Abstract-One of the difficulties in calculating the capacity of certain Poisson channels is that H(λ), the entropy of the Poisson distribution with mean λ, is not available in a simple form. In this work we derive upper and lower bounds for H(λ) that are asymptotically tight and easy to compute. The derivation of such bounds involves only simple probabilistic and analytic tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on the relative entropy D(n, p) between a binomial and a Poisson, thus refining the work of Harremoës and Ruzankin (2004). Bounds on the entropy of the binomial also follow easily.
We obtain closed form expressions for convolutions of scale transformations within a certain subset of Appell polynomials. This subset contains the Bernoulli, Apostol-Euler, and Cauchy polynomials, as well as various kinds of their generalizations, among others. We give a unified approach mainly based on a probabilistic generalization of the Stirling numbers of the second kind. Different illustrative examples, including reformulations of convolution identities already known in the literature, are discussed in detail.
We evaluate each Stieltjes constant γ m as a finite sum involving the first m + 1 Bernoulli numbers B k and the first m + 1 derivatives (−1) k α k of the alternating zeta function at the point 1. In turn, we compute each α k in an efficient way by means of a series with geometric rate 1/3. The coefficients of such a series are bounded and slowly decrease to zero. The computational significance of the preceding results is also discussed.
In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.
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