2004/02/25 by David Aldous, Aldous, David, Jim Pitman +1
Mathematics · #60F05 (Secondary) #60J65 (Primary) 60C05 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60C05 #msc:60F05 #msc:60J65
paper · pdf · doi:10.48550/arxiv.math/0402399
35 pages, 1 figure
arxiv created 2004/02/25 · openalex publication_date 2004/02/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1.