The Rényi entropy is a generalization of Shannon entropy to a one-parameter family of entropies. Tsallis entropy too is a generalization of Shannon entropy. The measure for Tsallis entropy is non-logarithmic. After the introduction of Shannon entropy , the conditional Shannon entropy was derived and its properties became known. Also, for Tsallis entropy, the conditional entropy was introduced a...

Two maximization problems of Rényi entropy rate are investigated: the maximization over all stochastic processes whose marginals satisfy a linear constraint, and the Burg-like maximization over all stochastic processes whose autocovariance function begins with some given values. The solutions are related to the solutions to the analogous maximization problems of Shannon entropy rate.

Csiszár’s forward -cutoff rate (given a fixed 0) for a discrete source is defined as the smallest number such that for every , there exists a sequence of fixed-length codes of rate with probability of error asymptotically vanishing as . For a discrete memoryless source (DMS), the forward -cutoff rate is shown by Csiszár [6] to be equal to the source Rényi entropy. An analogous concept of revers...

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

Rényi entropy dimension describes the rate of growth of coding cost in the process of lossy data compression in the case of exponential dependence between the code length and the cost of coding. In this paper we generalize the Csiszár estimation of the Rényi entropy dimension of the mixture of measures for the case of general probability metric space. This result determines the cost of encoding...

Rényi entropy of order α is a general measure of entropy. In this paper we derive estimations for the Rényi entropy of the mixture of sources in terms of the entropy of the single sources. These relations allow to compute the Rényi entropy dimension of arbitrary order of a mixture of measures. The key for obtaining these results is our new definition of the weighted Rényi entropy. It is shown t...

A task is randomly drawn from a finite set of tasks and is described using a fixed number of bits. All the tasks that share its description must be performed. Upper and lower bounds on the minimum ρ-th moment of the number of performed tasks are derived. The key is an analog of the Kraft Inequality for partitions of finite sets. When a sequence of tasks is produced by a source of a given Rényi ...

We consider the Rényi entropy function for weakly ψ-mixing systems. The first main result proves existence and regularity properties. The second main result of the paper is to get the decay rate for the large deviation of the return time to cylinder sets. We show it to be exponential with a rate given by the Rényi entropy function. Finally we also obtain bounds for the free energy.