On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local H\"older regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener integrals.Comment: 22 page

Directly Imaging Tidally Powered Migrating Jupiters

Upcoming direct-imaging experiments may detect a new class of long-period, highly luminous, tidally powered extrasolar gas giants. Even though they are hosted by ~ Gyr-"old" main-sequence stars, they can be as "hot" as young Jupiters at ~100 Myr, the prime targets of direct-imaging surveys. They are on years-long orbits and presently migrating to "feed" the "hot Jupiters." They are expected from "high-e" migration mechanisms, in which Jupiters are excited to highly eccentric orbits and then shrink semi-major axis by a factor of ~10-100 due to tidal dissipation at close periastron passages. The dissipated orbital energy is converted to heat, and if it is deposited deep enough into the atmosphere, the planet likely radiates steadily at luminosity L ~ 100-1000 L_Jup(2 x 10-7-2 x 10-6 L_Sun) during a typical ~ Gyr migration timescale. Their large orbital separations and expected high planet-to-star flux ratios in IR make them potentially accessible to high-contrast imaging instruments on 10 m class telescopes. ~10 such planets are expected to exist around FGK dwarfs within ~50 pc. Long-period radial velocity planets are viable candidates, and the highly eccentric planet HD 20782b at maximum angular separation ~0.''08 is a promising candidate. Directly imaging these tidally powered Jupiters would enable a direct test of high-e migration mechanisms. Once detected, the luminosity would provide a direct measurement of the migration rate, and together with mass (and possibly radius) estimate, they would serve as a laboratory to study planetary spectral formation and tidal physics.Comment: Updated to match the published version (with a figure

On Stein's Method for Infinitely Divisible Laws With Finite First Moment

We present, in a unified way, a Stein methodology for infinitely divisible laws (without Gaussian component) having finite first moment. Based on a correlation representation, we obtain a characterizing non-local Stein operator which boils down to classical Stein operators in specific examples. Thanks to this characterizing operator, we introduce various extensions of size bias and zero bias distributions and prove that these notions are closely linked to infinite divisibility. Combined with standard Fourier techniques, these extensions also allow obtaining explicit rates of convergence for compound Poisson approximation in particular towards the symmetric $\alpha$-stable distribution. Finally, in the setting of non-degenerate self-decomposable laws, by semigroup techniques, we solve the Stein equation induced by the characterizing non-local Stein operator and obtain quantitative bounds in weak limit theorems for sums of independent random variables going back to the work of Khintchine and L\'evy.Comment: 58 pages. Minor changes and new results in Sections 5 and

A stroll along the gamma

We provide the first in-depth study of the "smart path" interpolation between an arbitrary probability measure and the gamma-$(\alpha, \lambda)$ distribution. We propose new explicit representation formulae for the ensuing process as well as a new notion of relative Fisher information with a gamma target distribution. We use these results to prove a differential and an integrated De Bruijn identity which hold under minimal conditions, hereby extending the classical formulae which follow from Bakry, Emery and Ledoux's $\Gamma$-calculus. Exploiting a specific representation of the "smart path", we obtain a new proof of the logarithmic Sobolev inequality for the gamma law with $\alpha\geq 1/2$ as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with $\alpha\geq 1/2$.Comment: Typos correcte