A stochastic cubical tensor is a three-mode array (or hypermatrix) with nonnegative entries, whose $1$-mode fibers (i.e., columns) sum up to $1$. Such tensors appear in certain higher-order Markov chains and random walks with memory, exactly as stochastic matrices describe classical discrete Markov chains and random walks.
The interest in higher-order stochastic processes is significantly growing in recent years as they are much better at modeling high dimensional data and nonlinear dynamics in numerous applications. However, fundamental questions such as the convergence of the process towards a limiting distribution and the uniqueness of such a limit are still not well understood and are the subject of a rich recent literature.
We introduce a set of ergodicity coefficients for stochastic cubical tensors, by extending certain definitions known in the matrix case to the tensor setting. The proposed coefficients yield new explicit formulas that
(a) guarantee the uniqueness of nonlinear Perron 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 various higher-order Markov chains to the stationary distribution.
Joint work with Francesco Tudisco.