Stochastic LU factorizations, Darboux transformations and urn models

@article{Grnbaum2018StochasticLF,
  title={Stochastic LU factorizations, Darboux transformations and urn models},
  author={F. Alberto Gr{\"u}nbaum and Manuel Dom{\'i}nguez de la Iglesia},
  journal={J. Appl. Probab.},
  year={2018},
  volume={55},
  pages={862-886}
}
We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also… 
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