Random Sums Research Articles

Smoothness and Lévy concentration function inequalities for distributions of random diagonal sums

We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum S n = ∑ j = 1 n X j , π ( j ) S_n=\sum _{j=1}^nX_{j,\pi (j)} of a random n × n n\times n matrix X = ( X j , r ) X=(X_{j,r}) , where X j , r X_{j,r} are independent integer valued random variables, and π \pi denotes a uniformly distributed random permutation on { 1 , … , n } \{1,\dots ,n\} independent of X X . As a measure of smoothness, we consider the total variation distance between the distributions of S n S_n and 1 + S n 1+S_n . Our approach uses new auxiliary inequalities for a generalized normalized matrix hafnian and for inverse moments of non-negative random variables, which could be of independent interest. This approach is also used to prove upper bounds of the Lévy concentration function of S n S_n in the case of independent real valued random variables X j , r X_{j,r} .

Convergence rates in the limit theorems for random sums of [formula omitted]-orthogonal random variables

In the paper, upper bounds for the convergence rate in the limit theorems for random sums of m-orthogonal random variables are estimated using the K-functional method. Our results are extensions of some known results related to random sums.

Copula-based Bayesian estimation of probabilities for two dependent non homogenous compound Poisson processes

Bivariate copulas for two dependent non homogeneous compound Poisson processes (NCPP) are considered. In this scenario, the arrival times of events are assumed to follow two independent non homogeneous Poisson processes (NHPP), but the corresponding renewal variables are dependent through a bivariate copula. The article aims to provide a computational tool for the estimation of the probability that the random sum associated with the first NCPP is less than the other random sum by a truncated time. The proposed ML and Bayesian estimation methods in a general setting can be applied using bivariate copulas such as Gumbel (for upper tail dependency), Clayton (for lower tail dependency), and Frank (for symmetric dependency). Power Law (PL) intensity function for the NHPPs and Inverse-Gaussian as renewal distribution are considered as examples. Bayes estimates of NHPPs parameters use gamma priors, and for Bayes estimates of Inverse-Gaussian distribution, an MCMC algorithm is employed. Our method has a broad range of practical applications, especially in the insurance sector, where the renewal variable may represent the dollar amount associated with a claim. To illustrate our approach, we analyze injury claims in North Carolina and South Carolina over a specific period, using a dataset sourced from docs.databricks.com.

A neural network approach for the reliability analysis on failure of shallow foundations on cohesive soils

A collection of feed forward neural networks (FNN) for estimating the limit pressure load and the according displacements at limit state of a footing settlement is presented. The training procedure is through supervised learning with error loss function the mean squared error norm. The input dataset is originated from Monte Carlo simulations for a variety of loadings and stochastic uncertainty of the material of the clayey soil domain. The material yield function is the Modified Cam Clay model. The accuracy of the FNN’s is in terms of relative error no more than 10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-5}$$\\end{document} and this applies to all output variables. Furthermore, the epochs of the training of the FNN’s required for construction are found to be small in amount, in the order of magnitude of 90,000, leading to an alleviated data cost and computational expense. The input uncertainty of Karhunen Loeve random field sum appears to provide the most detrimental values for the displacement field of the soil domain. The most unfavorable situation for the displacement field result to limit displacements in the order of magnitude of 0.05 m, that may result to structural collapse if they appear to the founded structure. These series can provide an easy and reliable estimation for the failure of shallow foundation and therefore it can be a useful implement for geotechnical engineering analysis and design.

Paucity phenomena for polynomial products

AbstractLet be a polynomial with at least two distinct complex roots. We prove that the number of solutions to the equation (for any ) is asymptotically as . This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums match standard complex Gaussian moments as , where is the Steinhaus random multiplicative function.

Precise large deviations for non-centralized sums of partial sums and random sums of heavy-tailed END random variables

This paper obtains some asymptotic upper and lower bounds of precise large deviations for non-centralized sums of partial sums and random sums, where the summands are extended negatively dependent random variables with dominatedly-varying-tailed distributions supported on (−∞,+∞).

