Abstract
Initiated around the year 2007, the Malliavin–Stein approach to probabilistic approximations combines Stein’s method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin–Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin–Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.
Highlights
1 Introduction and overview The Malliavin–Stein method for probabilistic approximations was initiated in the paper [64], with the aim of providing a quantitative counterpart to the central limit theorems for random variables living in the Wiener chaos of a general separable Gaussian field
As formally discussed in the sections to follow, the basic idea of the approach initiated in [64] is that, in order to assess the discrepancy between some target law (Normal or Gamma, for instance), and the distribution of a nonlinear functional of a Gaussian field, one can fruitfully apply infinitedimensional integration by parts formulae from the Malliavin calculus of variations [57, 66, 77, 78] to the general bounds associated with the so-called Stein’s method for probabilistic approximations [66, 23]
Malliavin in the path-breaking reference [56], the Malliavin calculus is an infinite-dimensional differential calculus, whose operators act on smooth nonlinear functionals of Gaussian fields
Summary
We briefly introduce the main ingredients of Stein’s method for normal approximations in dimension one. For every smooth mapping f : R → R; heuristically, it follows that, if X is a random variable such that the quantity E[Xf (X) − f (X)] is close to zero for a large class of test functions f , the distribution of X should be close to Gaussian. The fact that such a heuristic argument can be made rigorous and applied in a wide array of probabilistic models was the main discovery of Stein’s original contribution [92], where the foundations of Stein’s method were first laid.
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