Linear functions on the classical matrix groups

@article{Meckes2005LinearFO,
  title={Linear functions on the classical matrix groups},
  author={Elizabeth S. Meckes},
  journal={Transactions of the American Mathematical Society},
  year={2005},
  volume={360},
  pages={5355-5366}
}
  • Elizabeth S. Meckes
  • Published 20 September 2005
  • Mathematics
  • Transactions of the American Mathematical Society
Let M be a random matrix in the orthogonal group On, distributed according to Haar measure, and let A be a fixed n x n matrix over R such that Tr(AA t ) = n. Then the total variation distance of the random variable Tr(AM) to a standard normal random variable is bounded by 2√3 n-1 and this rate is sharp up to the constant. Analogous results are obtained for M a random unitary matrix and A a fixed n x n matrix over C. The proofs are applications of a new abstract normal approximation theorem… Expand
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References

SHOWING 1-10 OF 25 REFERENCES
On the Eigenvalues of Random Matrices
Let M be a random matrix chosen from Haar measure on the unitary group U,,. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance 4normalExpand
ON RANDOM MATRICES FROM THE COMPACT CLASSICAL GROUPS
If M is a matrix taken randomly with respect to normalized Haar measure on U(n), O(n) or Sp(n), then the real and imaginary parts of the random variables Tr(Mk), k > 1, converge to independent normalExpand
Maxima of entries of Haar distributed matrices
Abstract.Let Γn=(γij) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also Wn:=max1≤i,j≤n|γij|. We obtain the limiting distributionExpand
Brownian motion and the classical groups
Let G be chosen from the orthogonal group On according to Haar measure, and let A be an ? ? ? real matrix with non-random entries satisfying TrAA1 = n. We show that TrAT converges in distribution toExpand
Finite de Finetti theorems in linear models and multivariate analysis
Let Xl,-.. , Xk be a sequence of random vectors. We give symmetry conditions on the joint distribution which imply that it is well approximated by a mixture of normal distributions. Examples includeExpand
How many entries of a typical orthogonal matrix can be approximated by independent normals
We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n , the p n x q n upper left block of a random matrix r n which is uniformly distributed on theExpand
The Central Limit Problem for Random Vectors with Symmetries
Abstract Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. InExpand
Invariant Integration over the Unitary Group
Integrals for the product of unitary-matrix elements over the U(n) group will be discussed. A group-theoretical formula is available to convert them into a multiple sum, but unfortunately the sumsExpand
A Combinatorial Central Limit Theorem
Let (Y n1,…,Y nn be a random vector which takes on the n! permutations of (1,…, n) with equal probabilities. Let c n(i,j), i,j = 1, …, n, be n real numbers. Sufficient conditions for the asymptoticExpand
How to Integrate A Polynomial Over A Sphere
  • G. Folland
  • Mathematics, Computer Science
  • Am. Math. Mon.
  • 2001
Several recent articles in the MONTHLY ([1], [2], [4]) have involved finding the area of n-dimensional balls or spheres or integrating polynomials over such sets. None of these articles, however,Expand
...
1
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3
...