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