Two remarks on generalized entropy power inequalities

This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed log-concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on

Topological genericity of nowhere differentiable functions in the disc algebra

In this paper we introduce a class of functions contained in the disc algebra A(D). We study functions f ∈ A1(D) which have the property that the continuous periodic function u = Ref{pipe}T, where T is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function f ∈ A(D) has the above property. Afterwards, we strengthen this result by proving that, generically, for every function f ∈ A(D), both continuous periodic functions u = Ref{pipe}T and ũ = Imf{pipe}T are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire's Category Theorem. © 2014 Springer Basel

Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

In this paper we examine functions in the disc algebra A(D) and the polydisc algebra A(DI), where I is a finite or countably infinite set. We prove that, generically, for every f∈A(D) the continuous periodic functions u=Ref|T and u~=Imf|T are nowhere differentiable on the unit circle T. Afterwards, we generalize this result by proving that, generically, for every f∈A(DI), where I is as above, the continuous periodic functions u=Ref|TI and u~=Imf|TI have no directional derivatives at any point of TI and every direction v∈RI with {norm of matrix}v{norm of matrix}∞=1. Finally, we describe how our proofs can be modified to give similar results for nowhere Hölder functions in these algebras. © 2014 Elsevier Inc

On the entropy and information of Gaussian mixtures

Publication status: PublishedAbstractWe establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if are independent scalar Gaussian mixtures, then the entropy of is concave in , thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix‐valued function acting on densities of mixtures in . As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.</jats:p

Sharp Rosenthal-type inequalities for mixtures and log-concave variables

Funder: Trinity College, CambridgeWe obtain Rosenthal‐type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremizers in log‐concave settings when the moments of summands are individually constrained