# 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} }

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…

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## References

SHOWING 1-10 OF 22 REFERENCES

### ON RANDOM MATRICES FROM THE COMPACT CLASSICAL GROUPS

- Mathematics
- 1997

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 normal…

### Maxima of entries of Haar distributed matrices

- Mathematics
- 2005

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 distribution…

### Brownian motion and the classical groups

- Mathematics
- 2002

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 to…

### Finite de Finetti theorems in linear models and multivariate analysis

- Mathematics
- 1992

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 include…

### How many entries of a typical orthogonal matrix can be approximated by independent normals

- Mathematics
- 2006

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 the…

### The Central Limit Problem for Random Vectors with Symmetries

- Mathematics
- 2007

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. In…

### Invariant Integration over the Unitary Group

- Mathematics
- 2003

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 sums…

### A Combinatorial Central Limit Theorem

- Mathematics
- 1951

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 asymptotic…

### PATTERNS IN EIGENVALUES: THE 70TH JOSIAH WILLARD GIBBS LECTURE

- Mathematics
- 2003

Typical large unitary matrices show remarkable patterns in their eigenvalue distribution. These same patterns appear in telephone encryption, the zeros of Riemann’s zeta function, a variety of…

### The semicircle law, free random variables, and entropy

- Mathematics
- 2000

Overview Probability laws and noncommutative random variables The free relation Analytic function theory and infinitely divisible laws Random matrices and asymptotically free relation Large…