Convolution, cumulants and infinitesimal generators in the formal power series ring

TL;DR

This paper generalizes convolution and cumulants to formal power series, establishing $t$-deformed probability laws, and explores their infinitesimal generators, connecting classical, free, and finite free probability theories.

Contribution

It introduces $t$-deformed convolution and cumulants in formal power series, extending probability concepts and deriving explicit formulas for their infinitesimal generators.

Findings

$t$-deformed laws recover classical, free, and finite free probability cases.

Explicit formulas for infinitesimal generators of $oxplus^t$-semigroups are derived.

Connections between $oxplus^t$-convolution semigroups and Lévy processes are established.

Abstract

We extend the notions of finite free convolution and finite free cumulants to the setting of formal power series by introducing their natural analogues, namely $t$-deformed convolution and $t$-deformed cumulants. In this framework, we establish $t$-deformed analogues of the law of large numbers and the central limit theorem, revealing structural parallels with classical, free, and finite free probability theories. We show that the case $t=-1$ recovers classical convolution at the level of moment generating functions, thereby connecting the theory directly to classical probability. We further investigate the infinitesimal generators associated with $\boxplus^t$-continuous semigroups, deriving explicit representation formulas that clarify how these generators describe the infinitesimal evolution of the semigroup. In the case $t = d$, our results yield explicit formulas for finite free infinitesimal generators. In the case $t = -1$, we relate these generators to those of one-dimensional L\'{e}vy processes by identifying the corresponding terms in their representations. This establishes a direct connection between $\boxplus^t$-convolution semigroups and classical L\'{e}vy-Khintchine-type generators.