关于Gauss过程增量的若干结果

Let {X(t);t≥0}be a Gaussian process with stationary increments,X(0)=0(a.s.),EX(t)=0 andσ~2(h)=EX(t+h)-X(t)~=EZ~2(h)=Coh~(2α),0<α≤1.In this paper,we first prove that the Levy's theorem of the modulus of continuityof the Wiener process is also true for {X(t);t≥0};i.e.Furthermore,we point out that some results on increments of the Wiener processesin[3]and[4]remain true for the increments of {X(t);t≥0}.     

指数索赔情形下一类相依两险种风险模型的破产概率

考虑一类稀疏过程下索赔相依的两险种风险模型:U(t)=u+ct-∑i=1N2(t)X_i-∑i=1N2(t)Y_(i),其中{N_1(t),t≥0}、{N_2(t),t≥0}分别表示两个险种的索赔次数,它们按下述方式相关:N_1(t)N_(11)(t)+N_(12)(t),N_2(t)=N_(22)(t)+N'_(12)(t),{N'_(12)(t),t≥0}是{N_(12)(t),t≥0}的一个p-稀疏.考虑下列两种情形:(Ⅰ){N_(11)(t),t≥0}、{N_(12)(t),t≥0}、{N_(22)(t),t≥0}均为Poisson过程;(Ⅱ){N_(11)(t),t≥0}、{N_(22)(t),t≥0}为Poisson过程,{N_(12)(t),t≥0}为Erlang(2)过程.在上述两种情形下,当两险种的单次索赔额均服从指数分布时,通过建立并求解生存概率所满足的微分方程,给出其破产概率的表达式.     

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   

TWO LIMIT THEOREMS ON ARIMA MODELS

Let the time series {X(t),t=1,2,…}satisfy φ(B)(1-B)~dX(t)=θ(B)e(t),where B is a backward shift operator,defined by BX(t)=X(t-1),and φ(z)=1+φz+…+φ_pz~p,θ(z)=1+θ_1z+…+θ_qz~q%,and all the roots of φ(z)lie outside the unit circle;{e(t)}is a sequence of iid random variables with mean zero and E|e(t)|~(4+r)<∞(r>0).In this paper,the limit properties of S_n=sum from t=1 X(t)~2/t~(2d)log n,where the integer d≥1,have been considered.     

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.     

Precise Asymptotics for Lévy Processes

Let {X（t）, t ≥ 0} be a Lévy process with EX（1） = 0 and EX^2（1） 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X（t）, t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.     

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

关于Lévy过程一个公开问题的部分回答

令{X_t,t∈R~ }是一Lévy过程,令γ_0=sup{α≥0:lim inf a~(-α)ET(a,1)<∞},这里T(a,1)=integral from 0 to 1 I{|X_t|≤a}dt.Taylor证明X_t的像集的填充维数等于γ0.由Pruitt和Taylor提出的一个公开问题是:等式γ_0=inf{α≥0:a~(-α)T(a,1)→∞a.s.,当a→0}是否成立?文中证明了:当{X_t,t∈R~ }是从属过程时,上述等式成立.     

关于L&#233;vy过程一个公开问题的部分回答

令{X_t,t∈R~+}是一L&#233;vy过程,令γ_0=sup{α≥0:lim inf a~(-α)ET(a,1)&lt;∞},这里T(a,1)=integral from 0 to 1 I{|X_t|≤a}dt.Taylor证明X_t的像集的填充维数等于γ0.由Pruitt和Taylor提出的一个公开问题是:等式γ_0=inf{α≥0:a~(-α)T(a,1)→∞a.s.,当a→0}是否成立?文中证明了:当{X_t,t∈R~+}是从属过程时,上述等式成立.     

A self-normalized law of the iterated logarithm for the geometrically weighted random series

Let {X, X_n; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX~2I(|X| ≤ x) is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞_(n=0)β~nX_n(0 β 1) is obtained, under some minimal conditions.     

