9 research outputs found
On the number of empty boxes in the Bernoulli sieve
The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with
random frequencies , where (W_k)_{k\in\mn} are
independent copies of a random variable taking values in . Assuming
that the number of balls equals , let denote the number of empty boxes
within the occupancy range. The paper proves that, under a regular variation
assumption, , properly normalized without centering, weakly converges to a
functional of an inverse stable subordinator. Proofs rely upon the observation
that is a perturbed random walk. In particular, some results for
general perturbed random walks are derived. The other result of the paper
states that whenever weakly converges (without normalization) the
limiting law is mixed Poisson.Comment: Minor corrections to Proposition 5.1 were adde
Conditional limit theorems for regulated fractional Brownian motion
We consider a stationary fluid queue with fractional Brownian motion input.
Conditional on the workload at time zero being greater than a large value ,
we provide the limiting distribution for the amount of time that the workload
process spends above level over the busy cycle straddling the origin, as
. Our results can be interpreted as showing that long delays occur
in large clumps of size of order . The conditional limit result
involves a finer scaling of the queueing process than fluid analysis, thereby
departing from previous related literature.Comment: Published in at http://dx.doi.org/10.1214/09-AAP605 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes
Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich
class of stationary stochastic processes for which the marginal distribution
and the dependence structure may be modeled independently. We show that they
can also display intermittency, a phenomenon affecting the rate of growth of
moments. To do so, we investigate the limiting behavior of integrated supOU
processes with finite variance. After suitable normalization four different
limiting processes may arise depending on the decay of the correlation function
and on the characteristic triplet of the marginal distribution. To show that
supOU processes may exhibit intermittency, we establish the rate of growth of
moments for each of the four limiting scenarios. The rate change indicates that
there is intermittency, which is expressed here as a change-point in the
asymptotic behavior of the absolute moments.Comment: Stochastic Processes and their Application
Poisson Shot Noise Traffic Model and Approximation of Significant Functionals
In Internet traffic modeling, many authors presented models based on particular fractal shot noise representations. The inconvenience of these approaches is the multitude of assumptions and the lack of tools to check them. In this paper we propose a unified model based on a general Poisson shot noise representation for the cumulative input process (CIP). We present a procedure of approximation of this process; then we give a procedure for controlling the bandwidth of Internet providers. The approximation and control go via limit theorems for functionals of the CIP, namely, the supremum process, the right inverse, and the storage mapping
Is network traffic approximated by stable Levy motion or fractional Brownian motion?
Cumulative broadband network traffic is often thought to be well modeled by fractional Brownian motion (FBM). However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection length distribution tails, then stable Levy motion is a sensible approximation to cumulative traffic over a time period. If connection rates are large relative to heavy tailed connection length distribution tails, then FBM is the appropriate approximation. The results are framed as limit theorems for a sequence of cumulative input processes whose connection rates are varying in such a way as to remove or induce long range dependence