On Structured Random Matrices Defined by Matrix Substitutions

Manuel L. Esquível () and Nadezhda P. Krasii
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Manuel L. Esquível: Department of Mathematics, FCT NOVA, and CMA New University of Lisbon, 2829-516 Caparica, Portugal
Nadezhda P. Krasii: Department of Higher Mathematics, Faculty of Informatics and Computer Engineering, Don State Technical University, 344003 Rostov-on-Don, Russia

Mathematics, 2023, vol. 11, issue 11, 1-29

Abstract: The structure of the random matrices introduced in this work is given by deterministic matrices—the skeletons of the random matrices—built with an algorithm of matrix substitutions with entries in a finite field of integers modulo some prime number, akin to the algorithm of one dimensional automatic sequences. A random matrix has the structure of a given skeleton if to the same number of an entry of the skeleton, in the finite field, it corresponds a random variable having, at least, as its expected value the correspondent value of the number in the finite field. Affine matrix substitutions are introduced and fixed point theorems are proven that allow the consideration of steady states of the structure which are essential for an efficient observation. For some more restricted classes of structured random matrices the parameter estimation of the entries is addressed, as well as the convergence in law and also some aspects of the spectral analysis of the random operators associated with the random matrix. Finally, aiming at possible applications, it is shown that there is a procedure to associate a canonical random surface to every random structured matrix of a certain class.

Keywords: random fields; random Matrices; random linear operators; notions of recurrence; symbolic dynamics; automata sequences (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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