Higher-order ergodicity coefficients
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Ergodicity coefficients for stochastic matrices provide valuable upper bounds for the magnitude of subdominant eigenvalues, allow to bound the convergence rate of methods for computing the stationary distribution and can be used to estimate the sensitivity of the stationary distribution to changes in the matrix. In this work we extend an important class of ergodicity coefficients defined in terms of the 1-norm to the setting of stochastic tensors. We show that the proposed higher-order ergodicity coefficients provide new explicit formulas that (a) guarantee the uniqueness of Perron Z-eigenvectors of stochastic tensors, (b) provide bounds on the sensitivity of such eigenvectors with respect to changes in the tensor and (c) ensure the convergence of different types of higher-order power methods to the stationary distribution of higher-order and vertex-reinforced Markov chains.
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