带漂移的布朗运动的极大游程与极小游程的联合分布

In this paper, we discuss the problem of extreme value for Brownian motion with positive drift. We obtain the joint distribution of the maximum excursion and the minimum excursion.     

The Distribution of the Sojourn Time for the Brownian Excursion

The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.     

生灭过程的Ito游程理论

当生灭过程不唯一,且附加的虚状态∞是"瞬时"且正则时,其轨道结构是异常复杂的.主要工作是利用Ito的游程理论来分析处理这种生灭过程,研究其轨道性质,并最终得到预解式.此预解式具有清楚的概率意义,能够直观地反映生灭过程的轨道结构.     

Time Change Approach to Generalized Excursion Measures, and Its Application to Limit Theorems

It is proved that generalized excursion measures can be constructed via time change of Itô’s Brownian excursion measure. A tightness-like condition on strings is introduced to prove a convergence theorem of generalized excursion measures. The convergence theorem is applied to obtain a conditional limit theorem, a kind of invariance principle where the limit is the Bessel meander.     

Ito游程理论与一般Markov链的平稳分布

在本文中,我们利用Ito游程理论给出一般Markov链的平稳分布,该公式包含了极小过程中断和含瞬时态的情形。最后我们给出一个含瞬时态的Markov链的计算例子。     

暂留Brown运动的极大游程与极小游程

对于暂留Ｂｒｏｗｎ运动,我们给出了极大游程与极小游程的若干定义,并且求出了极大游程（极小游程）的分布,以及极大游程（极小游程）与首达极大时（首达极小时）的联合分布．作为推论求出了首达极大时与首达极小时的分布。     

On the profile of random trees

Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer , respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic functions obtained by means of generating functions arising from the combinatorial setup and complex contour integration. Besides, an integral representation for the two-dimensional density of Brownian excursion local time is derived. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 421–451, 1997     

How K. Itô revolutionized the study of stochastic processes

The main facts of K. It?’s stochastic integration as well as excursion theory are presented, together with a number of applications.     

知识规律与规律的属性扰动

By employing the knowledge(R-element equivalence class)in one direction Srough sets and dual of one direction S-rough sets,the concept of knowledge law is given;the generation theorem of knowledge law,the excursion theorem of knowledge law,and the attribute disturbance discernible theorem of knowledge law are proposed.Knowledge law is a new characteristic of S-rough sets.     

Long Excursions of a Random Walk

In Csáki et al. ⁽¹⁾ and Révész and Willekens⁽⁹⁾ it was proved that the length of the longest excursion among the first n excursions of a plane random walk is nearly equal to the total sum of the lenghts of these excursions. In this paper several results are proved in the same spirit, for plane random walks and for random walks in higher dimensions.     

Itô’s excursion theory and its applications

A presentation of It?’s excursion theory for general Markov processes is given, with several applications to Brownian motion and related processes.     

Excursion decompositions for SLE and Watts' crossing formula

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if κ>4 and a.s. cutpoints if 4<κ<8. If κ>4, an appropriate version of SLE(κ) has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular SLE(κ) “away from its frontier”. For 4<κ<8, there is a two-sided analogue of this situation: a particular version of SLE(κ) has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this SLE “away from its cutpoints”. For κ=6, this overlaps Virág's results on “Brownian beads”. As a by-product of this construction, one proves Watts' formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.     

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本文中我们给出了生灭过程的轨道结构,指出轨道结构与构造理论之间的一一对应关系,并且利用Ito游程理论说明构造理论中各个参数的概率含义.     

An Extension of Vervaat's Transformation and Its Consequences

Vervaat⁽¹⁸⁾ proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s (0, 1), then we extend this result by determining a random time m _s such that when we exchange the pre-m _s-part and the post-m _s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m _s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

On the excursion theory for linear diffusions

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein’s representations that, e.g. the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss the Ornstein–Uhlenbeck processes.     

Excursion probabilities of isotropic and locally isotropic Gaussian random fields on manifolds

Let X = {X(p), p ∈ M} be a centered Gaussian random field, where M is a smooth Riemannian manifold. For a suitable compact subset \(D\subset M\), we obtain approximations to the excursion probabilities \(\mathbb {P}\{\sup _{p\in D} X(p) \ge u \}\), as \(u\to \infty \), for two cases: (i) X is smooth and isotropic; (ii) X is non-smooth and locally isotropic. For case (i), the expected Euler characteristic approximation is formulated explicitly; while for case (ii), it is shown that the asymptotics is similar to Pickands’ approximation on Euclidean space which involves Pickands’ constant and the volume of D. These extend the results in Cheng and Xiao (Bernoulli 22, 1113–1130 2016) from spheres to general Riemannian manifolds.     

Limit laws for embedded trees: Applications to the integrated superBrownian excursion

We study three families of labeled plane trees. In all these trees, the root is labeled 0 and the labels of two adjacent nodes differ by 0,1, or ?1. One part of the paper is devoted to enumerative results. For each family, and for all j ∈ ?, we obtain closed form expressions for the following three generating functions: the generating function of trees having no label larger than j; the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled j; and finally the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled at least, j. Strangely enough, all these series turn out to be algebraic, but we have no combinatorial intuition for this algebraicity. The other part of the paper is devoted to deriving limit laws from these enumerative results. In each of our families of trees, we endow the trees of size n with the uniform distribution and study the following random variables: M_n, the largest label occurring in a (random) tree; X_n(j), the number of nodes labeled j; and X(j), the number of nodes labeled j or more. We obtain limit laws for scaled versions of these random variables. Finally, we translate the above limit results into statements dealing with the integrated superBrownian excursion. In particular, we describe the law of the supremum of its support (thus recovering some earlier results obtained by Delmas) and the law of its distribution function at a given point. We also conjecture the law of its density (at a given point). © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

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??This paper establishes limsup type law of the iterated logarithm of the occupation measure, using the asymptotic equivalence relation between the occupation measure and the number of excursion process of a symmetric Cauchy process. Furthermore, by using the density theorem and the economic coverage method, it derives the exact Hausdorff measure for the range of a symmetric Cauchy process in \mathbb{R}.     

Asymptotic Properties of Ranked Heights in Brownian Excursions

Pitman and Yor^{(20, 21)} recently studied the distributions related to the ranked excursion heights of a Brownian bridge. In this paper, we study the asymptotic properties of the ranked heights of Brownian excursions. The heights of both high and low excursions are characterized by several integral tests and laws of the iterated logarithm. Our analysis relies on the distributions of the ranked excursion heights considered up to some random times.