352 research outputs found
Edgeworth expansions for errors-in-variables models
AbstractEdgeworth expansions for sums of independent but not identically distributed multivariate random vectors are established. The results are applied to get valid Edgeworth expansions for estimates of regression parameters in linear errors-in-variable models. The expansions for studentized versions are also developed. Further, Edgeworth expansions for the corresponding bootstrapped statistics are obtained. Using these expansions, the bootstrap distribution is shown to approximate the sampling distribution of the studentized estimators, better than the classical normal approximation
Limiting behavior of the eigenvalues of a multivariate F matrix
AbstractThe spectral distribution of a central multivariate F matrix is shown to tend to a limit distribution in probability under certain conditions as the number of variables and the degrees of freedom tend to infinity
Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices
We study the fluctuations of the matrix entries of regular functions of
Wigner random matrices in the limit when the matrix size goes to infinity. In
the case of the Gaussian ensembles (GOE and GUE) this problem was considered by
A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are
valid provided the off-diagonal matrix entries have finite fourth moment, the
diagonal matrix entries have finite second moment, and the test functions have
four continuous derivatives in a neighborhood of the support of the Wigner
semicircle law.Comment: minor corrections; the manuscript will appear in the Journal of
Statistical Physic
On determination of the order of an autoregressive model
AbstractTo determine the order of an autoregressive model, a new method based on information theoretic criterion is proposed. This method is shown to be strongly consistent and the convergence rate of the probability of wrong determination is established
Random matrices: Universality of local eigenvalue statistics up to the edge
This is a continuation of our earlier paper on the universality of the
eigenvalues of Wigner random matrices. The main new results of this paper are
an extension of the results in that paper from the bulk of the spectrum up to
the edge. In particular, we prove a variant of the universality results of
Soshnikov for the largest eigenvalues, assuming moment conditions rather than
symmetry conditions. The main new technical observation is that there is a
significant bias in the Cauchy interlacing law near the edge of the spectrum
which allows one to continue ensuring the delocalization of eigenvectors.Comment: 24 pages, no figures, to appear, Comm. Math. Phys. One new reference
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On the distinguishability of random quantum states
We develop two analytic lower bounds on the probability of success p of
identifying a state picked from a known ensemble of pure states: a bound based
on the pairwise inner products of the states, and a bound based on the
eigenvalues of their Gram matrix. We use the latter to lower bound the
asymptotic distinguishability of ensembles of n random quantum states in d
dimensions, where n/d approaches a constant. In particular, for almost all
ensembles of n states in n dimensions, p>0.72. An application to distinguishing
Boolean functions (the "oracle identification problem") in quantum computation
is given.Comment: 20 pages, 2 figures; v2 fixes typos and an error in an appendi
Circular Law Theorem for Random Markov Matrices
Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded
density, mean m, and finite positive variance sigma^2. Let M be the nxn random
Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its
sum. In particular, when X11 follows an exponential law, then M belongs to the
Dirichlet Markov Ensemble of random stochastic matrices. Our main result states
that with probability one, the counting probability measure of the complex
spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law
on the centered disk of radius sigma/m. The bounded density assumption is
purely technical and comes from the way we control the operator norm of the
resolvent.Comment: technical update via http://HAL.archives-ouvertes.f
Structured Random Matrices
Random matrix theory is a well-developed area of probability theory that has
numerous connections with other areas of mathematics and its applications. Much
of the literature in this area is concerned with matrices that possess many
exact or approximate symmetries, such as matrices with i.i.d. entries, for
which precise analytic results and limit theorems are available. Much less well
understood are matrices that are endowed with an arbitrary structure, such as
sparse Wigner matrices or matrices whose entries possess a given variance
pattern. The challenge in investigating such structured random matrices is to
understand how the given structure of the matrix is reflected in its spectral
properties. This chapter reviews a number of recent results, methods, and open
problems in this direction, with a particular emphasis on sharp spectral norm
inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and
Applications" (Springer
Local semicircle law and complete delocalization for Wigner random matrices
We consider Hermitian random matrices with independent identical
distributed entries. The matrix is normalized so that the average spacing
between consecutive eigenvalues is of order 1/N. Under suitable assumptions on
the distribution of the single matrix element, we prove that, away from the
spectral edges, the density of eigenvalues concentrates around the Wigner
semicircle law on energy scales . Up to the
logarithmic factor, this is the smallest energy scale for which the semicircle
law may be valid. We also prove that for all eigenvalues away from the spectral
edges, the -norm of the corresponding eigenvectors is of order
, modulo logarithmic corrections. The upper bound
implies that every eigenvector is completely delocalized, i.e., the maximum
size of the components of the eigenvector is of the same order as their average
size. In the Appendix, we include a lemma by J. Bourgain which removes one of
our assumptions on the distribution of the matrix elements.Comment: 14 pages, LateX file. An appendix by J. Bourgain was added. Final
version, to appear in Comm. Math. Phy
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