdecomposable distribution and a formulation of a Poisson random sum of power contractions. These results are established by incorporating a type of stochastic integral for a continuous in probability, homogeneous stochastic process with independent increments, and the same type of stochastic integral for a compound Poisson stochastic process with positive jumps. Interpretations of the results in treatment of risks threatening various systems are also provided.

A multi-year microlevel collective risk model

For a typical insurance portfolio, the claims process for a short period, typically one year, is characterized by observing frequency of claims together with the associated claims severities. The collective risk model describes this portfolio as a random sum of the aggregation of the claim amounts. In the classical framework, for simplicity, the claim frequency and claim severities are assumed to be mutually independent. However, there is a growing interest in relaxing this independence assumption which is more realistic and useful for the practical insurance ratemaking. While the common thread has been capturing the dependence between frequency and aggregate severity within a single period, the work of Oh et al. (2021) provides an interesting extension to the addition of capturing dependence among individual severities. In this paper, we extend these works within a framework where we have a portfolio of microlevel frequencies and severities for multiple years. This allows us to develop a factor copula model framework that captures various types of dependence between claim frequencies and claim severities over multiple years. It is therefore a clear extension of earlier works on one-year dependent frequency-severity models and on random effects model for capturing serial dependence of claims. We focus on the results using a family of elliptical copulas to model the dependence. The paper further describes how to calibrate the proposed model using illustrative claims data arising from a Singapore insurance company. The estimated results provide strong evidences of all forms of dependencies captured by our model.

Formulation and incorporation of a stochastic model for supporting optimal decision making in global risk governance

The structural elements of the stochastic model formulated by the present paper consist a discrete random variable, three sequences of discrete distributions, one of which is a sequence of Bernoulli random variables, a frequency of positive random variables, and finally a sequence of random variables taking values in the standard unit interval. The representation of the stochastic model as a random sum is another theoretical contribution of the present work. In addition to the formulation of the stochastic model and representing it as a random sum, the present paper interprets the stochastic model in the study of processes related to global risk severity reduction.

PageRank’s behavior under degree correlations

The focus of this work is the asymptotic analysis of the tail distribution of Google’s PageRank algorithm on large scale-free directed networks. In particular, the main theorem provides the convergence, in the Kantorovich–Rubinstein metric, of the rank of a randomly chosen vertex in graphs generated via either a directed configuration model or an inhomogeneous random digraph. The theorem fully characterizes the limiting distribution by expressing it as a random sum of i.i.d. copies of the attracting endogenous solution to a branching distributional fixed-point equation. In addition, we provide the asymptotic tail behavior of the limit and use it to explain the effect that in-degree/out-degree correlations in the underlying graph can have on the qualitative performance of PageRank.

Bivariate generalized Yule distribution: properties and applications

A bivariate generalized Yule distribution is introduced here as the distribution of the random sum of certain types of independent and identically distributed bivariate Bernoulli random variables. We study some important properties of the distribution through deriving explicit expressions for its probability mass function, factorial moments, and the probability generating function (p.g.f.) of its conditional distributions. Certain recurrence relations for its probabilities, raw moments and factorial moments are also developed. The method of maximum likelihood is employed for estimating the parameters of the distribution and applied to real life data sets for illustrating the relevance of the distribution.

Reliability Properties of the NDL Family of Discrete Distributions with Its Inference

The natural discrete Lindley (NDL) distribution is an intuitive idea that uses discrete analogs to well-known continuous distributions rather than using any of the published discretization techniques. The NDL is a flexible extension of both the geometric and the negative binomial distributions. In the present article, we further investigate new results of value in the areas of both theoretical and applied reliability. To be specific, several closure properties of the NDL are proved. Among the results, sufficient conditions that maintain the preservation properties under useful partial orderings, convolution, and random sum of random variables are introduced. Eight different methods of estimation, including the maximum likelihood, least squares, weighted least squares, Cramér–von Mises, the maximum product of spacing, Anderson–Darling, right-tail Anderson–Darling, and percentiles, have been used to estimate the parameter of interest. The performance of these estimators has been evaluated through extensive simulation. We have also demonstrated two applications of NDL in modeling real-life problems, including count data. It is worth noting that almost all the methods have resulted in very satisfactory estimates on both simulated and real-world data.

Edgeworth expansions for multivariate random sums

The sum of a random number of independent and identically distributed random vectors has a distribution which is not analytically tractable, in the general case. The problem has been addressed by means of asymptotic approximations embedding the number of summands in a stochastically increasing sequence. Another approach relies on fitting flexible and tractable parametric, multivariate distributions, as for example finite mixtures. Both approaches are investigated within the framework of Edgeworth expansions. A general formula for the fourth-order cumulants of the random sum of independent and identically distributed random vectors is derived and it is shown that the above mentioned asymptotic approach does not necessarily lead to valid asymptotic normal approximations. The problem is addressed by means of Edgeworth expansions. Both theoretical and empirical results suggest that mixtures of two multivariate normal distributions with proportional covariance matrices satisfactorily fit data generated from random sums where the counting random variable and the random summands are Poisson and multivariate skew-normal, respectively.

