Transport inequalities for random point measures

Abstract

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary σ-finite intensity measure) always satisfies a universal transport-entropy inequality à la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [31], originally proved for Poisson random measures with finite mass.

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

“…More recently Reitzner introduced a version of the convex distance inequality [94], Bachmann and Peccati [11] used modified log-Sobolev inequalities due to Wu to obtain concentration results with focus on geometric functionals, an approach which was subsequently extended by Bachmann [10], Bachmann and Reitzner [12]. Nourdin, Peccati, and Yang [91] proved restricted hypercontractive for certain classes of functions, whereas Gozlan, Herry, Peccati [59] obtained transportation type inequalities.…”

“…Non-negative U -statistics. Another application of Proposition 4.18 is related to geometric functionals of the Poisson process, specifically certain non-negative Ustatistics, investigated recently by several authors [10,11,12,59]. For a measurable kernel h : X m → [0, ∞), symmetric under permutation of arguments, let us define…”

“…Let us remark that the references [11,12,59] provide also bounds on the left tail of U . It does not seem that such a bound can be easily recovered from the moment approach, since it relies heavily on an another property of Poisson U -statistics with non-negative kernels, namely an appropriate notion of convexity, which allows for an application of certain correlation inequalities [11] or the Poisson convex distance inequality [12,59]. It is an interesting question what moment estimates can be obtained under an additional convexity assumption.…”

A Benamou–Brenier formulation of martingale optimal transport

“…In the context of non-stationary random measures [GHP21] prove classical functional inequalities in the setting of random point measures. The article [EH15] initiated the investigation of synthetic curvature properties of the configuration space w.r.t.…”