Stochastic Analysis for Poisson Point Processes

Stable limit theorems on the Poisson space

“…In particular, Proposition 2.2 gives a complete description of the carré du champ on the Poisson space. The reader can refer to the three monographs by Kingman [16]; Peccati & Reitzner [34]; and Last & Penrose [21] for more information about Poisson point processes.…”

“…The trendsetting work of Peccati, Solé, Taqqu & Utzet [32] extends the Malliavin-Stein approach beyond the scope of Gaussian fields to Poisson point processes. Despite being a very active field of research, the considered limit distributions are, most of the time, Gaussian [18,17,20,39,34,37,42,8,9,5] or, sometimes, Poisson [33] or Gamma [35]; to the best of our knowledge, prior to the present work, mixtures, have not been considered as limit distributions. The aim of this paper is to tackle this problem, by proving an array of new quantitative and stable limit theorems on the Poisson space, with a target distribution given either by a Gaussian mixture, that is the distribution of a centered Gaussian variable with random covariance; or a Poisson mixture, that is the distribution of a Poisson variable with random mean.…”

Malliavin–Stein method: a survey of some recent developments

“…The requirement that Z is Polish -together with several other assumptions adopted in the present section -is made in order to simplify the discussion; the reader is referred to [37,38] for statements and proofs in the most general setting. See also [50,51] for an exhaustive presentation of tools of stochastic analysis for functionals of Poisson processes, as well as [81] for a discussion of the relevance of variational techniques in the framework of modern stochastic geometry.…”

“…Since the publication of [64], the Malliavin-Stein method has generated several hundreds of papers, with ramifications in many (often unexpected) directions, including functional inequalities, random matrix theory, stochastic geometry, noncommutative probability and computer sciences. Many of hese developments largely exceed the scope of the present survey, and we invite the interested reader to consult the following general references (i)-(iii) for a more detailed presentation: (i) the webpage [1] is a constantly updated resource, listing all existing papers written around the Malliavin-Stein method; (ii) the monograph [66], written in 2012, contains a self-contained presentation of Malliavin calculus and Stein's method, as applied to functionals of general Gaussian fields, with specific emphasis on random variables belonging to a fixed Wiener chaos; (iii) the text [81] is a collection of surveys, containing an in-depth presentation of variational techniques on the Poisson space (including the Malliavin-Stein method), together with their application to asymptotic problems arising in stochastic geometry. The following more specific references (a)-(c) point to some recent developments that we find particularly exciting and ripe for further developments: (a) the papers [58,59,68,82,85,88,94] provide a representative overview of applications of Malliavin-Stein techniques to the study of nodal sets associated with Gaussian random fields on two-dimensional manifolds; (b) the papers [62,74] -and many of the reference therein -display a pervasive use of Malliavin-Stein techniques to determine rates of convergence in total variation in the Breuer-Major Theorem; (c) references [19,61] deal with the problem of tightness and functional convergence in the Breuer-Major theorem evoked at Point (b).…”