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

## TL;DR

This paper derives formulas for winning probabilities and game duration distributions in a family of multidimensional gambler models, extending known results from one-dimensional cases using advanced probabilistic and matrix techniques.

## Contribution

It introduces new formulas and sample-path constructions for multidimensional gambler models, leveraging intertwining matrices, Siegmund duality, and Kronecker products.

## Key findings

- Explicit formulas for winning probabilities in multidimensional models
- Distribution of game duration expressed via eigenvalues of related one-dimensional games
- Sample-path constructions for game duration in many cases

## 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 game duration (in terms of eigenvalues of underlying one-dimensional games). These formulas were known for one-dimensional case - initially proofs were purely analytical, later probabilistic construction has been given. Concerning the game duration, in many cases our approach yields sample-path constructions. We heavily exploit intertwining between (not necessary) stochastic matrices (for game duration results), a notion of Siegmund duality (for winning/ruin probabilities), and a notion of Kronecker products.