261 research outputs found
Precise large deviations for dependent regularly varying sequences
We study a precise large deviation principle for a stationary regularly
varying sequence of random variables. This principle extends the classical
results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying
sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of
centra limit theorems with infinite variance stable limits. We illustrate the
principle for \sv\ models, functions of a Markov chain satisfying a polynomial
drift condition and solutions of linear and non-linear stochastic recurrence
equations
The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains
We introduce the cluster index of a multivariate regularly varying stationary
sequence and characterize the index in terms of the spectral tail process. This
index plays a major role in limit theory for partial sums of regularly varying
sequences. We illustrate the use of the cluster index by characterizing
infinite variance stable limit distributions and precise large deviation
results for sums of multivariate functions acting on a stationary Markov chain
under a drift condition
The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution
In this paper, we deal with the asymptotic distribution of the maximum
increment of a random walk with a regularly varying jump size distribution.
This problem is motivated by a long-standing problem on change point detection
for epidemic alternatives. It turns out that the limit distribution of the
maximum increment of the random walk is one of the classical extreme value
distributions, the Fr\'{e}chet distribution. We prove the results in the
general framework of point processes and for jump sizes taking values in a
separable Banach space.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ255 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Stochastic volatility models with possible extremal clustering
In this paper we consider a heavy-tailed stochastic volatility model,
, , where the volatility sequence
and the i.i.d. noise sequence are assumed independent, is
regularly varying with index , and the 's have moments of order
larger than . In the literature (see Ann. Appl. Probab. 8 (1998)
664-675, J. Appl. Probab. 38A (2001) 93-104, In Handbook of Financial Time
Series (2009) 355-364 Springer), it is typically assumed that
is a Gaussian stationary sequence and the 's are regularly varying with
some index (i.e., has lighter tails than the 's), or
that is i.i.d. centered Gaussian. In these cases, we see that the
sequence does not exhibit extremal clustering. In contrast to this
situation, under the conditions of this paper, both situations are possible;
may or may not have extremal clustering, depending on the clustering
behavior of the -sequence.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ426 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The integrated periodogram of a dependent extremal event sequence
We investigate the asymptotic properties of the integrated periodogram
calculated from a sequence of indicator functions of dependent extremal events.
An event in Euclidean space is extreme if it occurs far away from the origin.
We use a regular variation condition on the underlying stationary sequence to
make these notions precise. Our main result is a functional central limit
theorem for the integrated periodogram of the indicator functions of dependent
extremal events. The limiting process is a continuous Gaussian process whose
covari- ance structure is in general unfamiliar, but in the iid case a Brownian
bridge appears. In the general case, we propose a stationary bootstrap
procedure for approximating the distribution of the limiting process. The
developed theory can be used to construct classical goodness-of-fit tests such
as the Grenander- Rosenblatt and Cram\'{e}r-von Mises tests which are based
only on the extremes in the sample. We apply the test statistics to simulated
and real-life data
Homogeneous mappings of regularly varying vectors
It is well known that the product of two independent regularly varying random
variables with the same tail index is again regularly varying with this index.
In this paper, we provide sharp sufficient conditions for the regular variation
property of product-type functions of regularly varying random vectors,
generalizing and extending the univariate theory in various directions. The
main result is then applied to characterize the regular variation property of
products of iid regularly varying quadratic random matrices and of solutions to
affine stochastic recurrence equations under non-standard conditions
The extremogram: A correlogram for extreme events
We consider a strictly stationary sequence of random vectors whose
finite-dimensional distributions are jointly regularly varying with some
positive index. This class of processes includes, among others, ARMA processes
with regularly varying noise, GARCH processes with normally or
Student-distributed noise and stochastic volatility models with regularly
varying multiplicative noise. We define an analog of the autocorrelation
function, the extremogram, which depends only on the extreme values in the
sequence. We also propose a natural estimator for the extremogram and study its
asymptotic properties under -mixing. We show asymptotic normality,
calculate the extremogram for various examples and consider spectral analysis
related to the extremogram.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ213 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series
We provide some asymptotic theory for the largest eigenvalues of a sample
covariance matrix of a p-dimensional time series where the dimension p = p_n
converges to infinity when the sample size n increases. We give a short
overview of the literature on the topic both in the light- and heavy-tailed
cases when the data have finite (infinite) fourth moment, respectively. Our
main focus is on the heavytailed case. In this case, one has a theory for the
point process of the normalized eigenvalues of the sample covariance matrix in
the iid case but also when rows and columns of the data are linearly dependent.
We provide limit results for the weak convergence of these point processes to
Poisson or cluster Poisson processes. Based on this convergence we can also
derive the limit laws of various function als of the ordered eigenvalues such
as the joint convergence of a finite number of the largest order statistics,
the joint limit law of the largest eigenvalue and the trace, limit laws for
successive ratios of ordered eigenvalues, etc. We also develop some limit
theory for the singular values of the sample autocovariance matrices and their
sums of squares. The theory is illustrated for simulated data and for the
components of the S&P 500 stock index.Comment: in Extremes; Statistical Theory and Applications in Science,
Engineering and Economics; ISSN 1386-1999; (2016
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