36,862 research outputs found
Limit theory for geometric statistics of point processes having fast decay of correlations
Let be a simple,stationary point process having fast decay of
correlations, i.e., its correlation functions factorize up to an additive error
decaying faster than any power of the separation distance. Let be its restriction to windows . We consider the statistic where denotes a score function
representing the interaction of with respect to . When depends
on local data in the sense that its radius of stabilization has an exponential
tail, we establish expectation asymptotics, variance asymptotics, and CLT for
and, more generally, for statistics of the re-scaled, possibly
signed, -weighted point measures , as . This gives the
limit theory for non-linear geometric statistics (such as clique counts,
intrinsic volumes of the Boolean model, and total edge length of the
-nearest neighbors graph) of -determinantal point processes having
fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear
statistics. It also gives the limit theory for geometric U-statistics of
-permanental point processes and the zero set of Gaussian entire
functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and
Takahashi (2003), which are also confined to linear statistics. The proof of
the central limit theorem relies on a factorial moment expansion originating in
Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast
decay of the correlations of -weighted point measures. The latter property
is shown to imply a condition equivalent to Brillinger mixing and consequently
yields the CLT for via an extension of the cumulant method.Comment: 62 pages. Fundamental changes to the terminology including the title.
The earlier 'clustering' condition is now introduced as a notion of mixing
and its connection to Brillinger mixing is remarked. Newer results for
superposition of independent point processes have been adde
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs
Central limit theorems for linear statistics of lattice random fields
(including spin models) are usually proven under suitable mixing conditions or
quasi-associativity. Many interesting examples of spin models do not satisfy
mixing conditions, and on the other hand, it does not seem easy to show central
limit theorem for local statistics via quasi-associativity. In this work, we
prove general central limit theorems for local statistics and exponentially
quasi-local statistics of spin models on discrete Cayley graphs with polynomial
growth. Further, we supplement these results by proving similar central limit
theorems for random fields on discrete Cayley graphs and taking values in a
countable space but under the stronger assumptions of {\alpha}-mixing (for
local statistics) and exponential {\alpha}-mixing (for exponentially
quasi-local statistics). All our central limit theorems assume a suitable
variance lower bound like many others in the literature. We illustrate our
general central limit theorem with specific examples of lattice spin models and
statistics arising in computational topology, statistical physics and random
networks. Examples of clustering spin models include quasi-associated spin
models with fast decaying covariances like the off-critical Ising model, level
sets of Gaussian random fields with fast decaying covariances like the massive
Gaussian free field and determinantal point processes with fast decaying
kernels. Examples of local statistics include intrinsic volumes, face counts,
component counts of random cubical complexes while exponentially quasi-local
statistics include nearest neighbour distances in spin models and Betti numbers
of sub-critical random cubical complexes.Comment: Minor changes incorporated based on suggestions by referee
Extremal dichotomy for uniformly hyperbolic systems
We consider the extreme value theory of a hyperbolic toral automorphism showing that if a H\"older observation
which is a function of a Euclidean-type distance to a non-periodic point
is strictly maximized at then the corresponding time series
exhibits extreme value statistics corresponding to an iid
sequence of random variables with the same distribution function as and
with extremal index one. If however is strictly maximized at a periodic
point then the corresponding time-series exhibits extreme value statistics
corresponding to an iid sequence of random variables with the same distribution
function as but with extremal index not equal to one. We give a formula
for the extremal index (which depends upon the metric used and the period of
). These results imply that return times are Poisson to small balls centered
at non-periodic points and compound Poisson for small balls centered at
periodic points.Comment: 21 pages, 4 figure
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
Clustering of extreme events created by multiple correlated maxima
We consider stochastic processes arising from dynamical systems by evaluating
an observable function along the orbits of the system. The novelty is that we
will consider observables achieving a global maximum value (possible infinite)
at multiple points with special emphasis for the case where these maximal
points are correlated or bound by belonging to the same orbit of a certain
chosen point. These multiple correlated maxima can be seen as a new mechanism
creating clustering. We recall that clustering was intimately connected with
periodicity when the maximum was achieved at a single point. We will study this
mechanism for creating clustering and will address the existence of limiting
Extreme Value Laws, the repercussions on the value of the Extremal Index, the
impact on the limit of Rare Events Points Processes, the influence on
clustering patterns and the competition of domains of attraction. We also
consider briefly and for comparison purposes multiple uncorrelated maxima. The
systems considered include expanding maps of the interval such as Rychlik maps
but also maps with an indifferent fixed point such as Manneville-Pommeau maps
Complete convergence and records for dynamically generated stochastic processes
We consider empirical multi-dimensional Rare Events Point Processes that keep
track both of the time occurrence of extremal observations and of their
severity, for stochastic processes arising from a dynamical system, by
evaluating a given potential along its orbits. This is done both in the absence
and presence of clustering. A new formula for the piling of points on the
vertical direction of bi-dimensional limiting point processes, in the presence
of clustering, is given, which is then generalised for higher dimensions. The
limiting multi-dimensional processes are computed for systems with sufficiently
fast decay of correlations. The complete convergence results are used to study
the effect of clustering on the convergence of extremal processes, record time
and record values point processes. An example where the clustering prevents the
convergence of the record times point process is given
Mark correlations: relating physical properties to spatial distributions
Mark correlations provide a systematic approach to look at objects both
distributed in space and bearing intrinsic information, for instance on
physical properties. The interplay of the objects' properties (marks) with the
spatial clustering is of vivid interest for many applications; are, e.g.,
galaxies with high luminosities more strongly clustered than dim ones? Do
neighbored pores in a sandstone have similar sizes? How does the shape of
impact craters on a planet depend on the geological surface properties? In this
article, we give an introduction into the appropriate mathematical framework to
deal with such questions, i.e. the theory of marked point processes. After
having clarified the notion of segregation effects, we define universal test
quantities applicable to realizations of a marked point processes. We show
their power using concrete data sets in analyzing the luminosity-dependence of
the galaxy clustering, the alignment of dark matter halos in gravitational
-body simulations, the morphology- and diameter-dependence of the Martian
crater distribution and the size correlations of pores in sandstone. In order
to understand our data in more detail, we discuss the Boolean depletion model,
the random field model and the Cox random field model. The first model
describes depletion effects in the distribution of Martian craters and pores in
sandstone, whereas the last one accounts at least qualitatively for the
observed luminosity-dependence of the galaxy clustering.Comment: 35 pages, 12 figures. to be published in Lecture Notes of Physics,
second Wuppertal conference "Spatial statistics and statistical physics
Rare Events for the Manneville-Pomeau map
We prove a dichotomy for Manneville-Pomeau maps : given
any point , either the Rare Events Point Processes (REPP),
counting the number of exceedances, which correspond to entrances in balls
around , converge in distribution to a Poisson process; or the point
is periodic and the REPP converge in distribution to a compound Poisson
process. Our method is to use inducing techniques for all points except 0 and
its preimages, extending a recent result by Haydn, Winterberg and Zweim\"uller,
and then to deal with the remaining points separately. The preimages of 0 are
dealt with applying recent results by Ayta\c{c}, Freitas and Vaienti. The point
is studied separately because the tangency with the identity map at
this point creates too much dependence, which causes severe clustering of
exceedances. The Extremal Index, which measures the intensity of clustering, is
equal to 0 at , which ultimately leads to a degenerate limit
distribution for the partial maxima of stochastic processes arising from the
dynamics and for the usual normalising sequences. We prove that using adapted
normalising sequences we can still obtain non-degenerate limit distributions at
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