On duality and the Spitzer–Pollaczek factorization for random walks

A new formulation of duality for pairs of stopping times is given. This formulation is constructive in that it provides a method for generating examples of dual times. We also use it to form the basis for a direct sample path proof of the Spitzer–Pollaczek factorization associated with a dual pair. The Spitzer–Pollaczek factorization relates, in a single expression, the distributions of a dual pair of times and the distribution of a random walk at each of these times. The accepted probabilistic derivation introduces an independent geometric time. The direct approach here omits this step and in doing so allows a separate treatment of the stopping time and the stopped random walk distributions and provides clear interpretations for the identities that arise. This novel look at duality makes clear further generalizations of the Spitzer–Pollaczek factorization which must hold and we conclude by proving a matrix factorization associated with a Markov-modulated random walk on Rd.