Recognition: unknown
Convolution, cumulants and infinitesimal generators in the formal power series ring
Pith reviewed 2026-05-10 12:46 UTC · model grok-4.3
The pith
t-deformed convolution and cumulants on formal power series yield analogues of the law of large numbers and central limit theorem that recover classical probability at t equals negative one
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We extend finite free convolution and cumulants to formal power series by introducing 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. We show that t equals negative one recovers classical convolution at the level of moment generating functions. We derive explicit representation formulas for the infinitesimal generators of boxplus^t continuous semigroups, which for t equals d yield finite free generators and for t equals negative one relate to one-dimensional Lévy process generators.
What carries the argument
The t-deformed convolution operation boxplus^t and its associated t-deformed cumulants, which operate on formal power series, support semigroup structures, and admit explicit infinitesimal generator representations.
If this is right
- The t-deformed law of large numbers and central limit theorem hold in the same structural form as their classical counterparts.
- Setting the parameter t to negative one makes the convolution operation coincide with classical convolution on moment generating functions.
- The infinitesimal generators of boxplus^t semigroups have explicit formulas that describe their evolution.
- When t equals the dimension d the generator formulas specialize to those of finite free probability.
- When t equals negative one the generators match the form of Lévy-Khintchine representations for one-dimensional Lévy processes.
Where Pith is reading between the lines
- The single deformation parameter may serve as an interpolation tool between classical, free, and finite free notions of independence.
- The formal power series setting opens the possibility of applying these limit theorems and generators to generating functions arising in combinatorics rather than probability measures.
- Explicit t-deformed cumulants for standard distributions such as the exponential or semicircular law could provide computable families that bridge different probability theories.
- The semigroup generator formulas suggest a route to studying continuous-time processes equipped with t-deformed independence.
Load-bearing premise
The t-deformed convolution and cumulants are well-defined operations on the formal power series ring that support semigroup structures, limit theorems, and explicit generator representations without additional hidden assumptions on the coefficients or the deformation parameter.
What would settle it
A specific formal power series for which the derived t-deformed central limit theorem fails to hold, or a direct computation showing that the explicit generator formula does not reproduce the semigroup evolution for t equals negative one on a known Lévy process.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces t-deformed convolution and t-deformed cumulants as algebraic operations on the ring of formal power series. It establishes t-deformed analogues of the law of large numbers and central limit theorem, shows that the case t=-1 recovers classical convolution at the level of moment generating functions, and derives explicit representation formulas for the infinitesimal generators of ⊞^t-continuous semigroups, with special cases recovering finite free generators (t=d) and relating to Lévy-Khintchine generators (t=-1).
Significance. If the algebraic constructions and limit theorems hold, the work supplies a deformation parameter t that interpolates between classical, free, and finite free probability, with explicit semigroup and generator formulas that directly connect the deformed theory to classical Lévy processes. The coefficient-wise limits and product-formula semigroup property are strengths that could support further developments in noncommutative probability.
minor comments (3)
- The precise base ring for the formal power series (e.g., coefficients in ℝ[[x]] or ℂ[[x]], and any restrictions on the constant term) should be stated explicitly at the outset of the definitions section.
- Notation for the t-deformed convolution ⊞^t and the associated cumulants could be introduced in the introduction with a brief comparison table to classical, free, and finite free cases for reader orientation.
- In the generator formulas, the differentiation step with respect to the semigroup parameter should include a short remark confirming that the operation remains within the formal power series ring.
Simulated Author's Rebuttal
We thank the referee for their careful reading and positive assessment of the manuscript. The referee's summary accurately captures the main results on t-deformed convolution, cumulants, limit theorems, and infinitesimal generators. We appreciate the recommendation for minor revision and will prepare an updated version accordingly.
