Two recursive decompositions of Brownian bridge

TL;DR

This paper explores two recursive decompositions of the Brownian bridge derived from different encoding methods of random mappings, revealing new identities and distributional properties of bridge fragments.

Contribution

It introduces two novel recursive decompositions of the Brownian bridge based on different encoding schemes of random mappings, extending identities and characterizations of path fragments.

Findings

Derived identities involving occupation measures of bridge fragments

Extended decompositions to Brownian and Bessel bridges

Characterized distributions of path fragments using Poisson excursion processes

Abstract

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.