The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)^α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.     

The Recursive Kernel Distribution Function Estimator Based on Negatively and Positively Associated Sequences

Let {X _j , j ≥ 1} be a strictly stationary negatively or positively associated sequence of real valued random variables with unknown distribution function F(x). On the basis of the random variables {X _j , j ≥ 1}, we propose a smooth recursive kernel-type estimate of F(x), and study asymptotic bias, quadratic-mean consistency and asymptotic normality of the recursive kernel-type estimator under suitable conditions.     

Precise Large Deviations for Sums of Independent Random Variables with Consistently Varying Tails

In this article, we investigate the precise large deviations for a sum of independent but not identical distributed random variables. {X _n , n ≥ 1} are independent non-negative random variables with distribution functions {F _n , n ≥ 1}. We assume that the average of right tails of distribution functions F _n is equivalent to some distribution function F with consistently varying tails. In applications, we apply our main results to a realistic example (Pareto-type distribution) and obtain a specific result.     

Exact Moment Convergence Rates of U-Statistics

Let U _n be a U-statistic based on a symmetric kernel h(x, y) and i.i.d. samples {X, X _n ; n ≥ 1}. In this article, the exact moment convergence rates in the first moment of U _n are obtained, which extend previous results concerning partial sums.     

The Distribution of Sums of I.I.D. Pareto Random Variables with Arbitrary Shape Parameter

Laplace transforms are used to derive an exact expression for the cdf of the sum of n i.i.d. Pareto random variables with common pdf f(x) = (α/β)(1 + x/β)^?α?1 for x > 0, where α > 0 and is not an integer, and β > 0. An attractive feature of this expression is that it involves an integral of non oscillating real-valued functions on the positive real line. Examples of values of cdfs are provided and are compared to those determined via simulations.     

Large Deviation Results for Wave Governed Random Motions Driven by Semi-Markov Processes

In this article, we present large deviation results for a model {ξ₁ + … + ξ _n : n ≥ 1} which is close to a random walk. More precisely, we consider independent random variables {ξ _n : n ≥ 1} such that {ξ _n : n ≥ 2} are i.i.d. and a different distribution for ξ₁ is allowed. We prove large deviation estimates for P(N _x  ≤ xT) and P(N _x < ∞) as x → ∞, where N _x : = inf {n ≥ 1: ξ₁ + … + ξ _n  ≥ x}. Moreover, we provide an asymptotically efficient simulation law for the estimation of P(N _x  ≤ xT) and P(N _x < ∞) by Monte Carlo simulation based on the importance sampling technique. These results will be adapted to wave governed random motions driven by semi-Markov processes and we present some simulations. Finally, we study the convergence of some large deviation rates for standard wave governed random motions based on a scaling presented in the literature (see Kac, 1974 Kac , M. ( 1974 ). A stochastic model related to the telegrapher's equation . Rocky Mountain Journal of Mathematics 4 : 497 – 509 .[Crossref] , [Google Scholar]; Orsingher, 1990 Orsingher , E. ( 1990 ). Probability law, flow function, maximum distribution of wave governed random motions and their connections with Kirchoff's laws . Stochastic Processes and their Applications 34 ( 1 ): 49 – 66 . [Google Scholar]).     

The Uniform Asymptotics of the Overshoot of a Random Walk with Light-Tailed Increments

Let {S _n : n ≥ 0} be a random walk with light-tailed increments and negative drift, and let τ(x) be the first time when the random walk crosses a given level x ≥ 0. Tang (2007 Tang , Q. ( 2007 ). The overshoot of a random walk with negative drift . Statist. Probab. Lett. 77 : 158 – 165 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained the asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, which is uniform for y ≥ f(x) for any positive function f(x) → ∞ as x → ∞. In this article, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for 0 ≤ y ≤ N for any positive number N will be given. Using the above two results, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for y ≥ 0, is presented.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

Statistical Discrimination Analysis Using the Maximum Function

The maximum of k functions defined on R ⁿ , n ≥ 1, by f _max (x) = max{f ₁ (x),…, f _k (x)}, ? x ? R ⁿ , can have important roles in Statistics, particularly in Classification. Through its relation with the Bayes error, which is the reference error in classification, it can serve to compute numerical bounds for errors in other classification schemes. It can also serve to define the joint L¹-distance between more than two densities, which, in turn, will serve as a useful tool in Classification and Cluster Analyses. It has a vast potential application in digital image processing too. Finally, its versatile role can be seen in several numerical examples, related to the analysis of Fisher's classical iris data in multidimensional spaces.     

On the Influence of Record Terms in the Addition of Independent Random Variables

We discuss some problems connected with the role of record values and maximal values generated by sequences of random variables X₁, X₂,…, X _n in the process of the growth of sums X₁ +···+ X_n, n = 1, 2,….     

On the strong convergence for weighted sums of ρ *-mixing random variables

A complete convergence result is obtained for weighted sums of identically distributed ρ ^*-mixing random variables with E|X ₁| ^α log(1 + |X ₁|) < ∞ for some 0 < α ≤ 2. This result partially extends the result of Sung (Stat Papers 52: 447–454, 2011) for negatively associated random variables to ρ ^*-mixing random variables. It also settles the open problem posed by Zhou et al. (J Inequal Appl, 2011, doi:10.1155/2011/157816).     

Characterization of Balanced Fractional 3 m Factorial Designs of Resolution III

We consider a fractional 3 ^m factorial design derived from a simple array (SA), which is a balanced array of full strength, where the non negligible factorial effects are the general mean and the linear and quadratic components of the main effect, and m ≥ 2. In this article, we give a necessary and sufficient condition for an SA to be a balanced fractional 3 ^m factorial design of resolution III. Such a design is characterized by the suffixes of indices of an SA.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

The Asymptotic Behavior of a Brownian Motion with a Drift from a Random Domain

Consider a Brownian motion with drift starting at an interior point of a random domain D in R ^d+1, d ≥ 1, let τ _D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of log P(τ _D  > T) are given for T → ∞, depending on the shape of the domain D and the order of the drift. The problem is motivated by the model in insurance and early works of Lifshits and Shi. The methods of proof are based on the calculus of variations and early works of Li, Lifshits and Shi in the drift free case.