Tightness for a family of recursion equations

In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.Comment: Published in at http://dx.doi.org/10.1214/08-AOP414 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A quenched invariance principle for certain ballistic random walks in i.i.d. environments

We prove that every random walk in i.i.d. environment in dimension greater than or equal to 2 that has an almost sure positive speed in a certain direction, an annealed invariance principle and some mild integrability condition for regeneration times also satisfies a quenched invariance principle. The argument is based on intersection estimates and a theorem of Bolthausen and Sznitman.Comment: This version includes an extension of the results to cover also dimensions 2,3, and also corrects several minor innacuracies. The previous version included a correction of a minor error in (3.21) (used for d=4); The correction pushed the assumption on moments of regeneration times to >

Slowdown in branching Brownian motion with inhomogeneous variance

We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form \sigma(t/T), where \sigma is a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_\sigma T is negative of order T^{-1/3}, where v_\sigma is the integral of the function \sigma over the interval [0,1]. In this paper, we refine we refine this result and show the existence of a function m_T, such that M_T-m_T converges in law, as T\to\infty. Furthermore, m_T=v_\sigma T - w_\sigma T^{1/3} - \sigma(1)\log T + O(1) with w_\sigma = 2^{-1/3}\alpha_1 \int_0^1 \sigma(s)^{1/3} |\sigma'(s)|^{2/3}\,\dd s. Here, -\alpha_1=-2.33811... is the largest zero of the Airy function. The proof uses a mixture of probabilistic and analytic arguments.Comment: A proof of convergence added in v2; details added and minor typos corrected in v

Large deviations for zeros of random polynomials with i.i.d. exponential coefficients

We derive a large deviation principle for the empirical measure of zeros of random polynomials with i.i.d. exponential coefficients.Comment: To appear in I.M.R.