Multidimensional random walks conditioned to stay ordered via generalized ladder height functions

DOI10.1007/978-3-030-57513-7_2zbMATH Open1473.60072arXiv1905.05693OpenAlexW2945793193MaRDI QIDQ2041473

Osvaldo Angtuncio-Hernández

Publication date: 22 July 2021

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 in [EK08] 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 $h$-transform 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 in [Ign18], it can be proved that our $h$-function is the only harmonic function which is zero outside the Weyl chamber $x=(x_1,ldots,x_d)inmathbb{R}^d:x_1<cdots<x_d$.

Full work available at URL: https://arxiv.org/abs/1905.05693

zbMATH Keywords

harmonic functionWeyl chamberordered random walksDoob h-transformmultidimensional ladder height function

Mathematics Subject Classification ID

Sums of independent random variables; random walks (60G50)

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