21,533 research outputs found
Asymptotic properties of random matrices and pseudomatrices
We study the asymptotics of sums of matricially free random variables called
random pseudomatrices, and we compare it with that of random matrices with
block-identical variances. For objects of both types we find the limit joint
distributions of blocks and give their Hilbert space realizations, using
operators called `matricially free Gaussian operators'. In particular, if the
variance matrices are symmetric, the asymptotics of symmetric blocks of random
pseudomatrices agrees with that of symmetric random blocks. We also show that
blocks of random pseudomatrices are `asymptotically matricially free' whereas
the corresponding symmetric random blocks are `asymptotically symmetrically
matricially free', where symmetric matricial freeness is obtained from
matricial freeness by an operation of symmetrization. Finally, we show that row
blocks of square, lower-block-triangular and block-diagonal pseudomatrices are
asymptotically free, monotone independent and boolean independent,
respectively.Comment: 33 pages, 2 figure
Gaussian fluctuations of characters of symmetric groups and of Young diagrams
We study asymptotics of reducible representations of the symmetric groups S_q
for large q. We decompose such a representation as a sum of irreducible
components (or, alternatively, Young diagrams) and we ask what is the character
of a randomly chosen component (or, what is the shape of a randomly chosen
Young diagram). Our main result is that for a large class of representations
the fluctuations of characters (and fluctuations of the shape of the Young
diagrams) are asymptotically Gaussian; in this way we generalize Kerov's
central limit theorem. The considered class consists of representations for
which the characters almost factorize and this class includes, for example,
left-regular representation (Plancherel measure), tensor representations. This
class is also closed under induction, restriction, outer product and tensor
product of representations. Our main tool in the proof is the method of genus
expansion, well known from the random matrix theory.Comment: 37 pages; version 3: conceptual change in the proof
Mode-Dependent Loss and Gain: Statistics and Effect on Mode-Division Multiplexing
In multimode fiber transmission systems, mode-dependent loss and gain
(collectively referred to as MDL) pose fundamental performance limitations. In
the regime of strong mode coupling, the statistics of MDL (expressed in
decibels or log power gain units) can be described by the eigenvalue
distribution of zero-trace Gaussian unitary ensemble in the small-MDL region
that is expected to be of interest for practical long-haul transmission.
Information-theoretic channel capacities of mode-division-multiplexed systems
in the presence of MDL are studied, including average and outage capacities,
with and without channel state information.Comment: 22 pages, 8 figure
The Trace Problem for Toeplitz Matrices and Operators and its Impact in Probability
The trace approximation problem for Toeplitz matrices and its applications to
stationary processes dates back to the classic book by Grenander and Szeg\"o,
"Toeplitz forms and their applications". It has then been extensively studied
in the literature.
In this paper we provide a survey and unified treatment of the trace
approximation problem both for Toeplitz matrices and for operators and describe
applications to discrete- and continuous-time stationary processes.
The trace approximation problem serves indeed as a tool to study many
probabilistic and statistical topics for stationary models. These include
central and non-central limit theorems and large deviations of Toeplitz type
random quadratic functionals, parametric and nonparametric estimation,
prediction of the future value based on the observed past of the process, etc.
We review and summarize the known results concerning the trace approximation
problem, prove some new results, and provide a number of applications to
discrete- and continuous-time stationary time series models with various types
of memory structures, such as long memory, anti-persistent and short memory
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