Fractionally integrated inverse stable subordinators
TLDR
In this article, the authors show that the fractionally integrated inverse stable subordinator (FIISS) is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean "inter-shot" distribution and regularly varying response function.About:
This article is published in Stochastic Processes and their Applications.The article was published on 2017-01-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Subordinator & Iterated logarithm.read more
Citations
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Transition densities of subordinators of positive order.
TL;DR: In this paper, the authors prove existence and asymptotic behavior of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity.
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A functional limit theorem for general shot noise processes
TL;DR: In this article, it was shown that a general shot noise process satisfies a functional limit theorem in the Skorokhod space with a locally Holder continuous Gaussian limit process, and that the response function is regularly varying at infinity.
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A functional limit theorem for general shot noise processes
Abstract: By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally H\"{o}lder continuous Gaussian limit process and that the response function is regularly varying at infinity we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann-Liouville process. We specialize our result for five particular counting processes. Also, we investigate H\"{o}lder continuity of the limit processes for general shot noise processes.
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Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates
TL;DR: In this article, the authors considered continuous-time Markov chains on integers with alternating rates and gave explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process.
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A functional limit theorem for random processes with immigration in the case of heavy tails
Alexander Marynych,Glib Verovkin +1 more
TL;DR: In this article, a functional limit theorem for the random process with immigration with immigration (Y(u))_{u\geq 0} was obtained for the case when the law of the positive variable is a stable law.
References
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Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
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Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Book
Stochastic-process limits : an introduction to stochastic-process limits and their application to queues
TL;DR: Experiencing Statistical Regularity, Random Walks in Applications, The Framework for Stochastic Process Limits, A Panorama of Stochastically-process limits, Heavy-Traffic Limits for Fluid Queues, Unmatched Jumps in the Limit Process Process, Fluid queuing with On-Off Sources, Single-Server Queues with on-off sources, Multi-server Queueing Networks, More on the Mathematical Framework, The Space D Useful Functions, Queuing Networks, The Spaces E and F- Appendices as mentioned in this paper
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Limit theorems for continuous-time random walks with infinite mean waiting times
TL;DR: In this article, the scaling limit of a continuous-time random walk is shown to be an operator Levy motion subordinated to the hitting time process of a classical stable subordinator.
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Semi-Stable Markov Processes. I
TL;DR: In this article, a real valued random function is called semi-stable if there is a constant > 0 (called the order of the process) such that for every a>O the random functions {x,t } and {a~x,} have the same joint distributions.