263 research outputs found
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.
A Central Limit Theorem for biased random walks on Galton-Watson trees
Let be a rooted Galton-Watson tree with offspring distribution
that has , mean and exponential tails.
Consider the -biased random walk on ;
this is the nearest neighbor random walk which, when at a vertex with
offspring, moves closer to the root with probability ,
and moves to each of the offspring with probability . It is
known that this walk has an a.s. constant speed
(where is the distance of from the root), with for and for . For all , we prove
a quenched CLT for |X_n|-n\v. (For the walk is positive
recurrent, and there is no CLT.) The most interesting case by far is
, where the CLT has the following form: for almost every ,
the ratio converges in law as to a
deterministic multiple of the absolute value of a Brownian motion. Our approach
to this case is based on an explicit description of an invariant measure for
the walk from the point of view of the particle (previously, such a measure was
explicitly known only for ) and the construction of appropriate
harmonic coordinates.Comment: 34 pages, 4 figure
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