Somewhat Stochastic Matrices

The standard theorem for stochastic matrices with positive entries is generalized to matrices with no sign restriction on the entries. The condition that column sums be equal to 1 is kept, but the positivity condition is replaced by a condition on the distances between columns

What chance-credence norms should be

We show a somewhat surprising result concerning the relationship between the Principal Principle and its allegedly generalized form. Then, we formulate a few desiderata concerning chance-credence norms and argue that none of the norms widely discussed in the literature satisfies all of them. We suggest that the New Principle comes out as the best contender

Steady-state Distributions of Somewhat Stochastic Matrices

International audienceSomewhat 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

Steady-state Distributions of Somewhat Stochastic Matrices

International audienceSomewhat 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

Steady-state Distributions of Somewhat Stochastic Matrices

International audienceSomewhat 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