On the law of homogeneous stable functionals

Letemplier, Julien; Simon, Thomas

doi:10.1051/ps/2018028

Cited by 8 publications

(33 citation statements)

References 56 publications

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“…• When α → 0 ⇔ a → m, our main result shows immediately that where the convergence follows from (2.5) in [24]. This is again an extension of the case m = 1 where Z −a a d −→ Γ 1 as a → 0.…”

Section: 22supporting

confidence: 61%

“…which is valid for every b ∈ (a, 1), it is not a subordination formula. As in Corollary 4 (a) of [24], it can also be shown that X m,β is a multiplicative factor of X m,α for every 0 < β < α < m.…”

Section: 31mentioning

confidence: 76%

“…The fact that the two above infinite products are actually a.s. convergent is an easy consequence of the martingale convergence theorem -see the beginning of Section 2.1 in [24] and the references therein. The proof of the above theorem relies on the Mellin transform of f, which is by (1) the solution to a functional equation of first order given as (10) below.…”

Section: Theorem the Equation (1) Has A Density Solution If And Onlymentioning

confidence: 98%

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Density solutions to a class of integro-differential equations

Jedidi

Simon

Wang

2018

Journal of Mathematical Analysis and Applications

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Abstract. We consider the integro-differential equation I α 0+ f = x m f on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if m > α. These density solutions extend the class of generalized one-sided stable distributions introduced in [29] and more recently investigated in [27]. We study various analytical aspects of these densities, and we solve the open problems about infinite divisibility formulated in [27].

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Section: 22supporting

confidence: 61%

Section: 31mentioning

confidence: 76%

Section: Theorem the Equation (1) Has A Density Solution If And Onlymentioning

confidence: 98%

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Density solutions to a class of integro-differential equations

Jedidi

Simon

Wang

2018

Journal of Mathematical Analysis and Applications

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“…Finally, if lim u→∞ u −β φ φ(u) < ∞ then we may choose β = β φ and apply the above argument to conclude that ν T φ is moment determinate. 22…”

Section: Lemma 43mentioning

confidence: 99%

The log-Lévy moment problem via Berg–Urbanik semigroups

Patie

Vaidyanathan

2020

Studia Math.

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We consider the Stieltjes moment problem for the Berg-Urbanik semigroups which form a class of multiplicative convolution semigroups on R + that is in bijection with the set of Bernstein functions. In [8], Berg and Durán proved that the law of such semigroups is moment determinate (at least) up to time t = 2, and, for the Bernstein function φ(u) = u, Berg [4] made the striking observation that for time t > 2 the law of this semigroup is moment indeterminate. We extend these works by estimating the threshold time T φ ∈ [2, ∞] that it takes for the law of such Berg-Urbanik semigroups to transition from moment determinacy to moment indeterminacy in terms of simple properties of the underlying Bernstein function φ, such as its Blumenthal-Getoor index. One of the several strategies we implement to deal with the different cases relies on a non-classical Abelian type criterion for the moment problem, recently proved by the authors in [27]. To implement this approach we provide detailed information regarding distributional properties of the semigroup such as existence and smoothness of a density, and, the large asymptotic behavior for all t > 0 of this density along with its successive derivatives. In particular, these results, which are original in the Lévy processes literature, may be of independent interest. française de Belgique. Both authors are grateful for the hospitality of the Laboratoire de Mathématiques et de leurs applications de Pau, where this work was initiated. 1 Remark 2.1. Note that all complete Bernstein functions satisfy the property φ t ∈ B J for all t ∈ (0, 1). Indeed, writing H = {z ∈ C; Im z > 0} for the upper half-plane, we recall that a Bernstein function φ is said to be a complete if its Lévy measure µ has a completely monotone density, or equivalently if Im φ(z) 0 for all z ∈ H. Such functions are also sometimes called Pick or Nevanlinna functions in the complex analysis literature. If φ is a complete Bernstein function, then for t ∈ (0, 1) and z ∈ H, Im φ t (z) = Im e t(log |φ(z)|+i arg φ(z)) = e t log |φ(z)| Im e it arg φ(z) 0, and hence φ t is a complete Bernstein function, and in particular its Lévy measure has a non-increasing density. In particular u → (u + m) α is a complete Bernstein function, for any m 0, α ∈ (0, 1), and thus u → (u + m) αt is also a complete Bernstein function, for any t ∈ (0, 1). We refer to [29, Chapter 16] for abundant examples of complete Bernstein functions and to [29, Chapter 6] for further details on the theory of complete Bernstein functions; see also [14] for some interesting mappings related to complete Bernstein functions.Remark 2.2. We mention that for Item (4) Patie and Savov, see [24, Proposition 4.4], have given sufficient conditions for the ratio of Bernstein functions to remain a Bernstein function, see also Proposition 4.1 below for another set of sufficient conditions. 4

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“…The case of M = 2 was first treated in [55] and then studied in depth in [59] in the context of the Mellin transform of certain functionals of the stable Lévy process. We extended the theory to general M in [78].…”

Section: N = M −mentioning

confidence: 99%

A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications

Ostrovsky

2018

Rev. Math. Phys.

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Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of 1/ f noises, and calculations of inverse participation ratios in the Fyodorov-Bouchaud model.

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On the law of homogeneous stable functionals

Cited by 8 publications

References 56 publications

Density solutions to a class of integro-differential equations

Density solutions to a class of integro-differential equations

The log-Lévy moment problem via Berg–Urbanik semigroups

A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications

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