LU-Factorization Versus Wiener-Hopf Factorization for Markov Chains

Abstract

Our initial motivation was to understand links between Wiener-Hopf factorizations for random walks and LU-factorizations for Markov chains as interpreted by Grassman (Eur. J. Oper. Res. 31(1):132---139, 1987). Actually, the first ones are particular cases of the second ones, up to Fourier transforms. To show this, we produce a new proof of LU-factorizations which is valid for any Markov chain with a denumerable state space equipped with a pre-order relation. Factors have nice interpretations in terms of subordinated Markov chains. In particular, the LU-factorization of the potential matrix determines the law of the global minimum of the Markov chain. For any matrix, there are two main LU-factorizations according as you decide to enter 1 in the diagonal of the first or of the second factor. When we factorize the generator of a Markov chain, one factorization is always valid while the other requires some hypothesis on the graph of the transition matrix. This dissymmetry comes from the fact that the class of sub-stochastic matrices is not stable under transposition. We generalize our work to the class of matrices with spectral radius less than one; this allows us to play with transposition and thus with time-reversal. We study some particular cases such as: skip-free Markov chains, random walks (this gives the WH-factorization), reversible Markov chains (this gives the Cholesky factorization). We use the LU-factorization to compute invariant measures. We present some pathologies: non-associativity, non-unicity; these can be cured by smooth assumptions (e.g. irreductibility).