Papers › An extension of Krishnan's central limit theorem to the Brown-Thompson groups
An extension of Krishnan's central limit theorem to the Brown-Thompson groups
Valeriano Aiello
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We extend a central limit theorem, recently established for the Thompson group F=F₂ by Krishnan, to the Brown-Thompson groups Fₚ, where p is any integer greater than or equal to $2$. The non-commutative probability space considered is the group algebra ℂ[Fₚ], equipped with the canonical trace. The random variables in question are aₙ:= (xₙ + xₙ⁻¹)/√(2), where {xᵢ}_(i≥0) represents the standard family of infinite generators. Analogously to the case of F=F₂, it is established that the limit distribution of sₙ = (a₀ + …+ aₙ₋₁)/√(n) converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones's oriented subgroup F⃗, such a central limit theorem does not hold.
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