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This paper studies the subgeometric convergence of the stationarydistribution in taking the infinite-level limit of a finite-level M/G/1-typeMarkov chain, that is, in letting the upper boundary level go to infinity. Thisstudy is performed through the fundamental deviation matrix, which is ablock-decomposition-friendly solution for the Poisson equation of the deviationmatrix. The fundamental deviation matrix yields a difference formula for therespective stationary distributions of the finite-level chain and thecorresponding infinite-level chain. The difference formula plays a crucial rolein deriving the main result of this paper: a subgeometric convergence formulafor the infinite-level limit of the stationary distribution of the finite-levelchain.

A subgeometric convergence formula for finite-level M/G/1-type Markov  chains: via a block-decomposition-friendly solution for the Poisson equation  of deviation matrix