Statistics of Blocks in k-Divisible Non-Crossing Partitions

Arizmendi, Octavio

doi:10.37236/2431

Cited by 6 publications

(9 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If c = 0, then F µ (z 0 ) = z 0 , which is possible only when µ = δ 0 , a contradiction. Hence we get (1).…”

Section: Multiplicative Monotone Convolution Corresponds To the Operatormentioning

confidence: 91%

See 1 more Smart Citation

Classical scale mixtures of Boolean stable laws

Arizmendi

2015

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We study Boolean stable laws, b α,ρ , with stability index α and asymmetry parameter ρ. We show that the classical scale mixture of b α,ρ coincides with a free mixture and also a monotone mixture of b α,ρ . For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary.We prove that any scale mixture of b α,ρ is both classically and freely infinitely divisible for α ≤ 1/2 and also for some α > 1/2. Furthermore, we show the multiplicative infinite divisibility of b α,1 with respect classical, free and monotone convolutions.Scale mixtures of Boolean stable laws include some generalized beta distributions of second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and Möller.We use a representation of b α,1 as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and M lotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that b α,1 has free divisibility indicator 0 for 1/2 < α.

show abstract

“…If c = 0, then F µ (z 0 ) = z 0 , which is possible only when µ = δ 0 , a contradiction. Hence we get (1).…”

Section: Multiplicative Monotone Convolution Corresponds To the Operatormentioning

confidence: 91%

“…In particular, they count the number of s-divisible and (s+1)-equal non-crossing partitions. For details on s-divisible non-crossing partitions, see Edelman [19], Stanley [42], Arizmendi [1] and Armstrong [7].…”

Section: Free Bessel Lawsmentioning

confidence: 99%

Classical scale mixtures of Boolean stable laws

Arizmendi

2015

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The first ones were counted by Kreweras [28]. Also bijections between them have been given in [1] and [19] . Moreover in [5] an order has been given to (ii) makings the objects in (ii) and (iii) isomorphic as posets and generalized to other Coxeter groups.…”

Section: Motivating Examplementioning

confidence: 99%

k-divisible random variables in free probability

Arizmendi

2018

Advances in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k.First, we consider the combinatorial convolution * in the lattices N C of noncrossing partitions and N C k of k-divisible non-crossing partitions and show that convolving k times with the zeta-function in N C is equivalent to convolving once with the zeta-function in N C k . Furthermore, when x is k-divisible, we derive a formula for the free cumulants of x k in terms of the free cumulants of x, involving k-divisible non-crossing partitions.Second, we prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (on a and s) of degree k − 2 on s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results.Next, we show that free multiplicative convolution between a measure concentrated in the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed.Finally, we concentrate on free additive powers of k-symmetric distributions and prove that µ t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping µ → µ k . We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.

show abstract

“…In the particular case of uniform k-divisible non-crossing partitions, Theorem 1.1(ii,iii) has been obtained by Ortmann [34,Section 2.3]. Also, Arizmendi [3] obtained by combinatorial means closed formulas for the expected number of blocks of given size in k-divisible non-crossing partitions.…”

Section: Introductionmentioning

confidence: 96%

“…However, uniform non-crossing partitions have attracted less attention. Arizmendi [3] finds the expected number of blocks of given size for non-crossing partitions of [n] with certain constraints on the block sizes, Ortmann [34] shows that the distribution of a uniform random block in a uniform non-crossing partition P n of [n] converges to a geometric random variable of parameter 1/2 as n → ∞, and Curien and Kortchemski [12] have obtained limit theorems concerning the length of the longest chord of P n .…”

Section: Introductionmentioning

confidence: 99%

Simply Generated Non-Crossing Partitions

Kortchemski¹,

Marzouk²

2017

Combinator. Probab. Comp.

View full text Add to dashboard Cite

We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing partitions with constraints on their block sizes. Our main tool is a bijection between non-crossing partitions and plane trees, which maps such simply generated non-crossing partitions into simply generated trees so that blocks of size k are in correspondence with vertices of outdegree k. This allows us to obtain limit theorems concerning the block structure of simply generated non-crossing partitions. We apply our results in free probability by giving a simple formula relating the maximum of the support of a compactly supported probability measure on the real line in term of its free cumulants.

show abstract

Statistics of Blocks in k-Divisible Non-Crossing Partitions

Cited by 6 publications

References 21 publications

Classical scale mixtures of Boolean stable laws

Classical scale mixtures of Boolean stable laws

k-divisible random variables in free probability

Simply Generated Non-Crossing Partitions

Contact Info

Product

Resources

About