起始为指数律的P—函数振荡问题

设y是标准p-函数类。对u>0令 y(u)={p∈yq≥0,p(t)=e~(-qt),0≤t≤u}在[9]Kingman证明了:如果p∈y(u)则p(t)≤e~(-1) e~(-qu)(t≥u),而在[4]中Griffeath进一步证明了:p(t)≤e~(-(1-e~(-qu)))(t≥u)。本文首先给出这一结果一个完全不同的新证明。然后证明下面的结果:如果p∈y(u),s≥u,p(t),m=P(s)则p(t)≤max(M,m e~(-1 m))(t≥u)。本文的第二个结果叙述如下:记 m(M,p)=inf{p(t):0≤t≤1,p(1)=M},p∈y I(M,u)=inf{m(M,p):p∈y(u)},I(M)=inf{m(M,p):p∈y} I~(M,u),v_0=inf{M>0:I(M)>0} v(M)=inf{M>0:I(M)>0}则v_0=v~。     

准马尔可夫矩阵能扩充为转移矩阵的充要条件

<正> 一个取值于{0,1,2,…,N}的随机过程 Y(t)(t≥0) 称为 n 阶准马尔可夫链,如果对任意 i=1,2,…,N,T>0,在事件{Y(T)=i}和(?)_T={Y(s);0≤s≤T}的条件下,过程 Y(T+t) (t≥0) 的有限维分布仅依赖于 i 而不依赖于 T 和(?)_T(见[1]).当此性质对 i=0也成立,Y(t)就是通常的马尔可夫链.     

一类带p-Laplacian的Sturm-Liouville型2m点边值问题三个正解的存在性

该文考虑了下面的具一维$p$\,-Laplacian算子的多点边值问题 $ \left\{ \begin{array}{rl} &;\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}01,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~ 0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0} <\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$ 通过运用锥上的不动点定理, 该文得到了至少三个正解的存在性. 有趣的是文中的边界条件是一个新型的Sturm-Liouville型边界条件, 这类边值问题到目前为止还很少被研究.     

2012年上海压轴题的一般性解决与追根溯源

今年上海理科的压轴题,有点竞赛题味道,符号较多,有较高的抽象度,是个比较有趣的试题.这里提供一个将问题全面解决的一般性方法供大家参考,欢迎方家指教:(2012上海理23)有一个集合X={-1,x1,x2,…,xn},其中02,且{-1,1,2,x}具有性质P,求     

BOUNDEDNESS OF INFINITE DELAY DIFFERENCE SYSTEMS IN TERMS OF TWO MEASURES

1 PreliminariesLet R (R--), Z (Z--) denote the sets of non-negative (non-positive) realnumbers and nonnegative (nonpositive) integers, respectively, X= {of: { --r,'' 1--2, --1, 0} - Rk}, where r is a non-negative integer or r = oo. DenoteF == {h: Z X Rk - R , h(n, x) is continuous in x, and inf{h(n, x)} = 0},K = {a E C(R ,R ) t a(u) is strictly increasing in u and a(0) = 0},n LQ = {ry E C(R , R ): there are constants a, L 2 1 such that Z n(s) < a,s=n 1for all n E Z }, and in this …     

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.     

对一道数学题的质疑

前不久,某地区高中毕业班统考数学试题(理科)第七题为已知数列{a_n},其前n项和为S_n(n∈N), (1)若S_n=1+pa_n(-1相似文献   

ASYMPTOTIC BEHAVIOR OF BOOTSTRAP SPECTRAL WINDOW ESTIMATION

ThisprojectissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionLetX~{X(t);t=0,FI,12,'}beazerthmeantunitvariancestationaryGaussianprocess,theautocovariancefunction7(u)=EX(n u)X(n)satisfySupposethatthespectraldensityfunctionisf(A)andspectraldistributionfunctionisF(A)oftheprocessX,wherewerestrictAwithin11(if=[--x,7]).Fromtheassumptionsabove,wecaneasilyseethatthef(A)isjustaprobabilitydensityfunction,andF(A)isaprobabilitydistributionfunction.LetX(1),'tX(n)betheobserv…     

Precise Rates in the Generalized Law of the Iterated Logarithm in R~m

Let {X, X_n, n ≥ 1} be a sequence of i.i.d. random vectors with EX =(0,..., 0)_(m×1) and Cov(X, X) = σ~2 Ⅰ_m, and set S_n =∑_(i=1)~n X_i, n ≥ 1. For every d 0 and a_n =o((log log n)~(-d)), the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of P(|S_n| ≥(ε + a_n)σn~(1/2)(log log n)~d).     

一类分数阶微分方程多点边值问题解的存在性

本文讨论下面一类分数阶微分方程多点边值问题 $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$其中$3<\alpha \leq 4$是一个实数.通过应用Mawhin重合度理论和构建适当的算子,得到了该边值问题解的存在性结果.