Abstract
This work studies mixtures of probability measures on and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.
Highlights
A mixture of two probability measures μ0 and μ1 on Rn is for some parameter p ∈ [0, 1] the probability measure μ p defined by μ p := pμ0 + (1 − p)μ1
This work establishes criteria to check in a simple way under which a mixture of measures satisfies a Poincaré PI($) or log–Sobolev inequality LSI(α) with constants $ and α, respectively, provided that each of the components satisfies one
The investigation of mixtures can be found in many different applications, and the main results of this work may be useful to the investigation of asymmetric Kalman filter estimates [10], the study of asymmetric mixtures in Marine Biology [11], Econometrics [12], Gradient-quadratic and fixed-point iteration algorithms [7] and estimates of multivariate Gaussian mixtures [13]
Summary
A mixture of two probability measures μ0 and μ1 on Rn is for some parameter p ∈ [0, 1] the probability measure μ p defined by μ p := pμ0 + (1 − p)μ1. This work establishes criteria to check in a simple way under which a mixture of measures satisfies a Poincaré PI($) or log–Sobolev inequality LSI(α) with constants $ and α, respectively, provided that each of the components satisfies one. The estimates on the Poincaré and log–Sobolev constants hold for the case, when the χ2 -distance of μ0 and μ1 is bounded (see Label (5) for its definition) For this to be true, at least one of the measures μ0 and μ1 needs to be absolutely continuous to the other, which is a necessary condition for the mixture having connected support.
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