148 research outputs found
Schur-Weyl duality and the heat kernel measure on the unitary group
We establish a convergent power series expansion for the expectation of a
product of traces of powers of a random unitary matrix under the heat kernel
measure. These expectations turn out to be the generating series of certain
paths in the Cayley graph of the symmetric group. We then compute the
asymptotic distribution of a random unitary matrix under the heat kernel
measure on the unitary group U(N) as N tends to infinity, and prove a result of
asymptotic freeness result for independent large unitary matrices, thus
recovering results obtained previously by Xu and Biane. We give an
interpretation of our main expansion in terms of random ramified coverings of a
disk. Our approach is based on the Schur-Weyl duality and we extend some of our
results to the orthogonal and symplectic cases
Finite Gel'fand pairs and their applications to Probability and Statistics
We present a general introduction to finite Gel’fand pairs and their associated spherical functions
yielding different characterizations, examine a few explicit examples, and, for each of these examples,
analyze the corresponding probabilistic problem, which will then be solved by applying the general results
and the machinery developed for a particular Gel’fand pair
Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution
We prove sharp rates of convergence to the Ewens equilibrium distribution for
a family of Metropolis algorithms based on the random transposition shuffle on
the symmetric group, with starting point at the identity. The proofs rely
heavily on the theory of symmetric Jack polynomials, developed initially by
Jack [Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971) 1-18], Macdonald
[Symmetric Functions and Hall Polynomials (1995) New York] and Stanley [Adv.
Math. 77 (1989) 76-115]. This completes the analysis started by Diaconis and
Hanlon in [Contemp. Math. 138 (1992) 99-117]. In the end we also explore other
integrable Markov chains that can be obtained from symmetric function theory.Comment: Published at http://dx.doi.org/10.1214/14-AAP1031 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
An Introduction to Wishart Matrix Moments
These lecture notes provide a comprehensive, self-contained introduction to
the analysis of Wishart matrix moments. This study may act as an introduction
to some particular aspects of random matrix theory, or as a self-contained
exposition of Wishart matrix moments. Random matrix theory plays a central role
in statistical physics, computational mathematics and engineering sciences,
including data assimilation, signal processing, combinatorial optimization,
compressed sensing, econometrics and mathematical finance, among numerous
others. The mathematical foundations of the theory of random matrices lies at
the intersection of combinatorics, non-commutative algebra, geometry,
multivariate functional and spectral analysis, and of course statistics and
probability theory. As a result, most of the classical topics in random matrix
theory are technical, and mathematically difficult to penetrate for non-experts
and regular users and practitioners. The technical aim of these notes is to
review and extend some important results in random matrix theory in the
specific context of real random Wishart matrices. This special class of
Gaussian-type sample covariance matrix plays an important role in multivariate
analysis and in statistical theory. We derive non-asymptotic formulae for the
full matrix moments of real valued Wishart random matrices. As a corollary, we
derive and extend a number of spectral and trace-type results for the case of
non-isotropic Wishart random matrices. We also derive the full matrix moment
analogues of some classic spectral and trace-type moment results. For example,
we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and
full matrix cases. Laplace matrix transforms and matrix moment estimates are
also studied, along with new spectral and trace concentration-type
inequalities
A selected survey of umbral calculus
We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of "magic rules" for lowering and raising indices, through its rebirth in the 1970’s as Rota’s school set it on a firm logical foundation using operator methods, to the current state of the art with numerous generalizations and applications. The survey itself is complemented by a fairly complete bibliography (over 500 references) which we expect to update regularly
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