Ordered exponential random walks

ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS(2023)

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Abstract
We study a d-dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function h for the killed process and then construct an ordered process using Doob's h-transform. Since these random walks are not nearest-neighbour, the harmonic function is not the Vandermonde determinant. The ordered process is related to the departure process of M/M/1 queues in tandem. We find asymptotics for the tail probabilities of the time until the components in exponential random walks become disordered and a local limit theorem. We find the distribution of the processes of smallest and largest particles as Fredholm determinants.
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Key words
Ordered random walks,Doob h-transform,Pfaffian,Weyl chamber,discrete harmonic function
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