On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulae for the distribution of the sum of n non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulae some special cases which have appeared in the literature.     

有趣的随机变量和的分布算法

根据实际问题.提出计算随机变量和的分布的两种方法,即多项式相乘法和概率母函数法.     

The Distribution of a Sum of Independent Binomial Random Variables

The distribution of a sum S of independent binomial random variables, each with different success probabilities, is discussed. An efficient algorithm is given to calculate the exact distribution by convolution. Two approximations are examined, one based on a method of Kolmogorov, and another based on fitting a distribution from the Pearson family. The Kolmogorov approximation is given as an algorithm, with a worked example. The Kolmogorov and Pearson approximations are compared for several given sets of binomials with different sample sizes and probabilities. Other methods of approximation are discussed and some compared numerically. The Kolmogorov approximation is found to be extremely accurate, and the Pearson curve approximation useful if extreme accuracy is not required.     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

On Large Deviations of a Sum of Independent Random Variables with Rapidly Decreasing Distribution Tails

Doklady Mathematics - Given a sum of a finite number of independent random variables, the asymptotic behavior of its distributions and densities at infinity is investigated in the case when the...     

随机变量和的模函数分布及其应用

介绍了一个在计算机科学、信息科学等学科中具有广泛应用的随机变量和的模函数,计算了其分布,并提供了该函数在图像信息安全领域的一个应用例子,验证了理论结果.     

关于DRCE随机变量渐近分布性质的探讨

本文讨论了ＤＲＣＥ随机变量渐近正态收敛速度的一致上、下界，并对一类特殊的ＤＲＣＥ随机变量，讨论了其退化情况的不变原理。     

On the Distribution of Random Samples from Finite Populations of Random Variables

This article deals with probability distributions of sums of simple random sample and Bernoulli sample when samples are selected from finite population of independent random variables. Random variables are quasi-lattice. Probability distributions from class ? and Poisson distribution are used for approximation. Analogue of Cornish-Fisher transformation is obtained in case of limit distributions from class ?.     

Renewed Looks at the Distribution of a Sum of Independent or Dependent Discrete Random Variables and Related Problems

Methodology and Computing in Applied Probability - Butler and Stephens (2017) have investigated the exact and approximate distributions of a sum S of independent binomial random variables with...     

An Inequality for the Sum of Independent Bounded Random Variables

We give a simple inequality for the sum of independent, bounded random variables. This inequality improves on the celebrated result of Hoeffding in a special case. It is optimal in the limit where the sum tends to a Poisson random variable.     

On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), under the extra condition that X₁ is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X₁ has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25     

独立随机变量序列加权和的相合性

设t0∈(0，1)，Wni(t0)是关于实变量t1，t2，…，tn的权函数；随机变量序列Y1，Y2，…，Yn，iid．本研究了随机变量序列加权和∑(i=1,n)Wni(t0)Yi的相合性．     

B-值随机变量序列加权和的收敛结果

本文讨论了双向无穷B-值随机变量序列加权和的弱大数定律、Lr收敛性、完全收敛性.并由此刻画了空间的几何性质.     

随机变量和的特征函数的性质与应用

运用随机变量和的特征函数定义研究了随机变量和的特征函数在原点处二阶导数与协方差矩阵的关系,并给了一个简单的应用.     

SLLN for Weighted Independent Identically Distributed Random Variables

For any sequence {a _k } with sup for some q>1, we prove that converges to 0 a.s. for every {X _n } i.i.d. with E(|X ₁|)< and E(X ₁)=0; the result is no longer true for q=1, not even for the class of i.i.d. with X ₁ bounded. We also show that if {a _k } is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {X _n { with finite absolute pth moment for some p> 1, converges a.s.     

On Operator-Valued Semicircular Random Variables

In this paper, we discuss some special properties of operator-valued semicircular random variables including representation of the Cauchy transform of a compactly supported probability measure in terms of their operator-valued Cauchy transforms and existence of nonzero discrete part of their associated distributions.     

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statuleviius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.     

On the Subexponential Property of a Class of Random Variables

Mathematical Notes -