Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

Anomalous scaling regime for one-dimensional Mott variable-range hopping

Abstract: We derive an anomalous, sub-diffusive scaling limit for a one-dimensional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this `blocking' mechanism with one of `trapping'. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.

Some properties of the Kilbas-Saigo function

Abstract: We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at $-\infty$, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

Abstract: Given a pure-jump subordinator (i.e. nondecreasing Levy process with no drift) with continuous Levy measure v, we derive a formula for the distribution function F s (x; t) at time t of the associated subordinator whose Levy measure is the restriction of ν to (0, s]. It will be expressed in terms of ν and the marginal distribution function F(.; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂ n F s (x; t)/∂x n . The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

The infinite extendibility problem for exchangeable real-valued random vectors

Abstract: We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on (mathbb{R}^{d}), which is: Can a given random vector (boldsymbol{X}=(X_{1},ldots ,X_{d})) be represented in distribution as the first (d) members of an infinite exchangeable sequence of random variables? This is the case if and only if (boldsymbol{X}) has a stochastic representation that is “conditionally iid” according to the seminal de Finetti’s Theorem. Of particular interest are cases in which the original motivation behind the model (boldsymbol{X}) is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when (boldsymbol{X}) takes values in ({0,1}^{d}), has a spherical law, a law with (ell _{1})-norm symmetric survival function, or a law with (ell _{infty})-norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on (mathbb{R}), like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when (boldsymbol{X}) has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Levy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than (mathbb{R}). Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.