40 research outputs found
Renorming divergent perpetuities
We consider a sequence of random variables defined by the recurrence
, , where is arbitrary and ,
, are i.i.d. copies of a two-dimensional random vector , and
is independent of . It is well known that if
and , then the sequence converges in distribution
to a random variable given by
, and usually
referred to as perpetuity. In this paper we consider a situation in which the
sequence itself does not converge. We assume that exists
but that it is non-negative and we ask if in this situation the sequence
, after suitable normalization, converges in distribution to a
non-degenerate limit.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ297 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On the distribution of parameters in random weighted staircase tableaux
Abstract. In this paper, we study staircase tableaux, a combinatorial object introduced due to its connections with the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. Due to their interesting connections, staircase tableaux have been the object of study in many recent papers. More specific to this paper, the distribution of various parameters in random staircase tableaux has been studied. There have been interesting results on parameters along the main diagonal, however, no such results have appeared for other diagonals. It was conjectured that the distribution of the number of symbols along the kth diagonal is asymptotically Poisson as k and the size of the tableau tend to infinity. We partially prove this conjecture; more specifically we prove it for the second main diagonal
A generatingfunctionology approach to a problem of Wilf
Wilf posed the following problem: determine asymptotically as
the probability that a randomly chosen part size in a randomly chosen
composition of n has multiplicity m. One solution of this problem was given by
Hitczenko and Savage. In this paper, we study this question using the
techniques of generating functions and singularity analysis.Comment: 12 page
Measuring the magnitude of sums of independent random variables
The final version of this paper appears in: "Annals of Probability" 29 (2001): 447-466. Print.This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms
Stability of equilibria of randomly perturbed maps
Abstract We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in R d . This condition can be used to stabilize weakly unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed linear and nonlinear maps in one-and two-dimensional spaces
Convergence to type I distribution of the extremes of sequences defined by random difference equation
We study the extremes of a sequence of random variables defined by
the recurrence , , where is arbitrary,
are iid copies of a non--degenerate random variable , , and
is a constant. We show that under mild and natural conditions on the
suitably normalized extremes of converge in distribution to a double
exponential random variable. This partially complements a result of de Haan,
Resnick, Rootz\'en, and de Vries who considered extremes of the sequence
under the assumption that .Comment: to appear in Stochastic Processes and their Application