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Ordered Random Walks


 
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1. Title Title of document Ordered Random Walks
 
2. Creator Author's name, affiliation, country Peter Eichelsbacher; University of Bochum
 
2. Creator Author's name, affiliation, country Wolfgang König; University of Leipzig
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Dyson's Brownian motions; Vandermonde determinant; Doob h-transform; non-colliding random walks; non-intersecting random processes; fluctuation theory.
 
3. Subject Subject classification 60G50, 60F17
 
4. Description Abstract We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Deutsche Forschungsgemeinschaft via SFB/TR 12
 
7. Date (YYYY-MM-DD) 2008-08-14
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/539
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-539
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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