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We consider UL and LU stochastic factorizations of the transition probabilitymatrix of a random walk on the integers, which is a doubly infinite tridiagonalstochastic Jacobi matrix. We give conditions on the free parameter of bothfactorizations in terms of certain continued fractions such that thisstochastic factorization is always possible. By inverting the order of thefactors (also known as a Darboux transformation) we get new families of randomwalks on the integers. We identify the spectral matrices associated with theseDarboux transformations (in both cases) which are basically conjugations by amatrix polynomial of degree one of a Geronimus transformation of the originalspectral matrix. Finally, we apply our results to the random walk with constanttransition probabilities with or without an attractive or repulsive force.

The spectral matrices associated with the stochastic Darboux transformations of random walks on the integers