The quantile transform of a simple walk

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.Comment: 46 pages, 20 figure

Locally Perturbed Random Walks with Unbounded Jumps

In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\mathbf Z^d$ $(d \ge 2)$ having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive $\sqrt{n \log n}$ scaling.Comment: 16 page

Strong approximation of fractional Brownian motion by moving averages of simple random walks

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to B.B. Mandelbrot and J.W. van Ness (1968) as a self-similar Gaussian process \WH (t) with stationary increments. Here self-similarity means that (a^{-H}\WH(at): t \ge 0) \stackrel{d}{=} (\WH(t): t \ge 0), where $H\in (0, 1)$ is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by P. R\'ev\'esz (1990) and then by the present author (1996). This approach is quite natural and elementary, and as such, can be extended to more general situations. Based on this, here we use moving averages of a suitable nested sequence of simple random walks that almost surely uniformly converge to fractional Brownian motion on compacts when H \in (\quart , 1). The rate of convergence proved in this case is O(N^{-\min(H-\quart,\quart)}\log N), where $N$ is the number of steps used for the approximation. If the more accurate (but also more intricate) Koml\'os, Major, Tusn\'ady (1975, 1976) approximation is used instead to embed random walks into ordinary Brownian motion, then the same type of moving averages almost surely uniformly converge to fractional Brownian motion on compacts for any $H \in (0, 1)$. Moreover, the convergence rate is conjectured to be the best possible $O(N^{-H}\log N)$, though only O(N^{-\min(H,\half)}\log N) is proved here.Comment: 30 pages, 4 figure

Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) which converges back to a Brownian motion with reduced diffusion coefficient at long times, after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed-down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in 2d. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte-Carlo simulations, we show that those three scenarios have distinctive effects on the apparent affinity of the reaction. While continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hinderance by obstacles both improve it. However, only in the case of slowed-down Brownian motion, the affinity is maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.Comment: Biophysical Journal (2013

Quantum random walks and minors of Hermitian Brownian motion

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam and van Moerbeke that the process of eigenvalues of two consecutive minors of an Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion