1.
Existence and spatial limit theorems for lattice and continuum particle systems - Penrose, Mathew D.
We give a general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site. We allow for jump events at a site that affect the state of its neighbours. We give a law of large numbers and functional central limit theorem for additive set functions taken over an increasing family of subcubes of Zd. We discuss application to marked spatial point processes with births, deaths and jumps of particles, in particular examples such as continuum and lattice ballistic deposition and a sequential model for random loose sphere packing.
2.
On exchangeable random variables and the statistics of large graphs and hypergraphs - Austin, Tim
De Finettis classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1, 2, 3], Hoover [41, 42], Kallenberg [44] and Kingman [47]. On the other hand, such exchangeable laws were first related to questions from combinatorics in an independent analysis by Fremlin and Talagrand [29], and again more recently in Tao [62], where they appear as a natural proxy for the leading order statistics of colourings of large graphs or hypergraphs. Moreover, this relation appears implicitly...
3.
Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples - James, Lancelot F.; Roynette, Bernard; Yor, Marc
[start-list]*In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution ( : GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.*To a GGC variable, one may associate a unique Thorin measure. Let G a positive r.v. and ?t(G) (resp. ?t(1/G)) the Generalized Gamma Convolution with Thorin measure t-times the law of G (resp. the law of 1/G). In Section 2, we compare the laws of ?t(G) and ?t(1/G).*In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent...
4.
Stochastic differential equations with jumps - Bass, Richard F.
This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions.
5.
Exponential functionals of Brownian motion, II: Some related diffusion processes - Matsumoto, Hiroyuki; Yor, Marc
This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévys and Pitmans theorems are discussed.
6.
Determinantal Processes and Independence - Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L2(D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These...
7.
General state space Markov chains and MCMC algorithms - Roberts, Gareth O.; Rosenthal, Jeffrey S.
This paper surveys various results about Markov chains on general (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling...
8.
Large deviations and stochastic calculus for large random matrices - Guionnet, Alice
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random...
9.
Nonclassical stochastic flows and continuous products - Tsirelson, Boris
Contrary to the classical wisdom, processes with independent values (defined properly) are much more diverse than white noises combined with Poisson point processes, and product systems are much more diverse than Fock spaces.
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This text is a survey of recent progress in constructing and investigating nonclassical stochastic flows and continuous products of probability spaces and Hilbert spaces.
10.
On the Markov chain central limit theorem - Jones, Galin L.
The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.
11.
The Skorokhod embedding problem and its offspring - Ob?ój, Jan
This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, based on one-dimensional potential theory, is made. Some new facts which appeared in a natural way when different solutions were cross-examined, are reported. Azéma and Yors and Roots solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.
12.
Probability & incompressible Navier-Stokes equations: An overview of some recent developments - Waymire, Edward C.
This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) 343366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 50035040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.
13.
A survey of results on random random walks on finite groups - Hildebrand, Martin
A number of papers have examined various aspects of random random walks on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this work. This article also provides a number of exercises. Some exercises involve straightforward computations; others involve proving details in proofs or extending results proved in the article. This article also describes some problems for further study.
14.
Exchangeable pairs and Poisson approximation - Chatterjee, Sourav; Diaconis, Persi; Meckes, Elizabeth
This is a survey paper on Poisson approximation using Steins method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collectors problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.
16.
Conformal restriction and related questions - Werner, Wendelin
This survey paper is based on mini-courses given in July 2003 at the University of St-Andrews, and at the ICMS in Edinburgh. Its goal is to give a self-contained and heuristic overview of the recent (i.e. pre-2003) results concerning conformal restriction properties for random planar curves.
17.
Exponential functionals of Lévy processes - Bertoin, Jean; Yor, Marc
This text surveys properties and applications of the exponential functional ?0texp(??s)ds of real-valued Lévy processes ?=(?t,t?0).
18.
Controlled diffusion processes - Borkar, Vivek S.
This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.
19.
Random trees and applications - Le Gall, Jean-François
We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous theorem giving the convergence in distribution of the contour process of conditioned Galton-Watson trees towards the normalized Brownian excursion. We also briefly discuss applications to combinatorial trees. In the continuous setting, we use the formalism of real trees, which yields an elegant formulation of the convergence of rescaled discrete trees towards continuous objects. We explain the coding of real trees by functions, which is a continuous version of the...
20.
Exponential functionals of Brownian motion, I: Probability laws at fixed time - Matsumoto, Hiroyuki; Yor, Marc
This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned.
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