Laplace functional

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In mathematics — specifically, in probability theory — the Laplace functional of a metric probability space is an extended-real-valued function that is closely connected to the concentration of measure properties of the space.

Definition[edit]

Let (Xdμ) be a metric probability space; that is, let (Xd) be a metric space and let μ be a probability measure on the Borel sets of (Xd). The Laplace functional of (Xdμ) is the function

E_{(X, d, \mu)} \colon [0, +\infty) \to [0, +\infty]

defined by

E_{(X, d, \mu)}(\lambda) := \sup \left\{ \left. \int_{X} e^{\lambda f(x)} \, \mathrm{d} \mu(x) \right| f \colon X \to \mathbb{R} \text{ is bounded, 1-Lipschitz and has } \int_{X} f(x) \, \mathrm{d} \mu(x) = 0 \right\}.

Properties[edit]

The Laplace functional of (Xdμ) can be used to bound the concentration function of (Xdμ). Recall that the concentration function of (Xdμ) is defined for r > 0 by

\alpha_{(X, d, \mu)}(r) := \sup \{ 1 - \mu(A_{r}) \mid A \subseteq X \text{ and } \mu(A) \geq \tfrac{1}{2} \},

where

A_{r} := \{ x \in X \mid d(x, A) \leq r \}.

In this notation,

\alpha_{(X, d, \mu)}(r) \leq \inf_{\lambda \geq 0} e^{- \lambda r / 2} E_{(X, d, \mu)}(\lambda).

References[edit]

  • Ledoux, Michel (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: American Mathematical Society. pp. x+181. ISBN 0-8218-2864-9.  MR 1849347