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First, we consider a nonnegative homogeneous block tri-diagonal matrix andobtain its convergence parameter, where some results in the field of matrixanalytic method are extended to the case where block matrices have countablyinfinite dimension. Second, we apply our results to a multi-dimensional QBDprocess and obtain lower bounds for the directional asymptotic decay rates ofthe stationary distribution.

Convergence parameters of nonnegative block tri-diagonal matrices and their application to multi-dimensional QBD processes