Related papers: On the self-decomposability of the Fr\'echet distr…

Let $\xi$ be a (possibly killed) subordinator with Laplace exponent $\phi$ and denote by $I_{\phi}=\int_0^{\infty}\mathrm{e}^{-\xi_s}\,\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{\psi_1}$…

Let $Y$ be a spectrally positive L\'evy process with $E Y_1<0$, $C$ an independent subordinator with finite expectation, and $X=Y+C$. A curious distributional equality proved in Huzak et al., Ann. Appl. Probab. 14 (2004) 1278--1397, states…

This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$ given…

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an…

Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via L\'evy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used…

This a free translation with additional explanations of {\em Processus \`a Accroissement Independants Chapitre I: La D\'ecomposition de Paul L\'evy}, by J.L. Bretagnolle, in {\em Ecole d'Et\'e de Probabilit\'es}, Lecture Notes in…

We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…

In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise,…

Understanding irreversibility in macrophysics from reversible microphysics has been the holy grail in statistical physics ever since the mid-19th century. Here the central question concerns the arrow of time, which boils down to deriving…

We give closed-form expressions for the Laurent series coefficients of the Gamma function near all its strictly negative singularities. These closed-form expressions are clearly self-similar. We briefly describe their algebraic and…

Distributional identities for a L\'evy process $X_t$, its quadratic variation process $V_t$ and its maximal jump processes, are derived, and used to make "small time" (as $t\downarrow0$) asymptotic comparisons between them. The…

We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability…

Consider a one dimensional critical branching L\'{e}vy process $((Z_t)_{t\geq 0}, \mathbb {P}_x)$. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction to some $\alpha$-stable…

We consider a spectrally positive L\'evy process $X$ that does not drift to $+\infty$, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process…

Suppose Xt is either a regular exponential type Levy process or a Levy process with a bounded variation jumps measure. The distribution of the extrema of Xt play a crucial role in many financial and actuarial problems. This article employs…

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the…

Let $\xi$ be a L\'{e}vy process and $I_\xi(t):=\int_{0}^te^{-\xi_s}\mathrm{d} s$, $t\geq 0,$ be the exponential functional of L\'{e}vy processes on deterministic horizon. Given that $\lim_{t\to \infty}\xi_t=-\infty$ we evaluate for general…

Let $(U_t,V_t)$ be a bivariate L\'evy process, where $V_t$ is a subordinator and $U_t$ is a L\'evy process formed by randomly weighting each jump of $V_t$ by an independent random variable $X_t$ having cdf $F$. We investigate the asymptotic…

We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is…

The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the $\beta$-Tracy-Widom distribution, which is the limiting distribution of the largest…