In this paper we present some interesting results which follow from the celebrated determinant formulas for noncoincidence probabilities of Markov processes discovered by Karlin and McGregor. The first theorem is a determinant formula for the probability that a Markov jump process will avoid a certain finite set of points. From this theorem a simple solution of the moving boundary problem for certain types of Markov processes can be obtained. The other theorems deal with noncoincidence probabilities of sets of random walks which need not be identically distributed. These formulas have interesting applications, especially in the theory of queues.
"On the Karlin-McGregor theorem and applications." Ann. Appl. Probab. 7 (2) 314 - 325, May 1997. https://doi.org/10.1214/aoap/1034625333