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Conference Papers Year : 2023

Steady-state Distributions of Somewhat Stochastic Matrices

Abstract

Somewhat stochastic matrices, S, are real square matrices whose individual row sum is 1 for each row. We define and develop an eigenvalue characterization of when Sk converges to a steady state distribution π, as k converges to ∞. This result generalizes previously known linear algebraic properties about n × n stochastic matrices to n × n somewhat stochastic matrices. As an application, an alternative method to calculate infinite-time gambler’s ruin on a finite state space using algebraic duality is described. Examples are presented. When the eigenvalues of certain classes of matrices are known, then the k or less steps gambler’s ruin probabilities can often be determined in terms of these eigenvalues using Siegmund duality. A corresponding approach to determine the absorbing probabilities in an infinite state space setting is still being explored.
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Dates and versions

hal-04389349 , version 1 (11-01-2024)

Licence

Public Domain

Identifiers

  • HAL Id : hal-04389349 , version 1

Cite

Alain Krinik, Gerardo Rubino, David Beecher, Heba Ayeda. Steady-state Distributions of Somewhat Stochastic Matrices. 2023 - 7th Conference on Stochastic Models, Institute of Mathematics, Polish Academy of Sciences, May 2023, Będlewo, Poland. pp.1-66. ⟨hal-04389349⟩
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