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．     

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.     

An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables

We provide precise bounds for tail probabilities, say {M _n x}, of sums M _n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.     

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     

Estimate from Below for Large-Deviation Probabilities of a Sum of Independent Random Variables with Finite Variances

The goal of this paper is to obtain nonasymptotic estimates from below for large deviation probabilities of a sum of independent random variables that generalize, refine, and are more natural than analogous estimates of other authors. Bibliography: 14 titles.     

Berry–Esseen Bound for a Sample Sum from a Finite Set of Independent Random Variables

Let {X ₁,...,X _N} be a set of N independent random variables, and let S _n be a sum of n random variables chosen without replacement from the set {X ₁,...,X _N} with equal probabilities. In this paper we give an estimate of the remainder term for the normal approximation of S _n under mild conditions.     

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.     

两两独立序列加权和的L~r收敛性

讨论了两两独立随机变量列加权和在满足r(1≤r<2)阶Ces`aro一致可积条件下的Lr收敛性,获得了与独立情形一致的结果.用相似的方法,对于其它相依或混合序列(如两两NQD列,φ-混合序列,ρ-混合序列)也有相同的结果.     

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...     

Prophet Regions for Discounted,Uniformly Bounded Random Variables

Abstract

Let X ₁, X ₂,… be any sequence of [0,1]-valued random variables. A complete comparison is made between the expected maximum E(max _j≤n Y _j ) and the stop rule supremum sup _t E Y _t for two types of discounted sequences: (i) Y _j  = b _j X _j , where {b _j } is a nonincreasing sequence of positive numbers with b ₁ = 1; and (ii) Y _j  = B ₁… B _j?1 X _j , where B ₁, B ₂,… are independent [0,1]-valued random variables that are independent of the X _j , having a common mean β. For instance, it is shown that the set of points {(x, y): x = sup _t E Y _{{(x, y): x=sup t E Y} and y = E(max _j≤n Y _j ), for some sequence X ₁,…,X _n and Y _j  = b _j X _j }, is precisely the convex closure of the union of the sets {(b _j x, b _j y): (x, y) ∈ C _j }, j = 1,…,n, where C _j  = {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ x[1 + (j ? 1)(1 ? x ^1/(j?1))]} is the prophet region for undiscounted random variables given by Hill and Kertz [8 Hill , T.P. , and R.P. Kertz . 1983 . Stop rule inequalities for uniformly bounded sequences of random variables . Trans. Amer. Math. Soc. 278 : 197 – 207 . [CSA]  [Google Scholar]]. As a special case, it is shown that the maximum possible difference E(max _j≤n β ^j?1 X _j ) ? sup _t E(β ^t?1 X _t ) is attained by independent random variables when β ≤ 27/32, but by a martingale-like sequence when β > 27/32. Prophet regions for infinite sequences are given also.     

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.     

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.     

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.     

Independent Random Variables in Lorentz Spaces

We investigate random variables in Lorentz spaces L_p,q. Conditionson the characteristic function are obtained which imply thata random variable belongs to the Lorentz space. Using them,we prove some estimates for the L_p,q-norm of sums of independentrandom variables. Some of these estimates are new for the spacesL_p.     

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...     

On Some Approximations for Sums of Independent Random Variables

Lithuanian Mathematical Journal - In this paper, we generalize a result on approximation of sums by sums with a fixed number of summands in [V. Bentkus, A new method for approximation in...     

On Stochastic Orders for Sums of Independent Random Variables

In this paper, it is shown that a convolution of uniform distributions (a) is more dispersed and (b) has a smaller hazard rate when the scale parameters of the uniform distributions are more dispersed in the sense of majorization. It is also shown that a convolution of gamma distributions with a common shape parameter greater than 1 is larger in (a) likelihood ratio order and (b) dispersive order when the scale parameters are more dispersed in the sense of majorization.     

随机变量绝对值的期望不等式

在任意两个随机变量独立同分布的条件下,得到有关绝对值的数学期望不等式,并利用测度论给予完整证明.     

Large Deviations for Sums of Independent Heavy-Tailed Random Variables

We obtain precise large deviations for heavy-tailed random sums , of independent random variables. are nonnegative integer-valued random variables independent of r.v. (X _i )i N with distribution functions F _i. We assume that the average of right tails of distribution functions F _i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given.     

独立的无界随机变量和的概率不等式

研究了在概率空间(Ω,T,P)上,独立的无界随机变量和尾部概率不等式,提出了一种用切割原始概率空间(Ω,T,P)的新型方法去处理独立的无界随机变量和。给出了独立的无界随机变量和的指数型概率不等式。作为结果的应用,一些有趣的例子被给出。这些例子表明:文中提出的方法和结果对研究独立的无界随机变量和的大样本性质是十分有用的。