José A. Adell

Sharp Bounds on the Entropy of the Poisson Law and Related Quantities

One of the difficulties in calculating the capacity of certain Poisson channels is that H(¿), the entropy of the Poisson distribution with mean ¿, is not available in a simple form. In this paper, we derive upper and lower bounds for H(¿) 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 Harremoe¿s and Ruzankin (2004). Bounds on the entropy of the binomial also follow easily.

Exact Kolmogorov and total variation distances between some familiar discrete distributions

We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters. In the Poisson case, such expressions are related with the Lambert function.

Asymptotic estimates for Stieltjes constants: a probabilistic approach

Let (γn)n ≥ 0 be the sequence of Stieltjes constants appearing in the Laurent expansion of the Riemann zeta function. We obtain explicit upper bounds for |γn|, whose order of magnitude is as n tends to infinity. To do this, we use a probabilistic approach based on a differential calculus for the gamma process.

Preservation of Moduli of Continuity for Bernstein-Type Operators

It is well known that many Bernstein-type operators preserve some properties of the functions on which they act, such as monotonicity, convexity, Lipschitz constants, etc. (cf. for instance [2]). In this paper, attention is focused on preservation of global smoothness, as measured by the usual moduli of continuity of first and second order. To the best of our knowledge, this problem has been studied by Kratz and Standtmuller in [11] for the first time. In this work the authors consider sequences (L n ) n≥1 of one-dimensional descrete operations satisfying certain moment assumptions and obtain estimates of the form

Sharp upper and lower bounds for the moments of Bernstein polynomials

\omega \left( {L_n f;h} \right) \leqslant c\omega \left( {f;h} \right),

Real inversion formulas with rates of convergence

(1) where ω(f;.) stands for the usual first modulus of continuity of function f and c is a positive constant which depends on the particular family of operations considered, but not upon f nor n and h. They provide the estimate c ≤ 4 in some important examples, such as Bernstein, Szasz and Baskakov operators.

Upper Estimates in Direct Inequalities for Bernstein-Type Operators

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let G s be the negative binomial distribution function with parameters α and a / s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s –p , where p = min(1, α ). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

RATES OF CONVERGENCE FOR THE ITERATES OF CESÀRO OPERATORS

We give upper and lower bounds for the moments and the uniform moments of Bernstein polynomials. Asymptotically, such bounds are best possible.