In previous papers, evolution of dependence and ageing, for vectors of non-negative random variables, have been separately considered. Some analogies between the two evolutions emerge however in those studies. In the present paper, we propose a unified approach, based on semigroup arguments, explaining the origin of such analogies and relations among properties of stochastic dependence and ageing.
We establish the optimal quantization problem for probabilities under constrained Rényi-α-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases α=0 (restricted codebook size) and α=1 (restricted Shannon entropy).