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1.
If suitably normalized maxima of an i.i.d. sample converge in distribution, the limiting distribution is known to be max-infinitely divisible and the common distribution of the sample is said to belong to its domain of attraction. We prove the existence of max-universal distributions belonging to the domain of attraction of every max-infinitely divisible law. The proof follows in the spirit of corresponding results for normalized sums of i.i.d. random variables originated by Doeblin and shows that necessarily the sampling size has to be rapidly increasing. Restricting the growth rate of the sampling size, we prove that one necessarily deals with max-semistable distributions and their domains of attraction. 2000 Mathematics subject classification Primary—60G70 Secondary—60E99, 60F05  相似文献   

2.
In this paper, we investigate the properties of the recently introduced measure of dependence called correlation cascade. We show that the correlation cascade is a promising tool for studying the dependence structure of infinitely divisible processes. We describe the ergodic properties (ergodicity, weak mixing, mixing) of stationary infinitely divisible processes in the language of the correlation cascade and establish its relationship with the codifference. Using the correlation cascade, we investigate the dependence structure of four fractional αα-stable stationary processes. We detect the property of long memory and verify the ergodic properties of the discussed processes.  相似文献   

3.
Processes with independent increments are proven to be the unique solutions of duality formulas. This result is based on a simple characterization of infinitely divisible random vectors by a functional equation in which a difference operator appears. This operator is constructed by a variational method and compared to approaches involving chaos decompositions. We also obtain a related characterization of infinitely divisible random measures.  相似文献   

4.
We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy–Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy–Itô decomposition. As a corollary, the semi-martingale property is proved.  相似文献   

5.
We characterize the finite variation property for stationary increment mixed moving averages driven by infinitely divisible random measures. Such processes include fractional and moving average processes driven by Lévy processes, and also their mixtures. We establish two types of zero–one laws for the finite variation property. We also consider some examples to illustrate our results.  相似文献   

6.
We compute the bi-free max-convolution which is the operation on bivariate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bipartite two-faced pairs of Hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely divisible laws, their determination becomes in this way a classical analysis question.  相似文献   

7.
8.
The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Lévy process {(X(t),N(t)),t≥0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t),N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model.  相似文献   

9.
Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.  相似文献   

10.
Summary Let {X n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS n=Xn,1+…+X n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f n(x)∼ defined on a stationary sequence {X j∼, whereX n.f=fn(Xj). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.  相似文献   

11.
A class of multivariate distributions that are mixtures of the positive powers of a max-infinitely divisible distribution are studied. A subclass has the property that all weighted minima or maxima belong to a given location or scale family. By choosing appropriate parametric families for the mixing distribution and the distribution being mixed, families of multivariate copulas with a flexible dependence structure and with closed form cumulative distribution functions are obtained. Some dependence properties of the class, as well as some characterizations, are given. Conditions for max-infinite divisibility of multivariate distributions are obtained.  相似文献   

12.
Summary We describe geometric properties of {W>}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=–b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>} is given. We then analyse the position and size of clusters and individual connected components of {W>} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.The research of this author was partially supported by NSF grant DMS-9103962, and, during the period of revision, by grant DAAL03-92-6-0323 from the Army Research Office  相似文献   

13.
Summary As a first step in the development of a general theory of set-indexed martingales, we define predictability on a general space with respect to a filtration indexed by a lattice of sets. We prove a characterization of the predictable -algebra in terms of adapted and left-continuous processes without any form of topology for the index set. We then define a stopping set and show that it is a natural generalization of the stopping time; in particular, the predictable -algebra can be characterized by various stochastic intervals generated by stopping sets.Research supported by a grant from the Natural Sciences and Engineering Research Council of CanadaResearch partially done while the second author was visiting the University of Ottawa. He wishes to thank the Department of Mathematics for its hospitality  相似文献   

14.
A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given pssMp XX, that never hits 0, we construct a pssMp XX that hits 0 in a finite time. The latter can be viewed as XX conditioned to hit 0 in a finite time, and we prove that this conditioning is determined by the pre-minimum part of XX. Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously.  相似文献   

15.
Summary We consider a point process with the Polish phase space (X,X) and a system of -fields (x),xX, generated by on certain sets (x)X. We define predictability for random processes indexed byX and for random measures onX and prove the existence and uniqueness of predictable and dual predictable projections under a regularity condition on . ForX= 2 + and under monotonicity assumptions on the sets x we will identify the predictable projections of some simple processes as regular versions of certain martingales.  相似文献   

16.
Summary We consider increasing processes {X(t)t0} of classL, that is, increasing self-similar processes with inswpendent increments. Leth(t) be an increasing positive function on (0,) withh(0+)=0 andh()=. By virtue of the zero-one laws, there existsc (resp.C) [0,] such that lim inf (resp. lim sup)X(t)/h(t)=c (resp.C) a.s. both ast tends to 0 and ast tends to . We decide a necessary and sufficient condition for the existence ofh(t) withc orC=1 and explicitly constructh(t) in caseh(t) exists withc orC=1. Moreover, we give a criterion to classify functionsh(t) withc (orC)=0 andh(t) withc (orC)= in caseh(t) does not exist withc (orC)=1.  相似文献   

17.
We find an analytical condition characterising when the probability that a Lévy Process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero. We show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results. For each α > 0 we find an analytical condition equivalent to and as where X is a Lévy Process and T r the time it first leaves an interval of radius r  相似文献   

18.
19.
Some dimension results for super-Brownian motion   总被引:4,自引:0,他引:4  
Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y t)t0 be a super-Brownian motion on d (d2) andH be a Borel subset of d . We determine the Hausdorff Dimension of {t0; SuppY tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of SuppY t as a function of the dimension ofB.  相似文献   

20.
We prove that the upward ladder height subordinator H associated to a real valued Lévy process ξ has Laplace exponent φ that varies regularly at ∞ (respectively, at 0) if and only if the underlying Lévy process ξ satisfies Sina?ˇ's condition at 0 (respectively, at ∞). Sina?ˇ's condition for real valued Lévy processes is the continuous time analogue of Sina?ˇ's condition for random walks. We provide several criteria in terms of the characteristics of ξ to determine whether or not it satisfies Sina?ˇ's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of H, here obtained, and which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to Veraverbeke and Grübel.  相似文献   

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