Spring Seminar Series - April 17, 2003
University
of Minnesota
School
of Statistics
College
of Liberal Arts
How Many Entries of a Typical Orthogonal Matrix Can Be Approximated
by Independent Normals?
Tiefeng Jiang
School of Statistics
University of Minnesota
Thursday, April 17, 2003
4:00 PM, 115
Ford Hall
Minneapolis, East Bank Campus
Social at 3:30 PM, 300
Ford Hall
Abstract
I
will present my solution to the open problem by Diaconis stated as follows:
What are the largest orders of $p_n$ and $q_n$ such that $Z_n,$ the $p_n\times
q_n$ left upper block of an $n$ by $n$ typical orthogonal matrix $\bold{\Gamma}_n,$
can be approximated by independent standard normals? This problem is
solved by two different approximation methods.
First, we show that the largest order of $p_n$ and $q_n$ are $o(\sqrt{n})$
in the sense of approximation by variation norm.
Second, suppose $\bold{\Gamma}_n=(\gamma_{ij})_{n\times n}$ is generated
by $\bold{Y}_n=(y_{ij})_{n\times n}$ through the Gram-Schmidt algorithm
where $\{y_{ij}; 1\leq i, j \leq n\}$ are i.i.d. standard normals. We show
the largest order of $m=m_n$ such that {\it $\epsilon_n(m):=\max_{1\leq
i \leq n, 1\leq j \leq m}|\sqrt{n}\gamma_{ij}-y_{ij}|$ goes to zero in
probability} is $o(n/\log n).$
A history of the problem from Mechanics, Statistics and Imaging Analysis
will also be presented.