## Algebraic central limit theorems in noncommutative probability

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##### Date

2022-01-24

##### Authors

Alahmade, Ayman

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University College Cork

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##### Abstract

Distributional symmetries and invariance principles in noncommutative probability theory provide sufficient conditions for the existence of central limit laws. In contrast to classical probability theory, there exist many different central limit laws for exchangeable sequences of noncommutative random variables and still little is known about their concrete form. This thesis goes one step further and investigates central limit laws for non-exchangeable spreadable sequences in the context of *-algebraic probability spaces. This provides first results on a new type of combinatorics underlying multivariate central limit theorems (CLTs).
The starting point of the thesis has been a quite simple family of spreadable sequences, which is parametrized by a unimodular complex parameter ω. Each sequence of this family is spreadable, but not exchangeable for ω different from ±1. Moreover, the sequences from this family provide CLTs, which interpolate between the normal distribution (ω = 1) and the symmetric Bernoulli distribution (ω = −1), but differ from q-Gaussian distributions (−1 < q < 1). An algebraic structure, which underlies the considered family, is identified and used to define so-called ‘ω-sequences of partial isometries’. These ω-sequences encode all information, as it is relevant for computations of *-algebraic CLTs. Explicit combinatorial formulas are established for CLTs associated to such ω-sequences, which involve the counting of oriented crossings of directed ordered pair partitions. The limiting distributions of certain multivariate CLTs associated to ω-sequences show some features as they are defining for ‘z-circular systems’ in the work of Mingo and Nica. This similarity, as well as the well-known relation between q-circular systems and q-semicircular systems (for −1 ≤ q ≤ 1), guides the introduction of ‘z-semicircular systems’ in this thesis. Finally, it is shown that the class of z-semicircular systems is stable under certain multivariate central limits. In other words, the moment formulas of z-semicircular systems are reproduced in large N-limit formulas of central limit type.

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##### Keywords

q-Gaussian random variables , Noncommutative probability theory , Central limit theorems , Quantum coin tosses , Distributional symmetries and invariance principles , Circular and semicircular systems

##### Citation

Alahmade, A. 2022. Algebraic central limit theorems in noncommutative probability. PhD Thesis, University College Cork.