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...