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Thomas Simon

Publications and source records attributed to Thomas Simon.

At least 37 records · Page 2Linked to original sources
arXiv2019

Fractional extreme distributions

We consider three classes of linear differential equations on distribution functions, with a fractional order $α\in [0,1].$ The integer case $α=1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $α-$stable subordinator. From the analytical viewpoint, this law is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case. By the stochastic representation, we can derive several analytical properties for the latter special functions, extending known features of the classical Mittag-Leffler function, and dealing with monotonicity, complete monotonicity, infinite divisibility, asymptotic behaviour at infinity, uniform hyperbolic bounds.

math.PR↗
arXiv2018

On a moment problem related to Bernstein functions

We give a simple proof of the moment-indeterminacy of the sequence $(n!)^t$ for $t > 2,$ using Lin's condition. Under a logarithmic self-decomposability assumption, the method conveys to power sequences defined as the rising factorials of a given Bernstein function, and to more general infinitely divisible moment sequences. We also provide a very short proof of the infinite divisibility of all the moment sequences recently investigated in Lin (2017), including Fuss-Catalan's.

math.PR↗
arXiv2018

Cramér's estimate for stable processes with power drift

We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.

math.PR↗
arXiv2018

Some properties of the free stable distributions

We investigate certain analytical properties of the free $α-$stable densities on the line. We prove that they are all classically infinitely divisible when $α\le 1$, and that they belong to the extended Thorin class when $α\leq 3/4.$ The Lévy measure is explicitly computed for $α=1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $α> 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.

math.PR↗
arXiv2017

On the law of homogeneous stable functionals

Let ${\mathcal A}$ be the ${\mathcal L}^q-$functional of a stable Lévy process starting from one and killed when crossing zero. We observe that ${\mathcal A}$ can be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transform, and an explicit computation on perpetuities of hypergeometric Lévy processes previously obtained by Kuznetsov and Pardo. This representation allows to retrieve several factorizations previously obtained by various authors, and also to derive new ones. We emphasize the connections between ${\mathcal A}$ and more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties of ${\mathcal A}$ related to infinite divisibility, self-decomposability, and the generalized Gamma convolutions.

math.PR↗
arXiv2017

Density solutions to a class of integro-differential equations

We consider the integro-differential equation ${\rm I}^α_{0+}f= x^m f$ 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 Schneider (1987) and more recently investigated in Pakes (2014). We study various analytical aspects of these densities, and we solve the open problems about infinite divisibility formulated in Pakes (2014).

math.CA↗
arXiv2016

Stieltjes functions of finite order and hyperbolic monotonicity

A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order, and are not necessarily smooth. We show that such functions have a unique integral representation, along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms of two Stieltjes transforms earlier obtained by the first author.

math.CA↗
arXiv2015

Diffusion hitting times and the Bell-shape

Consider a generalized diffusion on R with speed measure m, in the natural scale. It is known that the conditional hitting times have a unimodal density function. We show that these hitting densities are bell-shaped if and only if m has infinitely many points of increase between the starting point and the hit point. This result can be viewed as a visual corollary to Yamazato's general factorization for diffusion hitting times.

math.PR↗
arXiv2015

On the harmonic measure of stable processes

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real L{é}vy process. This gives a simple and unified proof of several results in the literature, old and recent. We also provide a full description of the corresponding Green functions. As a by-product, we compute the hitting probabilities of points and describe the non-negative harmonic functions for the stable process killed outside a finite interval.

math.PR↗
arXiv2014

Total positivity in stable semigroups

We characterize the total positivity in space-time of real strictly stable semigroups. In the positive case, this solves a problem which had been raised by Karlin. In the drifted Cauchy case, this concludes a study which we had initiated in a previous paper. The case of isotropic stable semigroups is also investigated. We apply these results to the bell-shape and monotone likelihood properties of certain real stable densities.

math.CA↗
arXiv2014

The area of a spectrally positive stable process stopped at zero

An identity in law for the area of a spectrally positive Lévy stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse Beta random variable and the square of a positive stable random variable. This identity entails that the stopped area is distributed as the perpetuity of a spectrally negative Lévy process, and is hence self-decomposable. We also derive a convergent series representation for the density, whose behaviour at zero is shown to be Fréchet-like.

math.PR↗
arXiv2014

Windings of the stable Kolmogorov process

We investigate the windings around the origin of the two-dimensional Markov process (X,L) having the stable Lévy process L and its primitive X as coordinates, in the non-trivial case when |L| is not a subordinator. First, we show that these windings have an almost sure limit velocity, extending McKean's result [McK63] in the Brownian case. Second, we evaluate precisely the upper tails of the distribution of the half-winding times, connecting the results of our recent papers [CP14, PS14].

math.PR↗
arXiv2014

On the infinite divisibility of inverse Beta distributions

We show that all negative powers B_{a,b}^-{s} of the Beta distribution are infinitely divisible. The case b<1 follows by complete monotonicity, the case b > 1, s > 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a Lévy perpetuity argument involving the hypergeometric series. We also observe that B_{a,b}^{-s} is self-decomposable whenever 2a + b + s + bs > 1, and that it is not always a generalized Gamma convolution. On the other hand, we prove that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering to a recent question of L. Bondesson.

math.PR↗
arXiv2014

Persistence of integrated stable processes

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Z. Shi (2003). Along the way, we investigate the law of the stable process L evaluated at the first time its integral X hits zero, when the bivariate process (X,L) starts from a coordinate axis. This extends classical formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion.

math.PR↗
arXiv2014

Comparing Fréchet and positive stable laws

Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_α$ the standard positive $α$-stable random variable. We prove that $\{(1-α) α^{γ_α} {\bf Z}_α^{-γ_α}, 0< α<1\}$ is decreasing for the optimal stochastic order and that $\{(1-α){\bf Z}_α^{-γ_α}, 0< α< 1\}$ is increasing for the convex order, with $γ_α= α/(1-α).$ We also show that $\{Γ(1+α) {\bf Z}_α^{-α}, 1/2\le α\le 1\}$ is decreasing for the convex order, that ${\bf Z}_α^{-α}\,\prec_{st}\, Γ(1-α) Ł$ and that $Γ(1+\a){\bf Z}_α^{-α} \,\prec_{cx}\,{\bf L}.$ This allows to compare ${\bf Z}_α$ with the two extremal Fréchet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_α$ and ${\bf Z}_α^{-α}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_α$ for $α$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.

math.PR↗
arXiv2013

Mittag-Leffler functions and complete monotonicity

We consider two operations on the Mittag-Leffler function which cancel the exponential term in the expansion at infinity, and generate a completely monotonic function. The first one is the action of a certain differential-difference operator, and leads to a characterization via some necktie domain. The second one is the subtraction of the exponential term itself multiplied by an incomplete Gamma function. These results extend previous works by various authors.

math.CA↗
arXiv2013

Further examples of GGC and HCM densities

We display several examples of generalized gamma convoluted and hyperbolically completely monotone random variables related to positive $α$-stable laws. We also obtain new factorizations for the latter, refining Kanter's and Pestana-Shanbhag-Sreehari's. These results give stronger credit to Bondesson's hypothesis that positive $α$-stable densities are hyperbolically completely monotone whenever $α\le1/2.$

math.ST↗