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.     

Deriving the Probability Density for Sums of Uniform Random Variables

A simple procedure for deriving the probability density function (pdf) for sums of uniformly distributed random variables is offered. This method is suited to introductory courses in probability and mathematical statistics. In our experience, deriving and working with the pdf for sums of random variables facilitates an understanding of the convergence properties of the density of such sums and motivates consideration of other algebraic manipulation for random variables.     

Randomly Weighted Sums of Pairwise Quasi Upper-Tail Independent Increments with Application to Risk Theory

This article investigates the asymptotic tail probabilities of randomly weighted sums and their maximum of pairwise quasi upper-tail independent increments with non negative and arbitrarily dependent random weights. An application of the obtained results to risk theory is proposed.     

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.     

Asymptotic Behavior of Random Time Ruin Probability Under Heavy-Tailed Claim Sizes and Dependence Structure

This article deals with the renewal risk model, in which there exists some asymptotic dependence relation between claim sizes and the inter-arrival times, and claim sizes are subexponential. Under this setting, we investigate the tail behaviour of random time ruin probability as the initial risk reserve x tends to infinity. We obtain the similar asymptotic formula as the previous results.     

Asymptotics and Criticality for a Correlated Bernoulli Process

A generalized binomial distribution that was proposed by Drezner & Farnum in 1993 comes from a correlated Bernoulli process with interesting asymptotic properties which differ strikingly in the neighbourhood of a critical point. The basic asymptotics and a short‐range/long‐range dependence dichotomy are presented in this note.     

Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

We develop accurate approximations for the delay distribution of the MArP/G/1 queue that capture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in a MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.     

Randomly weighted sums of linearly wide quadrant-dependent random variables with heavy tails

This paper investigates tail behavior of the randomly weighted sum ∑ⁿ_{k = 1}θ_kX_k and reaches an asymptotic formula, where X_k, 1 ? k ? n, are real-valued linearly wide quadrant-dependent (LWQD) random variables with a common heavy-tailed distribution, and θ_k, 1 ? k ? n, independent of X_k, 1 ? k ? n, are n non-negative random variables without any dependence assumptions. The LWQD structure includes the linearly negative quadrant-dependent structure, the negatively associated structure, and hence the independence structure. On the other hand, it also includes some positively dependent random variables and some other random variables. The obtained result coincides with the existing ones.     

On Complete Convergence for Weighted Sums of NA Random Fields

In this paper, we prove the complete convergence for the weighted sums of negatively associated random variables with multidimensional indices. The main result generalizes Theorem 2.1 in Kuczmaszewska and Lagodowski (2011 Kuczmaszewska, A., Lagodowski, Z.A. (2011). Convergence rates in the SLLN for some classes of dependent random field. J. Math. Anal. Appl. 380:571–584.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of weighted sums.     

Almost Sure Convergence of Weighted Sums for Negatively Associated Random Variables

In this article, let {X₁, …, X_n} be a sequence of negatively associated random variables and {a_ni, 1 ? i ? n, n ? 1} be a triangular array of constants. Several almost sure convergence theorems for the weighted sums ∑ⁿ_{i = 1}a_niX_i are established.     

Sum of a Random Number of Correlated Random Variables that Depend on the Number of Summands

The mean and variance of a sum of a random number of random variables are well known when the number of summands is independent of each summand and when the summands are independent and identically distributed (iid), or when all summands are identical. In scientific and financial applications, the preceding conditions are often too restrictive. Here, we calculate the mean and variance of a sum of a random number of random summands when the mean and variance of each summand depend on the number of summands and when every pair of summands has the same correlation. This article shows that the variance increases with the correlation between summands and equals the variance in the iid or identical cases when the correlation is zero or one.     

Parameter Estimation for the Tail Distribution of a Random Sequence

The tail Y_t = X_t – u of a random sequence {X_t ∈ , t ∈ } with identically distributed X_t is approximated by the generalized Pareto distribution according to the extreme value theory, wherein Y_t occurs in clusters because of the dependence in the random sequence. Nevertheless, the parameters of the generalized Pareto distribution are estimated by the same methods as in the case of independent and identically distributed Y_t, provided that there is independence between the clusters of Y_t. The estimation variances and confidence intervals can be estimated by the jackknife method. The approaches are theoretically discussed and verified by extensive numerical researches.     

Kernel Estimators for Additive Models with Dependent Observations from Random Fields

Nonparametric estimation of the regression function for additive models is investigated in cases where the observed data are dependent. An additive kernel estimator for the regression function under some general mixing conditions is proposed. Under the mixing conditions, the additive kernel estimator is shown to be asymptotically normal.     

An Improvement of Poisson Approximation for Sums of Dependent Bernoulli Random Variables

This article uses the Stein-Chen method to obtain new non uniform bounds on the error of the cumulative distribution function of sums of dependent Bernoulli random variables and the Poisson cumulative distribution function. The bounds obtained in the present study are sharper than those reported in Teerapabolarn and Neammanee (2006 Teerapabolarn, K., Neammanee, K. (2006). Poisson approximation for sums of dependent Bernoulli random variables. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis. 22:87–99. [Google Scholar]). Examples are provided to illustrate applications of the obtained results.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

Uniform Tail Asymptotics for the Sum of Two Correlated Classes with Stochastic Returns and Dependent Heavy Tails

Consider a continuous-time risk model with two correlated classes of insurance business and risky investments whose price processes are geometric Lévy processes. By assuming that the correlation comes from a common shock, and the claim sizes are heavy-tailed and pairwise quasi-asymptotically independent, we investigate the tail behavior of the sum of the stochastic present values of the two correlated classes, and a uniform asymptotic formula is obtained.     

Complete Convergence for Weighted Sums and Arrays of Rowwise Extended Negatively Dependent Random Variables

In this article, we study the complete convergence for weighted sums of extended negatively dependent random variables and row sums of arrays of rowwise extended negatively dependent random variables. We apply two methods to prove the results: the first of is based on exponential bounds and second is based on the generalization of the classical moment inequality for extended negatively dependent random variables.     

On the Alias Method for Generating Random Variables from a Discrete Distribution

The alias method of Walker is a clever, new, fast method for generating random variables from an arbitrary, specified discrete distribution. A simple probabilistic proof is given, in terms of mixtures, that the method works for any discrete distribution with a finite number of outcomes. A more efficient version of the table-generating portion of the method is described. Finally, a brief discussion on efficiency of the method is given. We believe that the generality, speed, and simplicity of the method make it attractive for use in generating discrete random variables.     

Asymptotics of maxima and sums for a type of strongly dependent isotropic Gaussian random fields

For a type of strongly dependent isotropic Gaussian random fields introduced by Mittal (1976 Mittal, Y. 1976. A class of isotropic covariances functions. Pacific Journal of Mathematics 64:517–38.[Crossref], [Web of Science ®] , [Google Scholar]), the joint limiting distribution of the maximum and the sum for the Gaussian random fields is derived. The asymptotic relation between the maximum and sum of the continuous time strongly dependent isotropic Gaussian random fields and the maximum and sum of this fields sampled at discrete time points is also obtained.     

Precise large deviations for the difference of two sums of END random variables with heavy tails

Assume that there are two types of insurance contracts in an insurance company, and the ith related claims are denoted by {X_ij, j ? 1}, i = 1, 2. In this article, the asymptotic behaviors of precise large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j are investigated, and under several assumptions, some corresponding asymptotic formulas are obtained.