Karlin and McGregor calculated the coincidence probabilities for $n$ particles independently executing a Markov process of a certain class. This note extends their result by allowing the particles to have different stopping times. Applied to a one-dimensional clustering problem, this gives a new solution computationally simpler than previous ones.
"A Generalization of the Karlin-McGregor Theorem on Coincidence Probabilities and an Application to Clustering." Ann. Probab. 5 (5) 814 - 817, October, 1977. https://doi.org/10.1214/aop/1176995725