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XIII Symposium on Probability and Stochastic ProcessesMultidimensional Random Walks Conditioned to Stay Ordered via Generalized Ladder Height Functions

XIII Symposium on Probability and Stochastic Processes: Multidimensional Random Walks Conditioned... [Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time n, and let n tend to infinity. A second method is conditioning instead to stay positive up to an independent geometric time, and send its parameter to zero. The multidimensional case (condition the components of a d-dimensional random walk to be ordered) was solved by Eichelsbacher and König (Ordered random walks, Electron J Probab 13(46):1307–1336, 2008. MR 2430709) using the first approach, but some moment conditions need to be imposed. Our approach is based on the second method, which has the advantage to require a minimal restriction, needed only for the finiteness of the h-function in certain cases. We also characterize when the limit is Markovian or sub-Markovian, and give several reexpresions of the h-function. Under some conditions given by Ignatiouk-Robert (Harmonic functions of random walks in a semigroup via ladder heights. ArXiv e-prints, 2018), it can be proved that our h-function is the only harmonic function which is zero outside the Weyl chamber {x=(x1,…,xd)∈ℝd:x1<⋯<xd}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{x=(x_1,\ldots , x_d)\in {\mathbb {R}}^d: x_1<\cdots < x_d\}$$ \end{document}.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

XIII Symposium on Probability and Stochastic ProcessesMultidimensional Random Walks Conditioned to Stay Ordered via Generalized Ladder Height Functions

Part of the Progress in Probability Book Series (volume 75)
Editors: López, Sergio I.; Rivero, Víctor M.; Rocha-Arteaga, Alfonso; Siri-Jégousse, Arno
Angtuncio-Hernández, Osvaldo
XIII Symposium on Probability and Stochastic Processes — Oct 17, 2020
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Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
ISBN
978-3-030-57512-0
Pages
39 –65
DOI
10.1007/978-3-030-57513-7_2
Publisher site
See Chapter on Publisher Site

Abstract

[Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time n, and let n tend to infinity. A second method is conditioning instead to stay positive up to an independent geometric time, and send its parameter to zero. The multidimensional case (condition the components of a d-dimensional random walk to be ordered) was solved by Eichelsbacher and König (Ordered random walks, Electron J Probab 13(46):1307–1336, 2008. MR 2430709) using the first approach, but some moment conditions need to be imposed. Our approach is based on the second method, which has the advantage to require a minimal restriction, needed only for the finiteness of the h-function in certain cases. We also characterize when the limit is Markovian or sub-Markovian, and give several reexpresions of the h-function. Under some conditions given by Ignatiouk-Robert (Harmonic functions of random walks in a semigroup via ladder heights. ArXiv e-prints, 2018), it can be proved that our h-function is the only harmonic function which is zero outside the Weyl chamber {x=(x1,…,xd)∈ℝd:x1<⋯<xd}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{x=(x_1,\ldots , x_d)\in {\mathbb {R}}^d: x_1<\cdots < x_d\}$$ \end{document}.]

Published: Oct 17, 2020

Keywords: Ordered random walks; Doob h-transform; Harmonic function; Weyl chamber; Multidimensional ladder height function; 60G50; 60F17

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