Somewhat stochastic matrices

TL;DR

This paper generalizes the classical theorem on regular stochastic matrices to include matrices without sign restrictions, replacing the regularity condition with a new condition based on the -distance between columns.

Contribution

It introduces a new condition based on -distance that extends the regularity theorem to a broader class of stochastic matrices without sign restrictions.

Findings

Generalization of the regular stochastic matrix theorem.

Condition based on -distance between columns.

Maintains column sum equal to 1.

Abstract

The standard theorem for regular stochastic matrices is generalized to matrices with no sign restriction on the entries. The condition that column sums be equal to 1 is kept, but the regularity condition is replaced by a condition on the $\ell_1$-distances between columns.