Total variation distance and compound poisson approximations for random sums

We derive new upper bounds for the total variation distance between compound Poisson distributions as well as between a random sum and a compound Poisson distribution, and as a result we also obtain upper bounds for the total variation distance between compound Poisson distributions and a sum of independent random variables. These bounds are generalizations and refinements of some well-known bounds in the literature. We also derive upper bounds for the total variation distance between negative binomial distributions of order k and between the negative binomial distributions of order k and compound Poisson distributions. Upper bounds for the total variation distances between the number of success runs of length k in binary Markovian trials and its limiting distributions for several enumeration schemes, are also given.

Compound Conway-Maxwell Poisson Gamma Distribution: Properties and Estimation

The distribution of a random sum of random events is called a compound distribution. It involves a counting (discrete) distribution to model the number of occurrences of the random event in a fixed time period and a continuous distribution to model the outcome of the random event. It has applications in the fields of actuarial sciences, meteorology etc. For example, in modelling insurance loss amounts through compound distributions, the number of claims and the claim amounts are used to calculate the total claim amount of a portfolio. The number of claims is modelled through a discrete distribution and the claim amounts are modelled through continuous distributions. Generally, Poisson distribution is used in compound models as the discrete distribution and such models are known as compound Poisson models. However, the equi-dispersion property of the Poisson distribution hinders its application in scenarios where the underlying count data is either over- or under-dispersed. In this paper, a two-parameter Poisson distribution, namely, Conway-Maxwell Poisson (CMP) distribution, which handles both over- and under-dispersed data, is considered as the counting distribution, and the corresponding compound CMP distribution is developed. Some mathematical properties of the distribution are derived and a methodology to estimate the parameters using the likelihood approach is proposed. A numerical illustration of the proposed methodology is given through a simulation study. An application of the compound CMP model is illustrated through transportation security administration (TSA) insurance claim data. Also, the estimation of the risk measures associated with the TSA claim data is discussed.

Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant division.

Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine.

Boundedness of Gaussian Random Sums on Rooted Homogeneous Trees

Boundedness of Gaussian Random Sums on Rooted Homogeneous Trees

Large deviations for sums associated with supercritical branching process in a random environment

In this paper, we study the random sum SZn of independent and identically distributed (i.i.d.) random variables {Xi}, where {Zn} is a supercritical branching process in an i.i.d. environment ξ, and X1 is of zero mean and finite variance. We shall prove the large deviations of SZn, first in the case that Z1 has linear fractional distribution, then in some general case.

On Stein factors for Laplace approximation and their application to random sums

In this article, we consider an integral Stein equation for Laplace distribution and solve it using the semigroup approach. Next, we derive regularity estimates for the solution of the Laplace Stein equation. Finally, we apply these estimates to obtain a convergence rate for Laplace approximation of the random geometric sums. We also compare our rate with the existing literature.

Proactive decision making by incorporation of discrete random sums

Adoption of proactivity provides theorists and practitioners with very significant information for undertaking research work in a wide range of scientific fields. In particular, the presence of proactivity substantially supports the implementation of activities under conditions of uncertainty. It constitutes a general recognition that proactive thinking facilitates strategic acting on the provision of operations for the protection of systems evolving in uncertain environments. Clearly, proactive decision making constitutes the most valuable factor for improving the resilience of systems threatened by a variety of risks. The present paper is primarily concerned with the formulation, investigation, and interpretation in proactive decision processes of two discrete stochastic models. The definitions and the representations as discrete random variables of many structural concepts are significant reasons for implementing discrete stochastic models in proactive treatment of future situations.

