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.     

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??In this paper, precise large deviations of nonnegative, non-identical distributions and negatively associated random variables are investigated. Under certain conditions, the lower bound of the precise large deviations for the non-random sum is solved and the uniformly asymptotic results for the corresponding random sum are obtained. At the same time, we deeply discussed the compound renewal risk model, in which we found that the compound renewal risk model can be equivalent to renewal risk model under certain conditions. The relative research results of precise large deviations are applied to the more practical compound renewal risk model, and the theoretical and practical values are verified. In addition, this paper also shows that the impact of this dependency relationship between random variables to precise large deviations of the final result is not significant.     

风险模型中重尾随机变量和的若干大偏差结果

进一步研究随机变量部分和与随机和的大偏差,其中S(n)=∑ni=1Xi,S(t)=∑N(t)i=1Xi(t>0).{Xn,n≥1}是一个独立同分布的随机变量(未必是非负的)序列具有共同的分布F(定义于R上)和有限期望μ=EX1.{N(t),t≥0}是一个非负的整数值的随机变量的更新计数过程且与{Xn,n≥1}相互独立.本文在假定F∈C条件下,进一步推广并改进了由Klüppelberg等和Kaiw等人给出的一些大偏差结果.这些结果可应用到某些金融保险方面的一些特定的问题中去.     

次线性期望下独立随机变量的样本轨道大偏差

本文得到次线性期望下独立同分布的随机变量的样本轨道大偏差. 在次线性期望下所得的结果推广了概率空间的相应结果.     

多维风险模型中独立和的局部大偏差

In this paper, we study the case of independent sums in multi-risk model. Assume that there exist k types of variables. The ith are denoted by {Xij, j ≥ 1}, which are i.i.d.with common density function fi(x) ∈ OR and finite mean, i = 1,..., k. We investigate local large deviations for partial sums k i=1Sni= k i=1 nij=1Xij.     

Large Deviations View Points for Heavy-Tailed Random Walks

Let X ₁, X ₂,... be a sequence of i.i.d. non-negative random variables with heavy tails. W e study logarithmic asymptotics for the distributions of the partial sums S _n = X ₁ + ··· + X _n . Our main interest is in the crude estimates P(S _n > n ^x ) n ^{–x + 1} for appropriate values of x where is a specific parameter. The related conjecture proposed by Gantert (Stat. Probab. Lett. 49, 113–118) is investigated.     

Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in Cramer zones and Linnik power zones for the distribution function of sums of independent nonidentically distributed random variables (r.v.). In this scheme of summation of r.v., the results are obtained first by mainly using the general lemma on large deviations considering asymptotic expansions for an arbitrary r.v. with regular behaviour of its cumulants [11]. Asymptotic expansions in the Cramer zone for the distribution function of sums of identically distributed r.v. were investigated in the works [1,2]. Note that asymptotic expansions for large deviations were first obtained in the probability theory by J. Kubilius [3].     

Large Deviations for Random Sums on Some Kind of Heavy-tailed Classes in Risk Models

This paper is a further investigation into the large deviations for random sums of heavy-tailed,we extended and improved some results in ref. [1] and [2]. These results can applied to some questions in Insurance and Finance.     

重尾随机变量和的大偏差

This paper is a further investigation of large deviations for sums of random variables S_n=sum form i=1 to n X_i and S(t)=sum form i=1 to N(t) X_i,(t≥0), where {X_n,n≥1) are independent identically distribution and non-negative random variables, and {N(t),t≥0} is a counting process of non-negative integer-valued random variables, independent of {X_n,n≥1}. In this paper, under the suppose F∈G, which is a bigger heavy-tailed class than C, proved large deviation results for sums of random variables.     

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.     

Large Deviations for Heavy-tailed Random Variables in Prospective-loss Process

In this paper, we study the precise large deviations for the prospectiveloss process with consistently varying tails. The obtained results improve some related known ones.     

特殊重尾随机变量随机和的大偏差

文献[1]对于一些经典重尾随机变量的随机和大偏差作了有意义的讨论,本文则讨论了另外一些同样有用的重尾随机和的大偏差.     

负相依随机变量之和的概率大偏差不等式

本文建立了负相依随机变量序列的概率大偏差不等式，并推广了以往文献的结果．     

同分布的NA序列加权和的强大数律

讨论了同分布NA随机变量序列加权和的强大数律,所得结果推广了Z．D．Bai和P．E．Cheng及S．H．Sung的结果．     

Large Deviations of Heavy-Tailed Sums with Applications in Insurance

First we give a short review of large deviation results for sums of i.i.d. random variables. The main emphasis is on heavy-tailed distributions. We stress more the methodology than the detailed calculations. Large deviation techniques are then applied to randomly indexed sums and shot noise processes. We also indicate the close relationship between large deviation results and the modeling of large insurance claims. This revised version was published online in July 2006 with corrections to the Cover Date.     

(D)族负象限相依随机变量和的精致大偏差

在负象限相依结构下,得到了支撵在(-∞,∞)上的(D)族随机变量非中心化以及中心化部分和的精致大偏差.同时,还在较弱的条件下,得到了相应的中心化随机和的精致大偏差.     

D族负象限相依随机变量和的精致大偏差

在负象限相依结构下, 得到了支撑在 (-∞,∞) 上的 D 族随机变量非中心化以及中心化部分和的精致大偏差. 同时, 还在较弱的条件下, 得到了相应的中心化随机和的精致大偏差.     

NA随机变量加权和的收敛性

利用Rosenthal型最大值不等式,得到了NA随机变量加权和的Marcinkiewicz-Zygmund强大数定律和完全收敛性,所获结果推广和改进了一些文献中相应的结果.     

The Law of the Logarithm for Weighted Sums of Independent Random Variables

Let X,X _n ;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b _n =B(n),n1. If