Formulation and Investigation of an Integral Equation for Characteristic Functions of Positive Random Variables

Functional equations of characteristic functions constitute power research tools for establishing new results in several significant areas of probability theory. The present paper makes use of the characteristic functions of two Poisson random sums and the concept of equality in distribution for introducing an important selfdecomposable distribution.

Bivariate Extended Yule Distribution

Here we introduce a bivariate version of the extended Yule distribution (EYD) of Martinez-Rodriguez et al (Computational Statistics and Data Analysis, 2011). The EYD has been found suitable for heavy tailed distributions. For emphasizing the practical relevance of the model, it is shown that the proposed distribution can be viewed as the distribution of the random sum of independently and identically distributed bivariate Bernoulli random variables. It is also observed that the proposed bivariate distribution is a very flexible distribution and it includes the bivariate geometric distribution as its special case. A genesis of the distribution and explicit closed form expressions for its probability mass function, factorial moments and the probability generating function of its conditional distributions are derived. Certain recurrence relations for probabilities, raw moments and factorial moments of the BEYD are also obtained. The method of maximum likelihood estimation is employed for estimating the parameters of the distribution. Some examples of application using data from different fields are included in order to illustrate all these procedures.

On the Rates of Convergence in Central Limit Theorems for Compound Random Sums of Independent Random Variables

On the Rates of Convergence in Central Limit Theorems for Compound Random Sums of Independent Random Variables

On the Order of Approximation in Limit Theorems for Negative–Binomial Sums of Strictly Stationary m-Dependent Random Variables

During the last several decades, the results related to geometric random sums of independent identically distributed random variables have become interesting results in probability theory and in insurance, risk theory, queuing theory and stochastic finance, etc. The negative–binomial random sums are extensions of geometric random sums. Up to the present, the negative–binomial random sums of independent identically distributed random variables have attracted much attention since they appear in many fields. However, for the strictly stationary m-dependent summands, such negative–binomial random sums are rarely studied. It seems that the complexity of the dependent structure has limited the use of classical tools based on independent random variables such as characteristic function. Up to now, a number of methods have been used to overcome the dependency such as the truncate method, Stein method, and Heinrich’s method. The paper deals with the order of approximation in weak limit theorems for normalized negative–binomial sums of strictly stationary m-dependent random variables generated by a sequence of independent, identically distributed random variables, using the moving average techniques. The orders of approximation in desired theorems are established in terms of the so-called Zolotarev ζ-metric. The obtained results are extensions and generalizations of several known results related to negative–binomial random sums and geometric random sums of independent, identically distributed random variables.

Convergence in Total Variation of Random Sums

Let (Xn) be a sequence of real random variables, (Tn) a sequence of random indices, and (τn) a sequence of constants such that τn→∞. The asymptotic behavior of Ln=(1/τn)∑i=1TnXi, as n→∞, is investigated when (Xn) is exchangeable and independent of (Tn). We give conditions for Mn=τn(Ln−L)⟶M in distribution, where L and M are suitable random variables. Moreover, when (Xn) is i.i.d., we find constants an and bn such that supA∈B(R)|P(Ln∈A)−P(L∈A)|≤an and supA∈B(R)|P(Mn∈A)−P(M∈A)|≤bn for every n. In particular, Ln→L or Mn→M in total variation distance provided an→0 or bn→0, as it happens in some situations.

Convergence rate of random geometric sum distributions to the Laplace law

В статье модифицированы метод Стейна и вспомогательная техника преобразования распределений случайных величин. Это позволило оценить скорость сходимости распределений нормированных геометрических сумм к закону Лапласа. В случае независимых слагаемых разработанный подход дает оптимальную оценку при использовании идеальной метрики порядка $3$. Новые результаты получены также с помощью метрик Колмогорова и Канторовича.

Convergence Rate of Random Geometric Sum Distributions to the Laplace Law

In this paper we modify the Stein method and the auxiliary technique of distributional transformations of random variables. This enables us to estimate the convergence rate of distributions of norm...

Deviations for random sums indexed by the generations of a branching process

Applying the results about harmonic moments of classical Galton-Watson process, we obtain the deviations for random sums indexed by the generations of a branching process. Our results show that the decay rates of large deviations and moderate deviations depend heavily on the degree of the heavy tail and the asymptotic distributions depend heavily on the normalizing constants. If the underlying Galton-Watson process belongs to the Schr?der case, both large deviation and moderate deviation probabilities show three decay rates, where the critical case depends heavily on the Schr?der index. Else if the Galton-Watson process belongs to the B?ttcher case, there are only two decay rate for both large deviation and moderate deviation probabilities. Simulations are also given to illustrate our results.

New error bounds for Laplace approximationviaStein’s method

We use Stein’s method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [ALEA Lat. Am. J. Probab. Math. Stat.11(2014) 571–587] for Stein’s method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.