Yvik Swan

Stein’s method for comparison of univariate distributions

We propose a new general version of Steins method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type operator}. The resulting Stein identity highlights the unifying theme behind the literature on Steins method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions : normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Frechet and Gumbel.

Integration by parts and representation of information functionals

We introduce a new formalism for computing expectations of functionals of arbitrary random vectors, by using generalised integration by parts formulae. In doing so we extend recent representation formulae for the score function introduced in [19] and also provide a new proof of a central identity first discovered in [7]. We derive a representation for the standardised Fisher information of sums of i.i.d. random vectors which we use to provide rates of convergence in information theoretic central limit theorems (both in Fisher information distance and in relative entropy) and a Stein bound for Fisher information distance.

A bound on the 2-Wasserstein distance between linear combinations of independent random variables

Abstract We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of l 2 ( N ∗ ) . We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.

A stroll along the gamma

We provide the first in-depth study of the Laguerre interpolation scheme between an arbitrary probability measure and the gamma distribution. We propose new explicit representations for the Laguerre semigroup as well as a new intertwining relation. We use these results to prove a local De Bruijn identity which hold under minimal conditions. We obtain a new proof of the logarithmic Sobolev inequality for the gamma law with α≥1/2 as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with α≥1/2.

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

In this paper we propose tight upper and lower bounds for the Wasserstein distance between any two {{univariate continuous distributions}} with probability densities

On Hodges and Lehmann's "6/pi result"

p_1

Entropy and the fourth moment phenomenon

and

Discrete Stein characterizations and discrete information distances

p_2

A Matrix-Analytic approach to the N-player ruin problem

having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio

Distances between probability distributions via characteristic functions and biasing

p_1/p_2