In this work we study the Hausdorff dimension of measures whose weight distribution satisfies a markov non-homogeneous property. In particular, we prove that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the Tricot dimensions (packing dimension) equal upper Rényi dimensions. As an application we get a continuity pr...

We establish the optimal quantization problem for probabilities under constrained Rényi-α-entropy of the quantizers. We determine the optimal quantizers and the optimal quan-tization error of one-dimensional uniform distributions including the known special cases α = 0 (restricted codebook size) and α = 1 (restricted Shannon entropy).

We formulate two types of extensions of the large deviation theory initiated by Bahadur in a non-regular setting. One can be regarded as a bound of the point estimation, the other can be regarded as the limit of a bound of the interval estimation. Both coincide in the regular case, but do not necessarily coincide in a non-regular case. Using the limits of relative Rényi entropies, we derive the...

In the past few years, several entropy-based tests have been proposed for testing either single SNP association or gene-gene interaction. These tests are mainly based on Shannon entropy and have higher statistical power when compared to standard χ2 tests. In this paper, we extend some of these tests using a more generalized entropy definition, Rényi entropy, where Shannon entropy is a special c...

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in exactly solvable models including spin chains. We use Fisher-Hartwig formula to calculate entanglement entropy [as well as Rényi entropy] of large block of spins in the ground state of XY spin chain. In the end of the paper we announce our recent re...

Numerous entropy-type characteristics (functionals) generalizing Rényi entropy are widely used in mathematical statistics, physics, information theory, and signal processing for characterizing uncertainty in probability distributions and distribution identification problems. We consider estimators of some entropy (integral) functionals for discrete and continuous distributions based on the numb...

Using Rényi entropy, an alternative statistics to Tsallis one for nonextensive systems at equilibrium is discussed. We show that it is possible to have the q-exponential distribution function for equilibrium nonextensive systems having nonadditive energy but additive entropy. PACS : 02.50.-r, 05.20.-y, 05.70.-a

The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance.

An extension of the entropy power inequality to the form N r (X +Y ) ≥ N r (X) +N r (Y ) with arbitrary independent summands X and Y in R is obtained for the Rényi entropy and powers α ≥ (r + 1)/2.

A classical upper bound for quantum entropy is identified and illustrated, 0 ≤ Sq ≤ ln(eσ2/ 2~), involving the variance σ2 in phase space of the classical limit distribution of a given system. A fortiori, this further bounds the corresponding information-theoretical generalizations of the quantum entropy proposed by Rényi.