196 research outputs found
Prediction theory for stationary functional time series
We survey aspects of prediction theory in infinitely many dimensions, with a view to the theory and applications of functional time series
Linear algebra and multivariate analysis in statistics: development and interconnections in the twentieth century
The most obvious points of contact between linear and matrix algebra and statistics are in the area of multivariate analysis. We review the way that, as both developed during the last century, the two influenced each other by examining a number of key areas. We begin with matrix and linear algebra, its emergence in the nineteenth century, and its eventual penetration into the undergraduate curriculum in the twentieth century. We continue with a similar account for multivariate analysis in statistics. We pick out the year 1936 for three key developments, and the early post-war period for three more. We then turn to some special results in linear algebra that we need. We briefly discuss four of the main contributors, and close with thirteen ‘case studies’, showing in a range of specific cases how these general algebraic methods have been put to good use and changed the face of statistics
Packing and Hausdorff measures of stable trees
In this paper we discuss Hausdorff and packing measures of random continuous
trees called stable trees. Stable trees form a specific class of L\'evy trees
(introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum
random tree (1991) which corresponds to the Brownian case. We provide results
for the whole stable trees and for their level sets that are the sets of points
situated at a given distance from the root. We first show that there is no
exact packing measure for levels sets. We also prove that non-Brownian stable
trees and their level sets have no exact Hausdorff measure with regularly
varying gauge function, which continues previous results from a joint work with
J-F Le Gall (2006).Comment: 40 page
Global clustering coefficient in scale-free networks
In this paper, we analyze the behavior of the global clustering coefficient
in scale free graphs. We are especially interested in the case of degree
distribution with an infinite variance, since such degree distribution is
usually observed in real-world networks of diverse nature.
There are two common definitions of the clustering coefficient of a graph:
global clustering and average local clustering. It is widely believed that in
real networks both clustering coefficients tend to some positive constant as
the networks grow. There are several models for which the average local
clustering coefficient tends to a positive constant. On the other hand, there
are no models of scale-free networks with an infinite variance of degree
distribution and with a constant global clustering.
In this paper we prove that if the degree distribution obeys the power law
with an infinite variance, then the global clustering coefficient tends to zero
with high probability as the size of a graph grows
On the Price of Anarchy of Highly Congested Nonatomic Network Games
We consider nonatomic network games with one source and one destination. We
examine the asymptotic behavior of the price of anarchy as the inflow
increases. In accordance with some empirical observations, we show that, under
suitable conditions, the price of anarchy is asymptotic to one. We show with
some counterexamples that this is not always the case. The counterexamples
occur in very simple parallel graphs.Comment: 26 pages, 6 figure
On small-noise equations with degenerate limiting system arising from volatility models
The one-dimensional SDE with non Lipschitz diffusion coefficient is widely
studied in mathematical finance. Several works have proposed asymptotic
analysis of densities and implied volatilities in models involving instances of
this equation, based on a careful implementation of saddle-point methods and
(essentially) the explicit knowledge of Fourier transforms. Recent research on
tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and
S.~Violante. Marginal density expansions for diffusions and stochastic
volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with
the rescaled variable : while
allowing to turn a space asymptotic problem into a small- problem
with fixed terminal point, the process satisfies a SDE in
Wentzell--Freidlin form (i.e. with driving noise ). We prove a
pathwise large deviation principle for the process as
. As it will become clear, the limiting ODE governing the
large deviations admits infinitely many solutions, a non-standard situation in
the Wentzell--Freidlin theory. As for applications, the -scaling
allows to derive exact log-asymptotics for path functionals of the process:
while on the one hand the resulting formulae are confirmed by the CIR-CEV
benchmarks, on the other hand the large deviation approach (i) applies to
equations with a more general drift term and (ii) potentially opens the way to
heat kernel analysis for higher-dimensional diffusions involving such an SDE as
a component.Comment: 21 pages, 1 figur
Upper bounds for number of removed edges in the Erased Configuration Model
Models for generating simple graphs are important in the study of real-world
complex networks. A well established example of such a model is the erased
configuration model, where each node receives a number of half-edges that are
connected to half-edges of other nodes at random, and then self-loops are
removed and multiple edges are concatenated to make the graph simple. Although
asymptotic results for many properties of this model, such as the limiting
degree distribution, are known, the exact speed of convergence in terms of the
graph sizes remains an open question. We provide a first answer by analyzing
the size dependence of the average number of removed edges in the erased
configuration model. By combining known upper bounds with a Tauberian Theorem
we obtain upper bounds for the number of removed edges, in terms of the size of
the graph. Remarkably, when the degree distribution follows a power-law, we
observe three scaling regimes, depending on the power law exponent. Our results
provide a strong theoretical basis for evaluating finite-size effects in
networks
Inversions of Levy Measures and the Relation Between Long and Short Time Behavior of Levy Processes
The inversion of a Levy measure was first introduced (under a different name)
in Sato 2007. We generalize the definition and give some properties. We then
use inversions to derive a relationship between weak convergence of a Levy
process to an infinite variance stable distribution when time approaches zero
and weak convergence of a different Levy process as time approaches infinity.
This allows us to get self contained conditions for a Levy process to converge
to an infinite variance stable distribution as time approaches zero. We
formulate our results both for general Levy processes and for the important
class of tempered stable Levy processes. For this latter class, we give
detailed results in terms of their Rosinski measures
Quantum Stochastic Processes and the Modelling of Quantum Noise
This brief article gives an overview of quantum mechanics as a {\em quantum
probability theory}. It begins with a review of the basic operator-algebraic
elements that connect probability theory with quantum probability theory. Then
quantum stochastic processes is formulated as a generalization of stochastic
processes within the framework of quantum probability theory. Quantum Markov
models from quantum optics are used to explicitly illustrate the underlying
abstract concepts and their connections to the quantum regression theorem from
quantum optics.Comment: 14 pages, invited article for the second edition of Springer's
Encyclopedia of Systems and Control (to appear). Comments welcom
A fluid model for a relay node in an ad-hoc network: the case of heavy-tailed input
Relay nodes in an ad hoc network can be modelled as fluid queues, in which the available service capacity is shared by the input and output. In this paper such a relay node is considered; jobs arrive according to a Poisson process and bring along a random amount of work. The total transmission capacity is fairly shared, meaning that, when n jobs are present, each job transmits traffic into the queue at rate 1/(n + 1) while the queue is drained at the same rate of 1/(n + 1). Where previous studies mainly concentrated on the case of exponentially distributed job sizes, the present paper addresses regularly varying jobs. The focus lies on the tail asymptotics of the sojourn time S. Using sample-path arguments, it is proven that P {S > x} behaves roughly as the residual job size, i.e., if the job sizes are regularly varying of index -nu, the tail of S is regularly varying of index 1 - nu. In addition, we address the tail asymptotics of other performance metrics, such as the workload in the queue, the flow transfer time and the queueing delay
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