6,322 research outputs found
Rounding of continuous random variables and oscillatory asymptotics
We study the characteristic function and moments of the integer-valued random
variable , where is a continuous random variables.
The results can be regarded as exact versions of Sheppard's correction. Rounded
variables of this type often occur as subsequence limits of sequences of
integer-valued random variables. This leads to oscillatory terms in asymptotics
for these variables, something that has often been observed, for example in the
analysis of several algorithms. We give some examples, including applications
to tries, digital search trees and Patricia tries.Comment: Published at http://dx.doi.org/10.1214/009117906000000232 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Transition asymptotics for reaction-diffusion in random media
We describe a universal transition mechanism characterizing the passage to an
annealed behavior and to a regime where the fluctuations about this behavior
are Gaussian, for the long time asymptotics of the empirical average of the
expected value of the number of random walks which branch and annihilate on
, with stationary random rates. The random walks are
independent, continuous time rate , simple, symmetric, with . A random walk at , binary branches at rate ,
and annihilates at rate . The random environment has coordinates
which are i.i.d. We identify a natural way to describe
the annealed-Gaussian transition mechanism under mild conditions on the rates.
Indeed, we introduce the exponents
, and assume
that for
small enough, where and
denotes the average of the expected value of the number of particles
at time and an environment of rates , given that initially there was
only one particle at 0. Then the empirical average of over a box of
side has different behaviors: if for some and large enough , a law of large
numbers is satisfied; if for some
and large enough , a CLT is satisfied. These statements are
violated if the reversed inequalities are satisfied for some negative
. Applications to potentials with Weibull, Frechet and double
exponential tails are given.Comment: To appear in: Probability and Mathematical Physics: A Volume in Honor
of Stanislav Molchanov, Editors - AMS | CRM, (2007
On the large time asymptotics of decaying Burgers turbulence
The decay of Burgers turbulence with compactly supported Gaussian "white
noise" initial conditions is studied in the limit of vanishing viscosity and
large time. Probability distribution functions and moments for both velocities
and velocity differences are computed exactly, together with the "time-like"
structure functions .
The analysis of the answers reveals both well known features of Burgers
turbulence, such as the presence of dissipative anomaly, the extreme anomalous
scaling of the velocity structure functions and self similarity of the
statistics of the velocity field, and new features such as the extreme
anomalous scaling of the "time-like" structure functions and the non-existence
of a global inertial scale due to multiscaling of the Burgers velocity field.
We also observe that all the results can be recovered using the one point
probability distribution function of the shock strength and discuss the
implications of this fact for Burgers turbulence in general.Comment: LATEX, 25 pages, The present paper is an extension of the talk
delivered at the workshop on intermittency in turbulent systems, Newton
Institute, Cambridge, UK, June 199
Extreme gaps between eigenvalues of random matrices
This paper studies the extreme gaps between eigenvalues of random matrices.
We give the joint limiting law of the smallest gaps for Haar-distributed
unitary matrices and matrices from the Gaussian unitary ensemble. In
particular, the kth smallest gap, normalized by a factor , has a
limiting density proportional to . Concerning the largest
gaps, normalized by , they converge in to a
constant for all . These results are compared with the extreme gaps
between zeros of the Riemann zeta function.Comment: Published in at http://dx.doi.org/10.1214/11-AOP710 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Kinetic models of opinion formation
We introduce and discuss certain kinetic models of (continuous) opinion
formation involving both exchange of opinion between individual agents and
diffusion of information. We show conditions which ensure that the kinetic
model reaches non trivial stationary states in case of lack of diffusion in
correspondence of some opinion point. Analytical results are then obtained by
considering a suitable asymptotic limit of the model yielding a Fokker-Planck
equation for the distribution of opinion among individuals
Self-intersection local times of random walks: Exponential moments in subcritical dimensions
Fix , not necessarily integer, with . We study the -fold
self-intersection local time of a simple random walk on the lattice up
to time . This is the -norm of the vector of the walker's local times,
. We derive precise logarithmic asymptotics of the expectation of
for scales that are bounded from
above, possibly tending to zero. The speed is identified in terms of mixed
powers of and , and the precise rate is characterized in terms of
a variational formula, which is in close connection to the {\it
Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation
principle for for deviation functions satisfying
t r_t\gg\E[\|\ell_t\|_p]. Informally, it turns out that the random walk
homogeneously squeezes in a -dependent box with diameter of order to produce the required amount of self-intersections. Our main tool is
an upper bound for the joint density of the local times of the walk.Comment: 15 pages. To appear in Probability Theory and Related Fields. The
final publication is available at springerlink.co
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