In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called tauepsilon ,phi algebra, a generalization of the more known tau algebra originally proposed by Bini and Capovani. We study the properties of eigenvalues and eigenvectors of the generator Tn,epsilon ,phi of the tauepsilon ,phi algebra. In particular, we derive the asymptotics for the outliers of Tn,epsilon ,phi and the associated eigenvectors; we obtain equations for the eigenvalues of Tn,epsilon ,phi, which provide also the eigenvectors of Tn,epsilon ,phi; and we compute the full eigendecomposition of Tn,epsilon ,phi in the specific case epsilonphi =1. We also present applications of our results in the context of queuing models, random walks, and diffusion processes, with a special attention to their implications in the study of wealth/income inequality and portfolio dynamics.
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