Abstract:
We introduce a differential calculus for linear operators represented by a family of finite signed measures, in particular, by stochastic processes. Such a calculus is based on the notions of g-derived operators and processes and g-integrating measures, g being a right-continuous nondecreasing function. Depending on the choice of g, this differential calculus works for non-smooth functions and under weak integrability conditions. For linear operators represented by stochastic processes, we provide a characterization criterion of g-differentiability in terms of characteristic functions of the random variables involved. Various illustrative examples are considered. Three kinds of applications will be discussed: In first place, we obtain exact values and sharp estimates for the total variation distance between binomial and Poisson distributions with the same mean . We give a simple efficient algorithm, whose complexity order is, to compute exact values. In doing this, the roots of the second Krawtchouk and Charlier polynomials play a fundamental role. In second place, we give sharp upper and lower bounds for the median of the distribution, thus providing an immediate proof of two conjectures by Chen and Rubin referring to the median of the Poisson distribution. We also prove and complete certain conjectures by Ramanujan concerning the remainder of a finite rational expansion of the exponential function. Finally, we consider the sequence of the Stieltjes constants appearing in the Laurent expansion of the Hurwitz and Riemann zeta functions. We obtain explicit upper bounds for useful for large values of n. On the other hand, we approximate each constant by means of a finite sum involving Bernoulli numbers. Such an approximation has a quasi-geometric rate of convergence.