On sums of independent random variables without power moments

In 1952 Darling proved the limit theorem for the sums of independent identically distributed random variables without power moments under the functional normalization. This paper contains an alternative proof of Darling’s theorem, using the Laplace transform. Moreover, the asymptotic behavior of probabilities of large deviations is studied in the pattern under consideration.     

Kolmogorov's strong law of large numbers for fuzzy random variables

In this paper, Kolmogorov's strong law of large numbers for sums of independent and level-wise identically distributed fuzzy random variables is obtained.     

Convex fuzzy random variables

A theory of fuzzy random variables is developed that applies to situations involving both randomness and fuzziness. The use of membership functions that are quasi-concave play an important role in the theory. The expectation of a fuzzy random variable is a fuzzy variable (fuzzy set). The usual linearity properties of probabilistic expectation carry over to fuzzy random variables. A special case of a fuzzy Law of Large Number is proven.     

On gaps and unoccupied urns in sequences of geometrically distributed random variables

This paper continues the study of gaps in sequences of n geometrically distributed random variables, as started by Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239], who concentrated on sequences which were gap-free. Now we allow gaps, and count some related parameters.Our terminology of gaps just means empty “urns” (within the range of occupied urns), if we think about an urn model. This might be called weak gaps, as opposed to maximal gaps, as in Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239]. If one considers only “gap-free” sequences, both notions coincide asymptotically, as n→∞.First, the probability p_n(r) that a sequence of length n has a fixed number r of empty urns is studied; this probability is asymptotically given by a constant p^*(r) (depending on r) plus some small oscillations. When , everything simplifies drastically; there are no oscillations.Then, the random variable ‘number of empty urns’ is studied; all moments are evaluated asymptotically. Furthermore, samples that have r empty urns, in particular the random variable ‘largest non-empty urn’ are studied. All moments of this distribution are evaluated asymptotically.The behavior of the quantities obtained in our asymptotic formulæ is also studied for p→0 resp. p→1, through a variety of analytic techniques.The last section discusses the concept called ‘super-gap-free.’ A sample is super-gap-free, if r=0 and each non-empty urn contains at least 2 items (and d-super-gap-free, if they contain ?d items). For the instance , we sketch how the asymptotic probability (apart from small oscillations) that a sample is d-super-gap-free can be computed.     

Saddlepoint approximation for moments of random variables

In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S − a)₊ ^m , a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and a > 0 is a constant. Numerical results are given to show the accuracy of this approximation method.     

A strong law of large numbers for fuzzy random variables

A limit of a sequence of fuzzy numbers is defined and its some properties are shown. Based on these concept and properties, an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown.     

Computing the moments of order statistics from nonidentically distributed phase-type random variables

In this paper, we derive a recurrence relation for the single moments of order statistics (o.s.) arising from n independent nonidentically distributed phase-type (PH) random variables (r.v.’s). This recurrence relation will enable one to compute all single moments of all o.s. in a simple recursive manner.     

舍选法的几何解释及其应用

本文给出了统计模拟中随机数生成之舍选法的几何解释 ,并将其应用到三角形分布和指数型分布的随机数生成算法中     

The strong law of large numbers for pairwise negatively dependent random variables

Necessary and sufficient conditions for the validity of the strong law of large numbers for pairwise negatively dependent random variables with infinite means are formulated.     

On the linear combination of normal and Laplace random variables

Summary  The exact distribution of the linear combination αX+βY is derived when X and Y are normal and Laplace random variables distributed independently of each other. A program in MAPLE is provided to compute the associated percentage points.     

On the exponential inequalities for negatively dependent random variables

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

Uniform nonintegrability of random variables

Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116: 27–37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability (W-UNI) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W-UNI, one of which is a W-UNI analogue of the celebrated de La Vallée Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.     

离散随机序列加权和的若干极限定理

设{Xn,n≥0}是任意离散随机变量序列,{ank,0≤k≤n,n≥0)是一常数阵列,我们引入随机序列渐近对数似然比的概念,作为表征随机序列的真实概率测度P与参考测度Q之间的差异的度量,用分析方法,得到了随机序列Jamison型加权和的若干随机偏差定理.     

On computing distributions of products of non-negative independent random variables

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative independent random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.     

An identity on the maximum of a set of random variables

It is shown that if X₁, X₂, …, X_n are symmetric random variables and max(X₁, …, X_n)⁺ = max(0, X₁, …, X_n), then E[max(X₁,…,X_n)⁺]=[max(X₁,X₁,+X₂,+X₁,+X₃,…X₁,+X_n)⁺], and in the case of independent identically distributed symmetric random variables, E[max(X₁, X₂)⁺] = E[(X₁)⁺] + (1/2)E[(X₁ + X₂)⁺], so that for independent standard normal random variables, E[max(X₁, X₂)⁺] = (1/√2π)[1 + (1/√2)].     

Using fuzzy random variables in life annuities pricing

This paper develops life annuity pricing with stochastic representation of mortality and fuzzy quantification of interest rates. We show that modelling the present value of annuities with fuzzy random variables allows quantifying their expected price and risk resulting from the uncertainty sources considered. So, we firstly describe fuzzy random variables and define some associated measures: the mathematical expectation, the variance, distribution function and quantiles. Secondly, we show several ways to estimate the discount rates to price annuities. Subsequently, the present value of life annuities is modelled with fuzzy random variables. We finally show how an actuary can quantify the price and the risk of a portfolio of annuities when their present value is given by means of fuzzy random variables.