Subexponential distribution functions in Rd
Subexponential distribution functions in Rd
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Rd 中的次指数分布函数
DOI：
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发表时间：
2006
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通讯作者：
E. Omey
中科院分区：
文献类型：
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作者：
E. Omey
∞n=0∞Pnf∗n(X)，其中F∗n(X)表示F(X)的n重卷积，其中F∗0(X)表示在0的单位质量。D.F.W(X)我称S为从属于F(X)的下级{pn}。与文中的单变量情形一样，我们假设N满足条件(A)：N有一个母函数P(Z)=E(Z N)，它在z=1处是解析的。本文讨论了1−F(X)的渐近性与1−F∗n(X)和1−W(X)的渐近性之间的关系。结果表明，与单变量情形一样，1−F∗n(X)渐近表现为n(1−F(X))，1−W(X)表现为E(N)(1−F(X))。为了说明这种精确的渐近行为，我们给出了一种形式的多元次指数。这篇论文的组织方式如下。在证券交易委员会。2，简要回顾了一元次指数D.F.的一些基本性质和定义。在证券交易委员会。3、引入并研究了S在SEC中提出的多元次指数D.F.4、讨论了正则变差与正则变差、正则变差在SEC中的关系。5、我们提供了一些扩展。在我们的主要结果中，我们得到了1−F∗n(X)和1−W(X)的一阶估计。在即将发表的一篇论文中，我们讨论了二阶估计。不作进一步评论，在本文中，我们将假设所有随机向量X、Y、Z等都是正的并且具有无限支撑度，即D.F.满足F(0+)=0和F(X)<；1，∀x∈Rd.我们还使用记号F(X)=1−F(X)，并且对于向量x和a，w e setx◦=min(Xi)a nda∗x=(a1x1，a2x2，...，a dxd)。2.一元次指数分布在一维情形下，已有许多文献研究了隶属D·F·S的尾部行为。在这方面，次指数D·F·S(记为：S)起着重要的作用。推广了S、Chover等人的类。[6，7]，引入了S(γ)类，其中γ≥0.为了定义这些类，设F(X)表示一个D.F.在满足F(0+)=0且F(X)<；1，∀x∈R的R中，设f(S)=E(e−Sx)表示X或F(X)的母函数。D.F.如果F(X)满足LIM，则F(X)属于次指数类S(记号：F∈S
∞n=0 ∞ pnF ∗n (x), where F ∗n (x) denotes the n-fold convolution of F (x) and where F ∗0 (x) denotes the unit mass at 0. The d.f. W (x )i s called subordinate to F (x) with subordinator {pn}. As in the univariate case in the paper, we shall assume that N satisfies condition (A): N has a generating function P (z )= E(z N ) that is analytic at z =1 . In the present paper, we discuss the relation between the asymptotic behavior of 1 − F (x) and that of 1 − F ∗n (x) and 1 − W (x). It turns out that, as in the univariate case, there are many cases in which 1 − F ∗n (x) asymptotically behaves as n(1 − F (x)) and 1 − W (x) behaves as E(N )(1 − F (x)). To specify the precise kind of asymptotic behavior, we present a form of multivariate subexponentiality. The paper is organized as follows. In Sec. 2, we briefly recall some basic properties and definitions concerning univariate subexponential d.f. In Sec. 3, we introduce and study multivariate subexponential d.f.’s. In Sec. 4, we discuss the relation with regular variation and, in Sec. 5, we provide some extensions. In our main results, we obtain first-order estimates for 1 − F ∗n (x) and 1 − W (x). In a forthcoming paper, we discuss second-order estimates. Without further comment, in the paper, we shall assume that all random vectors X, Y, Z, etc. are positive and have infinite support, i.e., the d.f. satisfies F (0+) = 0 and F (x) < 1, ∀x ∈ R d . We also use the notation F (x )=1 − F (x), and for vectors x and a ,w e setx ◦ = min(xi )a nda ∗ x =( a1x1 ,a 2x2 ,...,a dxd). 2. Univariate Subexponential Distributions In the one-dimensional case, many papers have been devoted to the tail behavior of subordinated d.f.’s. In doing so, the class of subexponential d.f.’s (notation: S) plays an important role. Extending the class S, Chover et al. [6, 7], introduced the class S(γ), where γ ≥ 0. To define these classes, let F (x) denote a d.f. in R such that F (0+) = 0 and F (x) < 1, ∀x ∈ R. Also, let f (s )= E(e −sX ) denote the generating function of X or F (x). The d.f. F (x) belongs to the subexponential class S (notation: F ∈ S) if it satisfies lim