Ordered Random Walks

By Peter Eichelsbacher, Wolfgang Konig. 2006

61 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5589a1d71f5715aa4c475b798d4366319b5ba477

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In Discrete Mathematics and Combinatorics

1. Vitali Wachtel
2. Denis Denisov
3. Kilian Raschel

In Mathematical Physics

1. Kilian Raschel
2. Denis Denisov
3. Vitali Wachtel
4. R. Doney
5. Matthew I. Roberts

In Statistics and Probability

1. Kilian Raschel
2. Vitali Wachtel
3. Denis Denisov

Citations in the corpus

• Random walks in the quarter plane, discrete harmonic functions and conformal mappings (2013, background): “Random processes conditioned on staying in cones of Z arouse a great interest in the mathematical community, as they appear in several distinct domains: quantum random walks [3, 5], random matrices [11], non-colliding random walks [8, 9, 12], etc.”
• Random walks in cones (2011, background): “For two further Weyl chambers, WC := {x∈Rd : 0 x1 x2 · · · xd} and WD := {x ∈Rd : |x1| x2 · · · xd} and random walks with independent coordinates König and Schmid [33] have proven versions of Theorems 1 and 3 under moment conditions which are weaker than E|X|p ∞.”
• Conditional limit theorems for ordered random walks (2009, methodology): “The authors of [9] used the Hölder inequality at many places in their proof.”