On Stochastic Dominance of Nilpotent Operators

Abstract

In this paper, motivated by Nica and Speicher [Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, Vol. 335 (Cambridge University Press, 2006)] and Kubo and Kuo [MRM-factors for the probability measures in the Meixner class, Infin. Dimens. Anal. Quantum Probab. Relat. Top.13 (2010) 525–550], we characterize a particular nilpotent case of a truncated forward shift operator by applying the technique of the random walks with repeated reflections and associated renewal equations. We also establish a stochastic order relationship by applying the crossing criteria.

Free W*-Dynamical Systems From p-Adic Number Fields and the Euler Totient Function

“…For useful themes from harmonic analysis of number fields, both commutative and noncommutative, see [23][24][25][26]. Some related themes from mathematical physics are found in [17,27,28]. In this section, we introduce basic definitions and backgrounds of the paper.…”

“…These studies are all motivated by well-known number-theoretic results (e.g., [13][14][15][16][17]) with help of free probability techniques (e.g., [8,11,12]). …”