87,657 research outputs found
On the Limiting Spectral Density of Random Matrices filled with Stochastic Processes
We discuss the limiting spectral density of real symmetric random matrices.
Other than in standard random matrix theory the upper diagonal entries are not
assumed to be independent, but we will fill them with the entries of a
stochastic process. Under assumptions on this process, which are satisfied,
e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures
on finite state spaces, or by Gaussian Markov processes, we show that the
limiting spectral distribution depends on the way the matrix is filled with the
stochastic process. If the filling is in a certain way compatible with the
symmetry condition on the matrix, the limiting law of the empirical eigenvalue
distribution is the well known semi-circle law. For other fillings we show that
the semi-circle law cannot be the limiting spectral density.Comment: 23 pages, 3 figure
Determinantal processes and completeness of random exponentials: the critical case
For a locally finite point set , consider the
collection of exponential functions given by . We examine the question whether
spans the Hilbert space , when
is random. For several point processes of interest, this belongs to a certain
critical case of the corresponding question for deterministic , about
which little is known. For the continuum sine kernel process,
obtained as the bulk limit of GUE eigenvalues, we establish that
is indeed complete. We also answer an analogous
question on for the Ginibre ensemble, arising as weak limits of
certain non-Hermitian random matrix eigenvalues. In fact we establish
completeness for any "rigid" determinantal point process in a general setting.
In addition, we partially answer two questions due to Lyons and Steif about
stationary determinantal processes on .Comment: To appear in Probability Theory and Related Field
Compression Limits for Random Vectors with Linearly Parameterized Second-Order Statistics
The class of complex random vectors whose covariance matrix is linearly
parameterized by a basis of Hermitian Toeplitz (HT) matrices is considered, and
the maximum compression ratios that preserve all second-order information are
derived --- the statistics of the uncompressed vector must be recoverable from
a set of linearly compressed observations. This kind of vectors arises
naturally when sampling wide-sense stationary random processes and features a
number of applications in signal and array processing.
Explicit guidelines to design optimal and nearly optimal schemes operating
both in a periodic and non-periodic fashion are provided by considering two of
the most common linear compression schemes, which we classify as dense or
sparse. It is seen that the maximum compression ratios depend on the structure
of the HT subspace containing the covariance matrix of the uncompressed
observations. Compression patterns attaining these maximum ratios are found for
the case without structure as well as for the cases with circulant or banded
structure. Universal samplers are also proposed to compress unknown HT
subspaces.Comment: 15 pages, 2 figures, 1 table, submitted to IEEE Transactions on
Information Theory on Nov. 4, 201
Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes
We propose a stochastic method for solving Schwinger-Dyson equations in
large-N quantum field theories. Expectation values of single-trace operators
are sampled by stationary probability distributions of the so-called nonlinear
random processes. The set of all histories of such processes corresponds to the
set of all planar diagrams in the perturbative expansions of the expectation
values of singlet operators. We illustrate the method on the examples of the
matrix-valued scalar field theory and the Weingarten model of random planar
surfaces on the lattice. For theories with compact field variables, such as
sigma-models or non-Abelian lattice gauge theories, the method does not
converge in the physically most interesting weak-coupling limit. In this case
one can absorb the divergences into a self-consistent redefinition of expansion
parameters. Stochastic solution of the self-consistency conditions can be
implemented as a "memory" of the random process, so that some parameters of the
process are estimated from its previous history. We illustrate this idea on the
example of two-dimensional O(N) sigma-model. Extension to non-Abelian lattice
gauge theories is discussed.Comment: 16 pages RevTeX, 14 figures; v2: Algorithm for the Weingarten model
corrected; v3: published versio
Entropic Equilibria Selection of Stationary Extrema in Finite Populations
We propose the entropy of random Markov trajectories originating and
terminating at a state as a measure of the stability of a state of a Markov
process. These entropies can be computed in terms of the entropy rates and
stationary distributions of Markov processes. We apply this definition of
stability to local maxima and minima of the stationary distribution of the
Moran process with mutation and show that variations in population size,
mutation rate, and strength of selection all affect the stability of the
stationary extrema
