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

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.