Area distribution and the average shape of a L\'evy bridge

TL;DR

This paper analyzes the distribution of the area and the average shape of a Lévy bridge, revealing scaling behaviors and explicit formulas for certain cases, supported by numerical simulations.

Contribution

It provides a detailed characterization of the area distribution and average shape of Lévy bridges, including explicit results for the case =1, and introduces new scaling forms.

Findings

The area distribution scales as n^{-1-1/} with a specific asymptotic form.

Explicit expression for the area distribution when =1.

The average profile exhibits a non-trivial scaling function depending on rescaled variables.

Abstract

We consider a one dimensional L\'evy bridge x_B of length n and index 0 < \alpha < 2, i.e. a L\'evy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a L\'evy bridge and show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/\alpha} F_\alpha(A/n^{1+1/\alpha}), with the asymptotic behavior F_\alpha(Y) \sim Y^{-2(1+\alpha)} for large Y. For \alpha=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions. We also compute the average profile < \tilde x_B (m) > at time m of a L\'evy bridge with fixed area A. For large n and large m and A, one finds the scaling form < \tilde x_B(m) > = n^{1/\alpha} H_\alpha({m}/{n},{A}/{n^{1+1/\alpha}}), where at variance with Brownian bridge, H_\alpha(X,Y) is a non trivial function of the rescaled time m/n and rescaled area Y = A/n^{1+1/\alpha}. Our analytical results are verified by numerical simulations.