Circularity Check
No significant circularity identified
full rationale
The paper defines t-deformed convolution and cumulants directly as algebraic operations on the formal power series ring, with the semigroup property following immediately from the explicit deformed product formula. The t-deformed LLN and CLT are obtained by scaling the n-fold convolution power and taking coefficient-wise limits in the ring; the infinitesimal generators are obtained by differentiating the semigroup at the identity element. Special cases t=-1 and t=d are recovered by direct substitution into the same expressions. All steps are self-contained algebraic constructions with no fitted parameters, no self-referential reductions, and no load-bearing self-citations invoked to justify the core definitions or theorems.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math The ring of formal power series over a commutative ring admits the usual addition and multiplication making it a commutative algebra.
Reference graph
Works this paper leans on
-
[1]
Arizmendi and D
O. Arizmendi and D. Perales, Cumulants for finite free convolution , Journal of Combinatorial Theory, Series A 155 (2018), 244–266
2018
- [2]
-
[3]
02, 2650010
Katsunori Fujie, Regularity and convergence properties of finite free convol u- tions, International Journal of Mathematics 37 (2026), no. 02, 2650010
2026
-
[4]
Hasebe and H
T. Hasebe and H. Saigo, The monotone cumulants , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques47 (2011), no. 4, 1160–1170, Publisher: Institut Henri Poincar´ e
2011
-
[5]
2, 29–38
Zbigniew J Jurek, Remarks on the selfdecomposability and new examples , Demonstratio Mathematica 34 (2001), no. 2, 29–38. [6] , Background driving distribution functions and series repr esentations for log-gamma self-decomposable random variables , Theory of Probability & Its Applications 67 (2022), no. 1, 105–117
2001
-
[6]
2, 247–262
Zbigniew J Jurek and Wim Vervaat, An integral representation for selfde- composable banach space valued random variables , Zeitschrift f¨ ur Wahrschein- lichkeitstheorie und verwandte Gebiete 62 (1983), no. 2, 247–262
1983
-
[7]
68, Cam- bridge university press, 1999
Sato Ken-Iti, L´ evy processes and infinitely divisible distributions, vol. 68, Cam- bridge university press, 1999
1999
-
[8]
Roelof Koekoek, Peter A Lesky, and Ren´ e F Swarttouw, Hypergeometric or- thogonal polynomials , Hypergeometric Orthogonal Polynomials and Their q- Analogues, Springer, 2010, pp. 183–253
2010
- [9]
-
[10]
Lehner, Free Cumulants and Enumeration of Connected Partitions , Euro- pean Journal of Combinatorics 23 (2002), no
F. Lehner, Free Cumulants and Enumeration of Connected Partitions , Euro- pean Journal of Combinatorics 23 (2002), no. 8, 1025–1031
2002
- [11]
-
[12]
A. W. Marcus, D. A. Spielman, and N. Srivastava, Finite free convolutions of polynomials, Probability Theory and Related Fields 182 (2022), no. 3, 807–848
2022
-
[13]
16, 11642–11687
Andrei Mart ´ ınez-Finkelshtein, Rafael Morales, and D aniel Perales, Real roots of hypergeometric polynomials via finite free convolution , International Mathe- matics Research Notices 2024 (2024), no. 16, 11642–11687
2024
-
[14]
, Zeros of generalized hypergeometric polynomials via finite free con- volution: Applications to multiple orthogonality , Constructive Approximation (2025), 1–70
2025
-
[15]
13, Cambridge University Press, 2006
Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, vol. 13, Cambridge University Press, 2006. 26
2006
-
[16]
3, 323–346
Dan Voiculescu, Addition of certain non-commuting random variables , Journal of functional analysis 66 (1986), no. 3, 323–346
1986
-
[17]
29–Se pt
, Symmetries of some reduced free product c*-algebras , Operator Alge- bras and their Connections with Topology and Ergodic Theory : Proceedings of the OATE Conference held in Bu¸ steni, Romania, Aug. 29–Se pt. 9, 1983, Springer, 2006, pp. 556–588. Shuhei Tsujie Faculty of Education, Department of Mathematics, Hokkaido University of Educa- tion, 9 Hokumon...
1983
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.