Local limit theorems in free probability theory

Wang, Jiun-Chau

doi:10.1214/09-aop505

Cited by 25 publications

(27 citation statements)

References 31 publications

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“…The results in [9] were extended to measures with unbounded support in [23], and a local almost everywhere convergence to freely stable densities was proved later in [18]. Our proofs in this paper rely on analytic subordination of free convolution and a technique of recasting weak convergence of measures into local uniform convergence of these subordination functions, where the latter has to do with the asymptotics of certain integral transforms (see Section 3).…”

Section: Introductionmentioning

confidence: 94%

Local limit theorems for multiplicative free convolutions

Anshelevich

Wang

Zhong

2014

Journal of Functional Analysis

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This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici-Pata bijection.

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Section: Introductionmentioning

confidence: 94%

Local limit theorems for multiplicative free convolutions

Anshelevich

Wang

Zhong

2014

Journal of Functional Analysis

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“…In particular, the preceding result generalizes the superconvergence for measures with finite variance in [16] to the entire free domain of attraction of the semicircular law.…”

Section: Stable Approximationmentioning

confidence: 54%

“…This phenomenon was called superconvergence in that paper. The assumption that X i be bounded was removed in subsequent work of the second author [16]. Even when the variables X i are not identically distributed, but are uniformly bounded, the support of S n was shown by Kargin [12] to converge to the interval [−2, 2] as n → ∞.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence to freely infinitely divisible distributions

2018

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We prove superconvergence results for all freely infinitely divisible distributions. Given a nondegenerate freely infinitely divisible distribution ν, let µn be a sequence of probability measures and let kn be a sequence of integers tending to infinity such that µ ⊞kn n converges weakly to ν. We show that the density dµ ⊞kn n /dx converges uniformly, as well as in all L p -norms for p > 1, to the density of ν except possibly in the neighborhood of one point. Applications include the global superconvergence to freely stable laws and that to free compound Poisson laws over the whole real line.

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“…Note that in the analytic theorems, the hypothesis on the distributions are identical to those in the usual CLT. On the other hand, in [9] and [26] the authors showed that the mode of convergence in the free CLT is actually much stronger than the classical convergence in distribution. In all these results, the limiting distribution is the semicircle law.…”

Section: Introductionmentioning

confidence: 99%

Two-state free Brownian motions

Anshelevich

2011

Journal of Functional Analysis

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In a two-state free probability space (A, ϕ, ψ), we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function R ϕ,ψ (z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states ϕ, ψ are normal, and ϕ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to ϕ = ψ), these processes only exist for finite time.

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Local limit theorems in free probability theory

Cited by 25 publications

References 31 publications

Local limit theorems for multiplicative free convolutions

Local limit theorems for multiplicative free convolutions

Superconvergence to freely infinitely divisible distributions

Two-state free Brownian motions

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