4,456 research outputs found
Terminal chords in connected chord diagrams
Rooted connected chord diagrams form a nice class of combinatorial objects.
Recently they were shown to index solutions to certain Dyson-Schwinger
equations in quantum field theory. Key to this indexing role are certain
special chords which are called terminal chords. Terminal chords provide a
number of combinatorially interesting parameters on rooted connected chord
diagrams which have not been studied previously. Understanding these parameters
better has implications for quantum field theory.
Specifically, we show that the distributions of the number of terminal chords
and the number of adjacent terminal chords are asymptotically Gaussian with
logarithmic means, and we prove that the average index of the first terminal
chord is . Furthermore, we obtain a method to determine any next-to
leading log expansion of the solution to these Dyson-Schwinger equations, and
have asymptotic information about the coefficients of the log expansions.Comment: 25 page
Higher-Order Improvements of the Sieve Bootstrap for Fractionally Integrated Processes
This paper investigates the accuracy of bootstrap-based inference in the case
of long memory fractionally integrated processes. The re-sampling method is
based on the semi-parametric sieve approach, whereby the dynamics in the
process used to produce the bootstrap draws are captured by an autoregressive
approximation. Application of the sieve method to data pre-filtered by a
semi-parametric estimate of the long memory parameter is also explored.
Higher-order improvements yielded by both forms of re-sampling are demonstrated
using Edgeworth expansions for a broad class of statistics that includes first-
and second-order moments, the discrete Fourier transform and regression
coefficients. The methods are then applied to the problem of estimating the
sampling distributions of the sample mean and of selected sample
autocorrelation coefficients, in experimental settings. In the case of the
sample mean, the pre-filtered version of the bootstrap is shown to avoid the
distinct underestimation of the sampling variance of the mean which the raw
sieve method demonstrates in finite samples, higher order accuracy of the
latter notwithstanding. Pre-filtering also produces gains in terms of the
accuracy with which the sampling distributions of the sample autocorrelations
are reproduced, most notably in the part of the parameter space in which
asymptotic normality does not obtain. Most importantly, the sieve bootstrap is
shown to reproduce the (empirically infeasible) Edgeworth expansion of the
sampling distribution of the autocorrelation coefficients, in the part of the
parameter space in which the expansion is valid
The Nagaev-Guivarc'h method via the Keller-Liverani theorem
The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller
and Liverani, has been exploited in recent papers to establish local limit and
Berry-Essen type theorems for unbounded functionals of strongly ergodic Markov
chains. The main difficulty of this approach is to prove Taylor expansions for
the dominating eigenvalue of the Fourier kernels. This paper outlines this
method and extends it by proving a multi-dimensional local limit theorem, a
first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type
theorem in the sense of Prohorov metric. When applied to uniformly or
geometrically ergodic chains and to iterative Lipschitz models, the above cited
limit theorems hold under moment conditions similar, or close, to those of the
i.i.d. case
Gaussian fluctuations for linear spectral statistics of large random covariance matrices
Consider a matrix ,
where is a nonnegative definite Hermitian matrix and is a random
matrix with i.i.d. real or complex standardized entries. The fluctuations of
the linear statistics of the eigenvalues are shown to be Gaussian, in the regime
where both dimensions of matrix go to infinity at the same pace and
in the case where is of class , that is, has three continuous
derivatives. The main improvements with respect to Bai and Silverstein's CLT
[Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general
entries with finite fourth moment, but whose fourth cumulant is nonnull, that
is, whose fourth moment may differ from the moment of a (real or complex)
Gaussian random variable. As a consequence, extra terms proportional to and
appear in the limiting variance and in the limiting bias, which not only
depend on the spectrum of matrix but also on its eigenvectors. Second, we
relax the analyticity assumption over by representing the linear statistics
with the help of Helffer-Sj\"{o}strand's formula. The CLT is expressed in terms
of vanishing L\'{e}vy-Prohorov distance between the linear statistics'
distribution and a Gaussian probability distribution, the mean and the variance
of which depend upon and and may not converge.Comment: Published at http://dx.doi.org/10.1214/15-AAP1135 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Strong Approximation of Empirical Copula Processes by Gaussian Processes
We provide the strong approximation of empirical copula processes by a
Gaussian process. In addition we establish a strong approximation of the
smoothed empirical copula processes and a law of iterated logarithm
Frequency of pattern occurrences in Motzkind words
In this work we show that the number of horizontal steps in a Motzkin word of length n, drawn at random under uniform distribution, has a Gaussian limit distribution. We also prove a local limit property for the same random variable which stresses its periodic behaviour. Similar results are obtained for the number of peaks in a word of given length drawn at random from the same language
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