• Corpus ID: 249209895

Quantization dimensions of compactly supported probability measures via R\'enyi dimensions

  title={Quantization dimensions of compactly supported probability measures via R\'enyi dimensions},
  author={Marc Kessebohmer and Aljoscha Niemann and Sanguo Zhu},
We provide a full picture of the upper quantization dimension in terms of the Rényi dimension, in that we prove that the upper quantization dimension of order r > 0 for an arbitrary compactly supported Borel probability measure ν is given by its Rényi dimension at the point qr where the Lq-spectrum of ν and the line through the origin with slope r intersect. In particular, this proves the continuity of r 7→ Dr(ν) as conjectured by Lindsay (2001). This viewpoint also sheds new light on the… 

Figures from this paper



Spectral dimensions of Krein-Feller operators in higher dimensions

A bstract . We study the spectral dimensions of Kre˘ın–Feller operators for finite Borel measures ν on the d -dimensional unit cube via a form approach. We introduce the notion of the spectral

Quantization and martingale couplings

  • B. JourdainG. Pagès
  • Mathematics
    Latin American Journal of Probability and Mathematical Statistics
  • 2022
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating

Lq spectra and Rényi dimensions of in-homogeneous self-similar measures

Let for i = 1, …, N be contracting similarities. Also, let (p1, …, pN, p) be a probability vector and let ν be a probability measure on with compact support. We show that there exists a unique

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact

$L^q$-spectrum of the Bernoulli convolution associated with the golden ratio

Based on the higher order self-similarity of the Bernoulli convolution measure for (p 5?1)=2 proposed by Strichartz et al, we derive a formula for the L q-spectrum, q > 0 of the measure. This formula

The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

Let be a family of contractive similitudes satisfying the open set condition. Let ν be a self-similar measure associated with . We study the quantization problem for the in-homogeneous self-similar

Asymptotics of the quantization errors for in-homogeneous self-similar measures supported on self-similar sets

We study the quantization for in-homogeneous self-similar measures µ supported on self-similar sets. Assuming the open set condition for the corresponding iterated function system, we prove the