The Distribution of Polynomials in Monotone Independent Elements

2023/11/10 by Banna, Marwa, Tseng, Pei-Lun
#46L53 #60B20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2311.05979

Abstract

Building on the work of Arizmendi and Celestino (2021), we derive the *-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers α and β, we derive explicitly the *-distribution of p_α,β=αab + βba whenever a and b are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

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