On the small-time behavior of subordinators
TLDR
In this article, it was shown that the Pareto law is the only possible weak limit for the convergence of a stochastic process near t = 0 of a decreasing function.Abstract:
We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.read more
Citations
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Lévy processes and infinitely divisible distributions
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
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Parametric Estimation of Lévy Processes
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Parametric estimation of L\'evy processes
TL;DR: The main purpose of as discussed by the authors is to present some theoretical aspects of parametric estimation of Levy processes based on high-frequency sampling, with a focus on infinite activity pure-jump models.
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Weak Convergence of Subordinators to Extremal Processes
Offer Kella,Andreas Löpker +1 more
TL;DR: In this article, it was shown that for certain subordinators, the process tends to an extremal process in the sense of convergence of the finite dimensional distributions in the space of functions equipped with Skorohod's $J_1$ metric.
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A Numerical Study of Small Parameter Behavior of Some Families of Distributions
TL;DR: This article proposes a procedure for the numerical assessment of the “goodness” of some easy-to-calculate limiting distributions, originally proposed in Bar-Lev and Enis, in various cases of the underlying distributions, some of which are inherently computationally challenging.
References
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Foundations of modern probability
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
Book
Lévy processes and infinitely divisible distributions
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Book
Introductory Lectures on Fluctuations of Lévy Processes with Applications
TL;DR: In this paper, the authors present decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short-and long-term behaviour.
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