Exchangeable random measures in the plane

A random measure on [0,1]², [0, 1]}₊ or ₊ ² is said to be separately exchangeable, if its distribution is invariant under arbitrary Lebesgue measure-preserving transformations in the two coordinates, and jointly exchangeable if is defined on [0,1]² or ₊ ² , and its distribution is invariant under mappings by a common measure-preserving transformation in both directions. In each case, we derive a general representation of in terms of independent Poisson processes and i.i.d. random variables.     

Approximations for Bivariate Extreme Values

We study the tail behavior of distributions in the domain of attraction of bivariate extreme value distributions (this includes bivariate extreme value distributions themselves). We provide results on finite approximations of the tail behavior and its analytical shape. The results could form a basis to improve current statistical modeling of bivariate extreme values.     

Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

We study distributions F on [0,) such that for some T , F ^*2(x, x+T] 2F(x, x+T]. The case T = corresponds to F being subexponential, and our analysis shows that the properties for T < are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.     

Bivariate gamma distributions, sums and ratios

Exact distributions of R = X +Y and W = X/(X +Y ) and the corresponding moment properties are derived when X and Y follow five flexible bivariate gamma distributions. The expressions turn out to involve several special functions.     

Local Limit Theorems for Sums of Power Series Distributed Random Variables and for the Number of Components in Labelled Relational Structures

A Local limit theorem for the distribution of the number of components in random labelled relational structures of size n (e.g., a type of random graphs on n vertices, random permutations of n elements, etc.) is proved as n→∞. The case when the corresponding exponential generating functions diverge at their radii of convergence is considered.     

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     

NOTES ON ERDOS' CONJECTURE

Let Xn,n 1,be a sequence of independent random variables satisfying P(Xn = O) = 1 － P(Xn = an) = 1 － 1/pn,where an,n 1,is a sequence of real numbers,andpn isthenthprime,setFN(x)=P(N∑Xnn=1 x).The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent,it is both sufficient and necessary that there exist three numbers x0 and x1 ＜ x2 such that limsup(FN(x2) －FN(x1)) ＞ 0 holds,and Lo =N→∞ lim FN(x0) exists.Moreover,the authors point ont that they can also obtain the same N→∞ result in the weakened case of liminfP(Xn = 0) ＞ 0.n→∞     

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)].     

On Rates and Limit Distributions

For regular parametric models, estimators converge uniformly at a rate n ^–1/2, and the limit distribution is normal with mean 0. The situation is different if the best possible rate is n ^–, with (0, 1/2), as common for nonparametric models. In this case, uniformly attainable normal limit distributions with mean 0 are impossible.     

多指标随机变量Marcinkiewicz大数定律完全收敛性和收敛速度

本文考虑指标在，ｄ≥１中的独立同分布随机变量序列，得到了有关大数定律的完全收敛性和收敛速度等一些结果．     

具有不同分布NA随机变量列的强大数律及应用

给出了具有不同分布的NA随机变量列满足的若干强大数律;作为应用,不仅将独立随机变量的一类强极限定理完整的推广到NA随机变量情形,而且关于NA随机变量的一些已有结果可以作为推论得出.     

Laws of Large Numbers for Arrays of Rowwise Negatively Associated Random Variables under Nonlinear Probabilities北大核心CSCD

In this paper, a strong law of large numbers for arrays of rowwise negatively associated random variables is obtained under nonlinear probabilities, from which Kolmogorov type and Marcinkiewicz–Zygmund type strong laws of large numbers are derived. And the notion of negative association is weaker than some existing notions of dependence in nonlinear probabilities. Furthermore, an extension of strong law of large numbers for arrays of rowwise independent random variables under nonlinear probabilities is obtained. As a special case, a Kolmogorov type strong law indicates that not only the cluster points of empirical averages lie in the interval between the lower expectation and upper expectation quasi-surely, but such an interval is also the smallest one that covers the empirical averages quasi-surely. Furthermore, the strong law also states that the upper and lower limits of the empirical averages will converge to the upper and lower expectations with upper probabilities one, respectively. © 2022 Chinese Academy of Sciences. All rights reserved.     

Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

On Asymptotic Proximity of Distributions

We consider some general facts concerning the convergence

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.