627,253 research outputs found
Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
We study the variance and the Laplace transform of the probability law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test
function of statistics is smooth enough and using the asymptotic formulas by
Deift et al for orthogonal polynomials with varying weights, we show first that
if the support of the Density of States of the model consists of two or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the Central
Limit Theorem would be valid, and we find the asymptotic form of the Laplace
transform of the probability law in certain cases
The LIL for canonical -statistics
We give necessary and sufficient conditions for the (bounded) law of the
iterated logarithm for canonical -statistics of arbitrary order ,
extending the previously known results for . The nasc's are expressed as
growth conditions on a parameterized family of norms associated with the
-statistics kernel.Comment: Published in at http://dx.doi.org/10.1214/07-AOP351 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Record statistics in random vectors and quantum chaos
The record statistics of complex random states are analytically calculated,
and shown that the probability of a record intensity is a Bernoulli process.
The correlation due to normalization leads to a probability distribution of the
records that is non-universal but tends to the Gumbel distribution
asymptotically. The quantum standard map is used to study these statistics for
the effect of correlations apart from normalization. It is seen that in the
mixed phase space regime the number of intensity records is a power law in the
dimensionality of the state as opposed to the logarithmic growth for random
states.Comment: figures redrawn, discussion adde
Inversion Symmetry and Critical Exponents of Dissipating Waves in the Sandpile Model
Statistics of waves of topplings in the Sandpile model is analysed both
analytically and numerically. It is shown that the probability distribution of
dissipating waves of topplings that touch the boundary of the system obeys
power-law with critical exponent 5/8. This exponent is not indeendent and is
related to the well-known exponent of the probability distribution of last
waves of topplings by exact inversion symmetry s -> 1/s.Comment: 5 REVTeX pages, 6 figure
An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation
The classical law of the iterated logarithm (LIL for short)as fundamental
limit theorems in probability theory play an important role in the development
of probability theory and its applications. Strassen (1964) extended LIL to
large classes of functional random variables, it is well known as the
invariance principle for LIL which provide an extremely powerful tool in
probability and statistical inference. But recently many phenomena show that
the linearity of probability is a limit for applications, for example in
finance, statistics. As while a nonlinear expectation--- G-expectation has
attracted extensive attentions of mathematicians and economists, more and more
people began to study the nature of the G-expectation space. A natural question
is: Can the classical invariance principle for LIL be generalized under
G-expectation space? This paper gives a positive answer. We present the
invariance principle of G-Brownian motion for the law of the iterated logarithm
under G-expectation
Signatures of dual scaling regimes in a simple avalanche model for magnetospheric activity
Recently, the paradigm that the dynamic magnetosphere displays sandpile-type phenomenology has been advanced, in which energy dissipation is by means of avalanches which do not have an intrinsic scale. This may in turn imply that the system is in a self-organised critical (SOC) state. Indicators of internal processes are consistent with this, examples are the power-law dependence of the power spectrum of auroral indices, and in situ magnetic field observations in the earth's geotail. However substorm statistics exhibit probability distributions with characteristic scales. In this paper we discuss a simple sandpile model which yields for energy discharges due to internal reorganisation a probability distribution that is a power-law, whereas systemwide discharges (flow of âsandâ out of the system) form a distinct group whose probability distribution has a well defined mean. When the model is analysed over its full dynamic range, two regimes having different inverse power-law statistics emerge. These correspond to reconfigurations on two distinct length scales: short length scales sensitive to the discrete nature of the sandpile model, and long length scales up to the system size which correspond to the continuous limit of the model. The latter are anticipated to correspond to large-scale systems such as the magnetosphere. Since the energy inflow may be highly variable, the response of the sandpile model is examined under strong or variable loading and it is established that the power-law signature of the large-scale internal events persists. The interval distribution of these events is also discussed
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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