  We consider UL (and LU) decompositions of the one-step transition probability
matrix of a random walk with state space the nonnegative integers, with the
condition that both upper and lower triangular matrices in the factorization
are also stochastic matrices. We give conditions on the free parameter of the
UL factorization in terms of certain continued fraction such that this
stochastic factorization is possible. By inverting the order of multiplication
(also known as a Darboux transformation) we get a new family of random walks
where it is possible the identify the spectral measures in terms of a Geronimus
transformation. The same can be done for the LU factorization but now without a
free parameter. Finally, we apply our results in two examples, the random walk
with constant transition probabilities and the random walk generated by the
Jacobi orthogonal polynomials. In both situations we give urn models associated
with all the random walks in question.
