Access the full text.
Sign up today, get DeepDyve free for 14 days.
References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.
[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
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.