LU-factorization and probability
Résumé
Our initial motivation was to understand links beetween $\mathtt{WH}$-factorizations for random walks and $\mathtt{LU}$-factorizations for Markov chains has interpreated by Grassman \cite{Grassmann87}. Actually, first ones are particular cases of second ones, up to Fourier transforms. We produce a new proof of $\mathtt{LU}$-factorizations which is valid for any Markov chain with a denumerable state space equiped with a pre-order relation. Factors have nice interpretations in term of subordinated Markov chains. In particular, the $\mathtt{LU}$-factorization of the potential Matrice determine the law of the global minimum of the Markov chain. For any matrice, there are two mains $\mathtt{LU}$-factorizations according you decide to entry 1 in the diagonal of the first or of the second factor. When we factorize the generator of a general Markov chain, one factorization is always valid while the other require some hypothesis on the graph of the transition matrix. This disymetry come 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 that one; this allow us to play with transposition and so with time reversal. We study some particular cases as: skip-free Markov chains, random walks (with gives the $\mathtt{WH}$-factorization), reversible Markov chains (wich gives the Cholesky factorization). We use the $\mathtt{LU}$-factorization to compute invariant measures. We exhibit some pathologies: non-associativity, non-unicity which can be cured by smooth assumptions (as irreductibility).
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