Large deviations for sums of independent random variables with dominatedly varying tails

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

广义复合二项风险模型的若干大偏差结果

This paper is a further investigation of large deviation for partial and random sums of random variables, where {Xn,n ≥ 1} is non-negative independent identically distributed random variables with a common heavy-tailed distribution function F on the real line R and finite mean μ∈ R. {N（n）,n ≥ 0} is a binomial process with a parameter p ∈ （0,1） and independent of {Xn,n ≥ 1}; {M（n）,n ≥ 0} is a Poisson process with intensity λ 〉 0, Sn = ΣNn i=1 Xi-cM（n）. Suppose F ∈ C, we futher extend and improve some large deviation results. These results can apply to certain problems in insurance and finance.     

Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^＋, {N（n）; n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C （0, 1）, n ≥ 0, let {M（n）; n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N（n）; n ≥ 0}, {Xn; n≥1} and {M（n）; n ≥ 0} are mutually independent. Write S（n） =N（n）∑i=1 Xi-cM（n）.Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.     

On necessary and sufficient conditions for the self-normalized central limit theorem In Honor of Professor Chuanrong Lu on His 85th Birthday

Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n~2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary.     

Local Precise Large and Moderate Deviations for Sums of Independent Random Variables

Let {X, X_k : k ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution F. In this paper, the authors establish some results on the local precise large and moderate deviation probabilities for partial sums S_n =sum from i=1 to n(X_i) in a unified form in which X may be a random variable of an arbitrary type,which state that under some suitable conditions, for some constants T 0, a and τ 1/2and for every fixed γ 0, the relation P(S_n- na ∈(x, x + T ]) ~nF((x + a, x + a + T ]) holds uniformly for all x ≥γn~τ as n→∞, that is, P(Sn- na ∈(x, x + T ]) lim sup- 1 = 0.n→+∞x≥γnτnF((x + a, x + a + T ])The authors also discuss the case where X has an infinite mean.     

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent(NOD) random variables are investigated. Let {X_n, n ≥ 1}be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.     

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.     

Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let {X,X_k:k≥1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX 0.Let r be a nonnegative integer-valued random variable,independent of {X,X_k:k≥1}.In this paper,the authors obtain the necessary and sufficient conditions for the random sums S_r =(?)X_n to have a consistently varying tail when the random number t has a heavier tail than the summands,i.e.,(P(Xx))/(P(r x))→0as x→∞     

A strong approximation of self-normalized sums

Let {X,Xn,n1} be a sequence of independent identically distributed random variables with EX=0 and assume that EX2I(|X|≤x) is slowly varying as x→∞,i.e.,X is in the domain of attraction of the normal law.In this paper a Strassen-type strong approximation is established for self-normalized sums of such random variables.     

Towards a unification of certain characterizations by conditional expectations

The paper presents a characterization of a general family of distributions by the form of the expectation of an appropriately truncated function of the random variable involved. The obtained result unifies results existing in the literature for specific distributions as well as new results that appear for the first time in this paper. A discrete version is also provided unifying existing characterizations of known discrete distributions.     

关于伽马分布及相关分布性质的一点研究

主要研究伽马分布的性质,并通过对伽马分布可加性的研究.得到由指数分布通过伽马分布构造卡方分布和均匀分布的方法,通过本文可以加深对伽马分布和其它常见连续性分布关系的认识.     

A Review of Results on Sums of Random Variables

Sums of random variables arise naturally in wireless communications and related areas. Here, we provide a review of the known results on sums of exponential, gamma, lognormal, Rayleigh and Weibull random variables. A discussion is provided of two applications. We expect that this review could serve as a useful reference and help to advance further research in this area.     

概率分布的若干特征性质

讨论了初等概率论中有关分布的特征性质,在现行的教材中均没有重点列出,但这些性质都是非常重要的.     

Asymptotic Distribution of a Kind of Dirichlet Distribution

The Dirichlet distribution that we are concerned with in this paper is very special, in which all parameters are different from each other. We prove that the asymptotic distribution of this kind of Dirichlet distributions is a normal distribution by using the central limit theorem and Slutsky theorem.     

统计学上三大分布推导方法

讨论了如何求随机变量函数分布的方法,然后用两种方法推出统计学上三个重要分布的概率分布密度函数.方法独特新颖.     

On singular matrix variate beta distribution

1.IntrodnctionThispaperextendsthestudyofthesingularmatrixvariatebetadistributionofrank1[1]tothecaseofageneralrank.Astherelateddistributiontonormalsampling,thematrixvariatebetadistribution(alsocalledthemultivariatebetadistribution)hasbeenstudiedextens...     

Exponential Mixture Representation of Geometric Stable Distributions

We show that every strictly geometric stable (GS) random variable can be represented as a product of an exponentially distributed random variable and an independent random variable with an explicit density and distribution function. An immediate application of the representation is a straightforward simulation method of GS random variables. Our result generalizes previous representations for the special cases of Mittag-Leffler and symmetric Linnik distributions.     

New examples of heavy-tailed O-subexponential distributions and related closure properties

Let L and S denote the classes of distributions with long tails and subexponential tails respectively. Let OS denote the class of distributions with O-subexponential tails, which means the distributions with the tails having the same order as the tails of their 2-fold convolutions. In this paper, we first construct a family of distributions without finite means in L∩OS?S. Next some distributions in L∩OS?S, which possess finite means or even finite higher moments, are also constructed. In connection with this, we prove that the class OS is closed under minimization of random variables. However, it is not closed under maximization of random variables.     

双曲分布及其在VaR模型分析中的应用

传统的计算V aR的R iskM etrics方法不能对市场风险分布的“厚尾”现象给出较为满意的刻画和计算方法.本文引入双曲分布及其算法并将双曲分布应用到V aR模型的计算之中,事实上通过对股票市场的实证研究表明,股票市场数据呈厚尾现象,用双曲分布对数据的拟合要比R iskM etrics方法假定的正态分布更符合金融市场数据的实际情况,故本文的结论与方法对金融风险管理和其他金融建模是有价值的.