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

We study classification of random sequences of characters selected from a given alphabet into two classes characterized by distinct character selection probabilities and length distributions. The classification is based on the sign of the log-likelihood score (LLS) consisting of a random sum and a random term depending on the length distributions for the two classes. For long sequences selected from a large alphabet, computing misclassification error rates is not feasible either theoretically or computationally. To mitigate this problem, we computed limiting distributions for two versions of the normalized LLS applicable to long sequences whose class-specific length follows a translated negative binomial distribution (TNBD). The two limiting distributions turned out to be plain or transformed Erlang distributions. This allowed us to establish the asymptotic accuracy of the likelihood-based classification of random sequences with TNBD length distributions. Our limit theorem generalizes a classic theorem on geometric random sums due to Rényi and is closely related to the published results of V. Korolev and coworkers on negative binomial random sums. As an illustration, we applied our limit theorem to the classification of DNA sequences contained in the genome of the bacterium Bacillus subtilis into two classes: protein-coding genes and standard noncoding open reading frames. We found that TNBDs provide an excellent fit to the length distributions for both classes and that the limiting distributions capture essential features of the normalized empirical LLS fairly well.

On Sum and Geometric Sum of independent New Quasi Lindley Random Variables and its Applications

The Laplace transformation method is used to drive the distribution of the sum Sn of n-fixed random variables, which has a new quasi Lindley distribution with two parameters θ and α, NQLD (θ,α). The sum of NQLD (SUNQLD) distribution is obtained in pdf and cdf formats. It is discussed how to calculate the random sum SN of a random number of NQLD random variables. The random sum of the NQLD distribution's pdf and cdf are calculated. When N has a geometric distribution, the geometric sum of NQLD distribution (GSN QLD) was introduced as an example of a random number of NQLD random variables. For all cases, some statistical measures are determined. The distribution's parameters are estimated using the maximum likelihood method. To test the viability and efficiency of the proposed distributions SNQLD and GSNQLD, lifetime count data sets from acute myeloid leukaemia are fitted. The results should become accepted knowledge in the fields of probability theory and its allied sciences. In addition, the histogram, fitted probability density function (pdf), and P-P plots for the analyzed real data set are presented.

Central limit theorem in uniform metrics for generalized Kac equations

The aim of this paper is to give explicit rates for the speed of convergence to equilibrium of the solution of the generalized Kac equation in two strong metrics: the total variation distance (TV) and the uniform metric between characteristic functions (χ0). A fundamental role in our study is played by the probabilistic representation of the solution of the generalized Kac equation as marginal law of a stochastic process which is a weighted random sum of i.i.d. random variables, where the weights are positive and dependent. Exponential bounds for the total variation distance between the solution and the gaussian stationary state of the Kac equation have been proved by Dolera, Gabetta and Regazzini (2009). In our more general setting the equilibrium states are scale mixtures of stable distributions and hence not necessarily gaussian. Therefore we develop new tools based on ideal metrics that are used in the literature for quantitative central limit theorems for i.i.d. random variables in the domain of attraction of a stable distribution. We obtain first exponential bounds in the so-called ”r-smoothed total variation” and in the weighted χr-metric for a suitable r, then we deduce rates of convergence with respect to the “corresponding” uniform metrics TV and χ0.

First-Passage Times for Random Partial Sums: Yadrenko’s Model for e and Beyond

M. I. Yadrenko discovered that the expectation of the minimum number N 1 of independent and identically distributed uniform random variables on (0, 1) that have to be added to exceed 1 is e. For any threshold a > 0, K. G. Russell found the distribution, mean, and variance of the minimum number Na of independent and identically distributed uniform random summands required to exceed a. Here we calculate the distribution and moments of Na when the summands obey the negative exponential and Lévy distributions. The Lévy distribution has infinite mean. We compare these results with the results of Yadrenko and Russell for uniform random summands to see how the expected first-passage time E ( N a ) , a > 0 , and other moments of Na depend on the distribution of the summand.

Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results

In this paper, the representability of the generalized Student’s distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws.

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

The distribution of extended discrete random sums and its application to waiting time distributions

The distribution of extended discrete random sums and its application to waiting time distributions