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.     

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.     

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.     

Large deviations and moderate deviations for kernel density estimators of directional data

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.     

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.     

Ergodicity of a class of nonlinear time series models in random environment domain

In this paper, we study the problem of a variety of p, onlinear time series model Xn＋ 1= TZn＋1（X（n）, … ,X（n - Zn＋l）, en＋1（Zn＋1）） in which {Zn} is a Markov chain with finite state space, and for every state i of the Markov chain, {en（i）} is a sequence of independent and identically distributed random variables. Also, the limit behavior of the sequence {Xn} defined by the above model is investigated. Some new novel results on the underlying models are presented.     

The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.     

ρ混合序列重对数律的精确率

§ 1　 Introduction and main resultsL et { X,Xn;n≥ 1} be a sequence of random variables with common distributionfunction F,mean0 and positive,finite variance,and set Sn= nk=1 Xk,n≥ 1.Also letlogx= ln(x∨e) ,log logx=log(logx) and(x) =2 xlog logx.Gut and Sp taru[2 ] studied theprecise asymptotics on the law of the iterated logarithm.One of their results is as follows.Theorem A.Spuuose that{ X ,Xn;n≥ 1} is a sequence of i.i.d.random variables with EX= 0 and0 相似文献   

The maximum and minimum degrees of random bipartite multigraphs

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows:let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a₁,a₂,...,a_n} and B = {b₁,b₂,...,b_m}, in which the numbers t_ai,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_ai,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_c,d,A, the number of vertices in A with degree between c and d of G_n,m∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_n,m∈ζ（n,m;{pk}） has maximum degree D（n）in A? under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A?     

负相协随机变量非随机和的差的精确大偏差

In this paper,we study precise large deviation for the non-random difference sum from j=1 to n_1(t) X_(1j)-sum from j=1 to n_2(t) X_(2j),where sum from j=1 to n_1(t) X_(1j) is the non-random sum of {X_(1j),j≥1} which is a sequence of negatively associated random variables with common distribution F_1(x),and sum from j=1 to n_2(t) X_(2j) is the non-random sum of {X_(2j),j≥1} which is a sequence of independent and identically distributed random variables,n_1(t) and n_2(t) are two positive integer functions.Under some other mild conditions,we establish the following uniformly asymptotic relation lim t→∞ sup x≥r(n_1(t))~(p+1)|(P(∑~(n_1(t)_(j=1)X_(1j)-∑~(n_2(t)_(j=1)X_(2j)-(μ_1n_1(t)-μ_2n_2(t)x))/(n_1(t)F_1(x))-1|=0.     

截断和随机乘积的渐近性质

对一列独立同分布平方可积的随机变量序列{Xn,n≥1},当随机变量的分布具有中尾分布时,讨论了其截断和Tn(a)的随机乘积的渐近正态性质,其中Tn(a)=Sn-Sn(a),n=1,2,…,Sn(a)=n∑ j=1 XjI{Mn-a＜Xj≤Mn},a为某一大于零的常数'Mn=max 1≤k≤n{Xk}.     

Complete Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables

Let $\{X_n,n\geq1\}$ be a sequence of negatively superadditive dependent (NSD, in short) random variables and $\{a_{nk}, 1\leq k\leq n, n\geq1\}$ be an array of real numbers. Under some suitable conditions, we present some results on complete convergence for weighted sums $\sum_{k=1}^na_{nk}X_k$ of NSD random variables by using the Rosenthal type inequality. The results obtained in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.     

NA阵列加权乘积和的完全收敛性

设{X_(ni):1≤i≤n,n≥1}为行间NA阵列,g(x)是R~+上指数为α的正则变化函数,r>0,m为正整数,{a_(ni):1≤i≤n,n≥1}为满足条件(?)|a_(ni)|=O((g(n))~1)的实数阵列,本文得到了使sum from n=1 to ∞n~(r-1)Pr(|■multiply from j=1 to m a_(nij) X_(nij)|>ε)<∞,■ε>0成立的条件,推广并改进了Stout及王岳宝和苏淳等的结论。     

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.     

随机环境下上临界分值过程的偏差不等式

令$\{Z_{n}, n\ge 0\}$为独立同分布随机环境下的上临界分支过程$\xi=(\xi_n)_{n\geq 0}$.本文给出了$\ln (Z_{n+n_{0}}/Z_{n_{0}})$的一些偏差不等式及其在构造置信区间上的一些应用.     

CONVERGENCE ON RANDOMLY TRIMMED SUMS WITH A DEPENDENT SAMPLE

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＲｅｓｕｌｔｓＬｅｔ｛Ｘｎ，ｎ１｝ｂｅａｓｅｑｕｅｎｃｅｏｆｒａｎｄｏｍｖａｒｉａｂｌｅｓｗｉｔｈａｃｏｍｍｏｎｄｉｓｔｒｉｂｕｔｉｏｎｆｕｎｃｔｉｏｎＦ（ｘ）ａｎｄｌｅｔＸｎ１Ｘｎ２…Ｘｎｎｂｅｔｈｅｏｒ...     

两两NQD阵列加权和的收敛性

令{X_(nk)；1≤k≤m_n↑∞,n∈N}为零均值的行两两NQD阵列,{a_(nk)；1≤k≤m_n↑∞,n∈N}为非负(或非正)实数阵列,研究两两NQD阵列加权和S_(nm_n)= (?) a_(nk)X_(nk)的收敛性．     

和的乘积的重对数律

设{X,X_n,n≥1}是独立的或φ -混合的或 ρ -混合的正的平稳随机变量序列，或$\{X,X_n,n≥1}$是正的随机变量序列使得{X_n-EX,n≥1\} 是平稳遍历的鞅差序列，记S_n=\sum\limitsⁿ_{j=1}X_j, n≥1 . 该文在条件EX=μ> 0 及0 Var(X)<∞下，证明了部分和的乘积$\prod\limits^n_{j=1}S_j/n!\mu^n$在合适的正则化因子下的某种重对数律.     

同分布 ρ 混合序列的最大值不等式及其应用

设X_n,n≥1是同分布的ρ混合序列, 记S_n=∑ⁿ_{i=1 Xi}. 该文讨论了$\max\limits_{1\leq i\leq n}\frac{|S_i|}{i}$ $(n\geq1)$的分布函数的上界. 作为应用,获得了随机变量$\sup\limits_{n\geq1}\frac{|S_n|}{n}$的1阶矩及$p(>1)$阶矩分别存在有限的充分必要条件,这是一个与独立同分布场合相一致的结果.