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 Lévy 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.

We applied new methods which take available knowledge of visual field physiology and pathophysiology into account, and employ modern computer-intensive mathematical methods for real time estimates of threshold values and threshold error estimates. In this way it was possible to design a family of testing algorithms which significantly reduced perimetric test time without any loss of quality in results.

} are i.i.d. IWt-valued random pairs. A special case of interest is when { Y,,} is derived from a first order ARCH process. Parameters of the limit law are exhibited; some are hard to calculate explicitly but easy to simulate.

[1] In the past, the concepts of return levels and return periods have been standard and important tools for engineering design. However, these concepts are based on the assumption of a stationary climate and do not apply to a changing climate, whether local or global. In this paper, we propose a refined concept, Design Life Level, which quantifies risk in a nonstationary climate and can serve as the basis for communication. In current practice, typical hydrologic risk management focuses on a standard (e.g., in terms of a high quantile corresponding to the specified probability of failure for a single year). Nevertheless, the basic information needed for engineering design should consist of (i) the design life period (e.g., the next 50 years, say 2015-2064); and (ii) the probability (e.g., 5% chance) of a hazardous event (typically, in the form of the hydrologic variable exceeding a high level) occurring during the design life period. Capturing both of these design characteristics, the Design Life Level is defined as an upper quantile (e.g., 5%) of the distribution of the maximum value of the hydrologic variable (e.g., water level) over the design life period. We relate this concept and variants of it to existing literature and illustrate how they, and some useful complementary plots, may be computed and used. One practically important consideration concerns quantifying the statistical uncertainty in estimating a high quantile under nonstationarity.

Let $(X_{n,i})_{1\le i\le n,n\in\mathbb{N}}$ be a triangular array of
row-wise stationary $\mathbb{R}^d$-valued random variables. We use a "blocks
method" to define clusters of extreme values: the rows of $(X_{n,i})$ are
divided into $m_n$ blocks $(Y_{n,j})$, and if a block contains at least one
extreme value, the block is considered to contain a cluster. The cluster starts
at the first extreme value in the block and ends at the last one. The main
results are uniform central limit theorems for empirical processes
$Z_n(f):=\frac{1}{\sqrt {nv_n}}\sum_{j=1}^{m_n}(f(Y_{n,j})-Ef(Y_{n,j})),$ for
$v_n=P\{X_{n,i}\neq0\}$ and $f$ belonging to classes of cluster functionals,
that is, functions of the blocks $Y_{n,j}$ which only depend on the cluster
values and which are equal to 0 if $Y_{n,j}$ does not contain a cluster.
Conditions for finite-dimensional convergence include $\beta$-mixing, suitable
Lindeberg conditions and convergence of covariances. To obtain full uniform
convergence, we use either "bracketing entropy" or bounds on covering numbers
with respect to a random semi-metric. The latter makes it possible to bring the
powerful Vapnik--\v{C}ervonenkis theory to bear. Applications include
multivariate tail empirical processes and empirical processes of cluster values
and of order statistics in clusters. Although our main field of applications is
the analysis of extreme values, the theory can be applied more generally to
rare events occurring, for example, in nonparametric curve estimation.Comment: Published in at http://dx.doi.org/10.1214/09-AOS788 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

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