On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields
Many statistical applications require establishing central limit theorems for
sums, integrals, or for quadratic forms of functions of a stationary process. A
particularly important case is that of Appell polynomials, since the Appell
expansion rank" determines typically the type of central limit theorem
satisfied by these functionals. We review and extend here to multidimensional
indices a functional analysis approach to this problem proposed by Avram and
Brown (1989), based on the method of cumulants and on integrability assumptions
in the spectral domain; several applications are presented as well
The Distortion Rate Function of Cyclostationary Gaussian Processes
A general expression for the distortion rate function (DRF) of
cyclostationary Gaussian processes in terms of their spectral properties is
derived. This expression can be seen as the result of orthogonalization over
the different components in the polyphase decomposition of the process. We use
this expression to derive, in a closed form, the DRF of several cyclostationary
processes arising in practice. We first consider the DRF of a combined sampling
and source coding problem. It is known that the optimal coding strategy for
this problem involves source coding applied to a signal with the same structure
as one resulting from pulse amplitude modulation (PAM). Since a PAM-modulated
signal is cyclostationary, our DRF expression can be used to solve for the
minimal distortion in the combined sampling and source coding problem. We also
analyze in more detail the DRF of a source with the same structure as a
PAM-modulated signal, and show that it is obtained by reverse waterfilling over
an expression that depends on the energy of the pulse and the baseband process
modulated to obtain the PAM signal. This result is then used to study the
information content of a PAM-modulated signal as a function of its symbol time
relative to the bandwidth of the underlying baseband process. In addition, we
also study the DRF of sources with an amplitude-modulation structure, and show
that the DRF of a narrow-band Gaussian stationary process modulated by either a
deterministic or a random phase sine-wave equals the DRF of the baseband
process.Comment: First revision for the IEEE Transactions on Information Theor
On Hidden States in Quantum Random Walks
It was recently pointed out that identifiability of quantum random walks and
hidden Markov processes underlie the same principles. This analogy immediately
raises questions on the existence of hidden states also in quantum random walks
and their relationship with earlier debates on hidden states in quantum
mechanics. The overarching insight was that not only hidden Markov processes,
but also quantum random walks are finitary processes. Since finitary processes
enjoy nice asymptotic properties, this also encourages to further investigate
the asymptotic properties of quantum random walks. Here, answers to all these
questions are given. Quantum random walks, hidden Markov processes and finitary
processes are put into a unifying model context. In this context, quantum
random walks are seen to not only enjoy nice ergodic properties in general, but
also intuitive quantum-style asymptotic properties. It is also pointed out how
hidden states arising from our framework relate to hidden states in earlier,
prominent treatments on topics such as the EPR paradoxon or Bell's
inequalities.Comment: 26 page
Modeling of Stationary Periodic Time Series by ARMA Representations
This is a survey of some recent results on the rational circulant covariance
extension problem: Given a partial sequence of covariance
lags emanating from a stationary
periodic process with period , find all possible rational
spectral functions of of degree at most or, equivalently, all
bilateral and unilateral ARMA models of order at most , having this partial
covariance sequence. Each representation is obtained as the solution of a pair
of dual convex optimization problems. This theory is then reformulated in terms
of circulant matrices and the connections to reciprocal processes and the
covariance selection problem is explained. Next it is shown how the theory can
be extended to the multivariate case. Finally, an application to image
processing is presented
Characterization of the tail behavior of a class of BEKK processes: A stochastic recurrence equation approach
We provide new, mild conditions for strict stationarity and ergodicity of a
class of BEKK processes. By exploiting that the processes can be represented as
multivariate stochastic recurrence equations, we characterize the tail behavior
of the associated stationary laws. Specifically, we show that the each
component of the BEKK processes is regularly varying with some tail index. In
general, the tail index differs along the components, which contrasts most of
the existing literature on the tail behavior of multivariate GARCH processes.Comment: Keywords: Regular variation, GARCH, BEKK, stochastic recurrence
equation. Statistics classified with econometrics, JEL: C32 and C58, 30 page
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