80 research outputs found
Limit theorems for strongly mixing stationary random measures
AbstractCharacterization theorems are obtained for the possible limits in distribution of a family of stationary random measures {ζT} satisfying a strong mixing condition, with necessary and sufficient conditions for convergence. The application of these results to ‘exceedance random measures’ is shown to provide a unifying viewpoint for obtaining results in extremal theory for stationary processes
Ground State Energy of the One-Dimensional Discrete Random Schr\"{o}dinger Operator with Bernoulli Potential
In this paper, we show the that the ground state energy of the one
dimensional Discrete Random Schroedinger Operator with Bernoulli Potential is
controlled asymptotically as the system size N goes to infinity by the random
variable \ell_N, the length the longest consecutive sequence of sites on the
lattice with potential equal to zero. Specifically, we will show that for
almost every realization of the potential the ground state energy behaves
asymptotically as in the sense that the ratio of
the quantities goes to one
Countable Random Sets: Uniqueness in Law and Constructiveness
The first part of this article deals with theorems on uniqueness in law for
\sigma-finite and constructive countable random sets, which in contrast to the
usual assumptions may have points of accumulation. We discuss and compare two
approaches on uniqueness theorems: First, the study of generators for
\sigma-fields used in this context and, secondly, the analysis of hitting
functions. The last section of this paper deals with the notion of
constructiveness. We will prove a measurable selection theorem and a
decomposition theorem for constructive countable random sets, and study
constructive countable random sets with independent increments.Comment: Published in Journal of Theoretical Probability
(http://www.springerlink.com/content/0894-9840/). The final publication is
available at http://www.springerlink.co
The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics
We prove that the distributional limit of the normalised number of returns to
small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical
systems is compound Poisson. The returns to small balls around a fixed point in
the phase space correspond to the occurrence of rare events, or exceedances of
high thresholds, so that there is a connection between the laws of Return Times
Statistics and Extreme Value Laws. The fact that the fixed point in the phase
space is a repelling periodic point implies that there is a tendency for the
exceedances to appear in clusters whose average sizes is given by the Extremal
Index, which depends on the expansion of the system at the periodic point.
We recall that for generic points, the exceedances, in the limit, are
singular and occur at Poisson times. However, around periodic points, the
picture is different: the respective point processes of exceedances converge to
a compound Poisson process, so instead of single exceedances, we have entire
clusters of exceedances occurring at Poisson times with a geometric
distribution ruling its multiplicity.
The systems to which our results apply include: general piecewise expanding
maps of the interval (Rychlik maps), maps with indifferent fixed points
(Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic
Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble
The paper studies the spectral properties of large Wigner, band and sample
covariance random matrices with heavy tails of the marginal distributions of
matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at
Giens, France on September 13-17, 200
Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution
In this paper we perform an analytical and numerical study of Extreme Value
distributions in discrete dynamical systems. In this setting, recent works have
shown how to get a statistics of extremes in agreement with the classical
Extreme Value Theory. We pursue these investigations by giving analytical
expressions of Extreme Value distribution parameters for maps that have an
absolutely continuous invariant measure. We compare these analytical results
with numerical experiments in which we study the convergence to limiting
distributions using the so called block-maxima approach, pointing out in which
cases we obtain robust estimation of parameters. In regular maps for which
mixing properties do not hold, we show that the fitting procedure to the
classical Extreme Value Distribution fails, as expected. However, we obtain an
empirical distribution that can be explained starting from a different
observable function for which Nicolis et al. [2006] have found analytical
results.Comment: 34 pages, 7 figures; Journal of Statistical Physics 201
Size Doesn't Matter: Towards a More Inclusive Philosophy of Biology
notes: As the primary author, O’Malley drafted the paper, and gathered and analysed data (scientific papers and talks). Conceptual analysis was conducted by both authors.publication-status: Publishedtypes: ArticlePhilosophers of biology, along with everyone else, generally perceive life to fall into two broad categories, the microbes and macrobes, and then pay most of their attention to the latter. ‘Macrobe’ is the word we propose for larger life forms, and we use it as part of an argument for microbial equality. We suggest that taking more notice of microbes – the dominant life form on the planet, both now and throughout evolutionary history – will transform some of the philosophy of biology’s standard ideas on ontology, evolution, taxonomy and biodiversity. We set out a number of recent developments in microbiology – including biofilm formation, chemotaxis, quorum sensing and gene transfer – that highlight microbial capacities for cooperation and communication and break down conventional thinking that microbes are solely or primarily single-celled organisms. These insights also bring new perspectives to the levels of selection debate, as well as to discussions of the evolution and nature of multicellularity, and to neo-Darwinian understandings of evolutionary mechanisms. We show how these revisions lead to further complications for microbial classification and the philosophies of systematics and biodiversity. Incorporating microbial insights into the philosophy of biology will challenge many of its assumptions, but also give greater scope and depth to its investigations
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