Discrete chaotic calculus and covariance identities ∗

We show that for the binomial process (or Bernoulli random walk) the orthogonal functionals constructed in Kroeker [14] for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals. This allows to construct a chaotic calculus based on gradient and divergence operators and structure equations, and to establish a Clark representation formula. As an application we obtain simple infinite dimensional proofs of covariance identities on the discrete cube. AMS Subject Classification: 60H05, 60E15, 05E35.