The Local Coefficient of Ergodicity of a Nonnegative Matrix

Abstract

The local coefficient of ergodicity $\tau(T,Y',w)$ of a nonnegative column-allowable matrix T at a fixed positive vector Y is defined as the supremum of d(X'T,Y'T)/d(X',Y') for X not colinear to Y and d(X',Y')\leq w$ (d is the projective distance in the positive quadrant). A near-closed-form expression is given for $\tau(T,Y',w)$. If T' is scrambling (i.e., no two rows of T' are orthogonal), then for any Y>0, $w < \infty$ we have $\tau(T,Y',w)<1$. When Y is a positive left eigenvector of T and Xo>0, these results can be used to prove the convergence in direction of X'oTp to Y'. Results are illustrated with a numerical example.