Intrinsic Entropies of Log-Concave Distributions

DOI10.1109/TIT.2017.2757502zbMATH Open1390.94626arXiv1702.01203MaRDI QIDQ4566620

Publication date: 27 June 2018

Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)

Abstract: The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable

X

, the sequence of the

l f l o o r n h e t a f l o o r^{e x t t h}

intrinsic volumes of the typical sets of

X

in dimensions

n g e q 1

grows exponentially with a well-defined rate. We denote this rate by

h_{X} (h e t a)

, and call it the

h e t a^{e x t t h}

intrinsic entropy of

X

. We show that

h_{X} (h e t a)

is a continuous function of

h e t a

over the range

[0, 1]

, thereby providing a smooth interpolation between the values 0 and

h (X)

at the endpoints 0 and 1, respectively.

Full work available at URL: https://arxiv.org/abs/1702.01203

Mathematics Subject Classification ID

Measures of information, entropy (94A17)