Let f (·, t) be the probability density function which represents the solution of Kac's equation at time t, with initial data f0, and let gσ be the Gaussian density with zero mean and variance σ 2 , σ 2 being the value of the second moment of f0. This is the first study which proves that the total variation distance between f (·, t) and gσ goes to zero, as t → +∞, with an exponential rate equal to −1/4. In the present paper, this fact is proved on the sole assumption that f0 has finite fourth moment and its Fourier transform ϕ0 satisfies |ϕ0(ξ)| = o(|ξ| −p ) as |ξ| → +∞, for some p > 0. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.

In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186–201] it is proved that the total variation distance between the solution f(⋅, t) of Kac’s equation and the Gaussian density (0, σ2) has an upper bound which goes to zero with an exponential rate equal to −1/4 as t→+∞. In the present paper, we determine a lower bound which decreases exponentially to zero with this same rate, provided that a suitable symmetrized form of f0 has nonzero fourth cumulant κ4. Moreover, we show that upper bounds like ̅Cδe−(1/4)tρδ(t) are valid for some ρδ vanishing at infinity when ∫ℝ|v|4+δf0(v) dv<+∞ for some δ in [0, 2[ and κ4=0. Generalizations of this statement are presented, together with some remarks about non-Gaussian initial conditions which yield the insuperable barrier of −1 for the rate of convergence

The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. The beginning of the story dates back to a pioneering work by Hilbert, who first formalized the concept of linearization of the collision operator and pointed out the importance of its eigenvalues with respect to a certain asymptotic behavior of the Boltzmann equation. Under really mild conditions on initial data -close to being necessary -and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1.1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form Ce Λ b t , Λ b being the least negative of the aforesaid eigenvalues and C a constant which depends only on a few simple numerical characteristics (e.g. moments) of the initial datum. The validity of this quantification was conjectured, about fifty years ago, in a paper by Henry P. McKean but, in spite of several attempts, the best answer known up to now consists in a bound with a rate which can be made arbitrarily close to Λ b , to the cost of the "explosion" of the constant * Supported in part by MIUR-2008MK3AFZ † Also affiliated with CNR-IMATI, Milano, Italy.C. Moreover, its deduction is subject to restrictive hypotheses on the initial datum, besides the Grad angular cutoff condition. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean. Our work represents in fact a confirmation of this analogy, since the techniques we develop here crucially rely on certain formulations of the central limit theorem. The proof of Theorem 1.1 starts by assuming the Grad angular cutoff and proceeds with these steps: 1) A new representation, in Theorem 1.2, for the solution of the Boltzmann equation as expectation of a random probability distribution of a weighted random sum of independent and identically distributed random vectors. 2) An upper bound for the total variation distance of interest expressed as sum of expectations of the total variation distance, between the aforesaid random probability distribution and the limiting Maxwellian law, over two appropriate events U and U c . 3) The proof that the probability of U approaches zero, as time goes to infinity, at an exponential rate equal to Λ b . 4) An extension of a classical Beurling inequality which, combined with newBerry-Esseen-like inequalities, leads to the validity of the desired exponential rate Λ b of decay also for the expectation over U c . Then, the conclusion can be extended to the case of weak cutoff hypothesis by a standard truncation argument. To complete this description of the paper, we mention the use of the aforesaid representation to characterize, in Theorem 1.3, the domain of attraction of the Maxwellian limit.Mathematics subj...

In Bayesian statistics, a continuity property of the posterior distribution with respect to the observable variable is crucial as it expresses well-posedness, i.e., stability with respect to errors in the measurement of data. Essentially, this requires to analyze the continuity of a probability kernel or, equivalently, of a conditional probability distribution with respect to the conditioning variable.Here, we tackle this problem from a theoretical point of view. Let (X, d X ) and (T, d T ) be two metric spaces, endowed with Borel σ-algebras X and T , respectively. Given a probability kernel π(•|•) : T × X → [0, 1], we provide general conditions ensuring the Lipschitz continuity of the mapping X ∋ x → π(•|x) ∈ P(T), when the space of probability measures P(T) is endowed with a metric arising within the optimal transport framework, such as a Wasserstein metric. The majority of our results give a solution in the case that X and T are finite-dimensional and the kernel is of the form π(dθ|x) = g(x, θ)π(dθ) for some suitable function g : X × T → [0, +∞) (i.e., a dominated kernel). In this setting, the Lipschitz constant enjoys an explicit upper bound in terms of Fisher-information functionals and weighted Poincaré constants, obtained by exploiting the dynamic formulation of the optimal transport. Moreover, some of our statements deal with kernels whose support is varying with x, which are less frequently analyzed in statistical theory. We also provide some results for more abstract settings, including infinite-dimensional spaces and non dominated kernels.Finally, we give some illustrations on noteworthy classes of probability kernels, and we apply the main results to improve on some open questions in Bayesian statistics, dealing with the approximation of posterior distributions by mixtures and posterior consistency.

In the setting of dominated statistical models, we provide conditions yielding strong continuity of the posterior distribution with respect to the observed data. We show some applications, with special focus on exponential models.2010 Mathematics Subject Classification. 62F15.