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One of the difficulties in calculating the capacity of certain Poisson channels is that <i>H</i>(¿), the entropy of the Poisson distribution with mean <i>¿</i>, is not available in a simple form. In this paper, we derive upper and lower bounds for <i>H</i>(¿) that are asymptotically tight and easy to compute. The derivation of such bounds involves only… (More)
In this paper, we consider the sequence (θn)n≥0 solving the Ramanujan equation en 2 = n ∑ k=0 nk k! + nn n! (θn − 1), n = 0, 1, . . . . The three main achievements are the following. We introduce a continuous– time extension θ(t) of θn and show its close connections with the medians λn of the Γ(n+1, 1) distributions and the Charlier polynomials. We give… (More)
1. INTRODUCTION. In his 1974 book entitled " The Life and Times of the Central Limit Theorem, " Adams  describes this theorem as " one of the most remarkable results in all of mathematics " and " a dominating personality in the world of probability and statistics. " More than three decades later, his description is not only still pertinent but has also… (More)
We obtain explicit upper estimates in direct inequalities with respect to the usual sup-norm distance for Bernstein-type operators. Our approach combines analytical and probabilistic techniques based on representations of the operators in terms of stochastic processes. We illustrate our results by considering some classical families of operators, such as… (More)
We obtain the best possible constants in preservation inequalities concerning the usual first modulus of continuity for the classical Szász–Mirakyan operator. The probabilistic representation of this operator in terms of the standard Poisson process is used.