Ergodic behavior for Markov chains can be determined by studying the properties of the corresponding sequence of stochastic transition kernels. Dobrushin's ergodic coefficient has been useful for this purpose. In this paper we define pointwise strongly and weakly ergodic behavior for sequences of nonnegative kernels and use Dobrushin's ergodic coefficient to give sufficient conditions for these two types of behavior. Applications are given to sequential probability ratio tests.
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Madsen et al. (1973) studied this question.
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