Markov Property of Monotone Lévy Processes

Franz, Uwe; Muraki, Naofumi

doi:10.1142/9789812701503_0003

Cited by 13 publications

(9 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…and similarly, the anti-monotone convolution µ 1 µ 2 is defined as the probability measure on R such that F µ 1 µ 2 (z) = F µ 2 (F µ 1 (z)) , for z ∈ C + ; see [14].…”

Section: (C) Monotone Convolution and Monotone Cumulantsmentioning

confidence: 99%

On the non-classical infinite divisibility of power semicircle distributions

Arizmendi¹,

Pérez‐Abreu²

2010

COSA

View full text Add to dashboard Cite

Abstract. The family of power semicircle distributions defined as normalized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian distribution. A review of some results including a genesis and the so-called Poincaré's theorem is presented. The moments of these distributions are related to the super Catalan numbers and their Cauchy transforms in terms of hypergeometric functions are derived. Some members of this class of distributions play the role of the Gaussian distribution with respect to additive convolutions in non-commutative probability, such as the free, the monotone, the anti-monotone and the Boolean convolutions. The infinite divisibility of other power semicircle distributions with respect to these convolutions is studied using simple kurtosis arguments. A connection between kurtosis and the free divisibility indicator is found. It is shown that for the classical Gaussian distribution the free divisibility indicator is strictly less than 2.

show abstract

“…and similarly, the anti-monotone convolution µ 1 µ 2 is defined as the probability measure on R such that F µ 1 µ 2 (z) = F µ 2 (F µ 1 (z)) , for z ∈ C + ; see [14].…”

Section: (C) Monotone Convolution and Monotone Cumulantsmentioning

confidence: 99%

On the non-classical infinite divisibility of power semicircle distributions

Arizmendi¹,

Pérez‐Abreu²

2010

COSA

View full text Add to dashboard Cite

show abstract

“…For a comprehensive introduction to upsilon transformations we refer to Barndorff-Nielsen et al (2008). It is remarkable that in this case µ (s, ·) is not infinitely divisible in the classical sense, but according to Franz and Muraki (2004), µ (s, ·) is infinitely divisible with respect to the monotone convolution. In fact, such a distribution plays the role of the Gaussian distribution under this operation, i.e.…”

Section: A Class Of Lévy Measures Obtained By Lévy Mixingmentioning

confidence: 97%

On the class of distributions of subordinated Lévy processes and bases

Sauri

Veraart

2017

Stochastic Processes and their Applications

View full text Add to dashboard Cite

This article studies the class of distributions obtained by subordinating Lévy processes and Lévy bases by independent subordinators and meta-times. To do this we derive properties of a suitable mapping obtained via Lévy mixing. We show that our results can be used to solve the so-called recovery problem for general Lévy bases as well as for moving average processes which are driven by subordinated Lévy processes.

show abstract

“…Monotone convolution. The monotone convolution was defined in [15] and extended to unbounded measures in [10]. The monotone convolution of two probability measures µ 1 , µ 2 on R is defined as the probability measure…”

Section: Free Convolutionmentioning

confidence: 99%

Convergence of the fourth moment and infinite divisibility: quantitative estimates.

Arizmendi

Jaramillo

2014

Electron. Commun. Probab.

View full text Add to dashboard Cite

We give an estimate for the Kolmogorov distance between an infinitely divisible distribution (with mean zero and variance one) and the standard Gaussian distribution in terms of the difference between the fourth moment and 3. In a similar fashion we give an estimate for the Kolmogorov distance between a freely infinitely divisible distribution and the Semicircle distribution in terms of the difference between the fourth moment and 2.

show abstract

Markov Property of Monotone Lévy Processes

Cited by 13 publications

References 10 publications

On the non-classical infinite divisibility of power semicircle distributions

On the non-classical infinite divisibility of power semicircle distributions

On the class of distributions of subordinated Lévy processes and bases

Convergence of the fourth moment and infinite divisibility: quantitative estimates.

Contact Info

Product

Resources

About