L 1 ergodic behavior of non-negative kernels

Some results that have been obtained in the study of strongly and weakly ergodic behavior of non-homogeneous stochastic kernels are generalized to the case of non-negative kernels. The first generalization simply involves extending the definitions of weakly and strongly ergiodic behavior to the case of non-negative kernels and using the ergodic coefficient which was first defined for stochastic kernels by Dobrushin and extended to non-negative kernels by Blum and Reichaw. It happens that this straightforward extension excludes many cases of non-negative kernels which do exhibit a types of ergodic behavior. In order to study these cases a definition ofL ₁ weakly and strongly ergodic behavior is given in which normalizing by constants is allowed. Sufficient conditions for these types of ergodic behavior are given.