evy-She er and IID-She er polynomials with applications to stochastic integrals

In [11] an unusual connection between orthogonal polynomials and martingales has been studied. There, all orthogonal She er polynomials, were linked to a unique L evy process, i.e., a continuous time stochastic process with stationary and independent increments. The connection between the polynomials and the L evy process is expressed by a martingale relation. As an application of these martingales we show that the Charlier polynomials are the counterparts for Itô’s integral with respect to a variant of the Poisson process of the customary powers. A simpler approach is possible when trying to obtain discrete time martingales from a She er set. We illustrate this by for example relating Krawtchouk polynomials to partial sums of Bernoulli IID variables. c © 1998 Elsevier Science B.V. All rights reserved. AMS classi cation: 60G42; 60J30; 60H05; 33C45; 11B83