Linear functions on the classical matrix groups

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a random matrix in the orthogonal group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {O}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , distributed according to Haar measure, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a fixed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> × </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper T normal r left-parenthesis upper A upper A Superscript t Baseline right-parenthesis equals n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>t</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Tr}(AA^t)=n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Then the total variation distance of the random variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper T normal r left-parenthesis upper A upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">T</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Tr}(AM)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a standard normal random variable is bounded by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 2 StartRoot 3 EndRoot Over n minus 1 EndFraction"> <mml:semantics> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {2\sqrt {3}} {n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and this rate is sharp up to the constant. Analogous results are obtained for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a random unitary matrix and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a fixed <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> × </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The proofs are applications of a new abstract normal approximation theorem which extends Stein’s method of exchangeable pairs to situations in which continuous symmetries are present.