Abstract
We examine a new path transform on 1-dimensional simple random walks and Brownian motion, the quantile transform. This transformation relates to identities in fluctuation theory due to Wendel, Port, Dassios and others, and to discrete and Brownian versions of Tanaka’s formula. For an $n$-step random walk, the quantile transform reorders increments according to the value of the walk at the start of each increment. We describe the distribution of the quantile transform of a simple random walk of $n$ steps, using a bijection to characterize the number of pre-images of each possible transformed path. We deduce, both for simple random walks and for Brownian motion, that the quantile transform has the same distribution as Vervaat’s transform. For Brownian motion, the quantile transforms of the embedded simple random walks converge to a time change of the local time profile. We characterize the distribution of the local time profile, giving rise to an identity that generalizes a variant of Jeulin’s description of the local time profile of a Brownian bridge or excursion.
Highlights
Given a simple walk with increments of ±1, one observes that the step immediately following the maximum value attained must be a down step, and the step immediately following the minimum value attained must be an up step
We characterize the image of the quantile transform on simple (Bernoulli) random walks, which we call quantile walks, and we find the multiplicity with which each quantile walk arises
By passing to a Brownian limit, that the quantile transform of certain Bernoulli walks converge to an expression involving Brownian local times
Summary
Given a simple walk with increments of ±1, one observes that the step immediately following the maximum value attained must be a down step, and the step immediately following the minimum value attained must be an up step. The bijection from the discrete setting results in an identity, Theorem 8.18, describing local times of Brownian motion up to a fixed time, as a function of level. This identity generalizes a theorem of Jeulin[32]. Aldous[3], too, made use of this identity to study Brownian motion conditioned on its local time profile; and Aldous, Miermont, and Pitman[1], while working in the continuum random tree setting, discovered a version of Jeulin’s result for a more general class of Lévy processes.
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