Absorption time and absorption probabilities for a family of multidimensional gambler models

Pawel Lorek, Piotr Markowski

Abstract:

For a family of multidimensional gambler models we provide formulas for the winning
probabilities in terms of parameters of the system and for the distribution of a game duration in
terms of eigenvalues of underlying one-dimensional games. These formulas were known for the
one-dimensional case – initially proofs were purely analytical, recently probabilistic constructions
have been given. Concerning the game duration, in many cases our approach yields sample-path
constructions. We heavily exploit intertwining between (not necessarily) stochastic matrices (for
game duration results), a notion of Siegmund duality (for winning/ruin probabilities), and a notion
of Kronecker products.