Random circulant matrices

213 Pages·2018·25.302 MB·English

by  Bose, Arup, Saha, Koushik

Random Circulant
Matrices

Random Circulant
Matrices
By
Arup Bose
Koushik Saha
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Bose, Arup, author. | Saha, Koushik,
author.
Title: Random circulant matrices / Arup Bose and Koushik Saha.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018.
Identifiers: LCCN 2018028758 | ISBN 9781138351097 (hardback)
Subjects: LCSH: Random matrices--Problems, exercises, etc. |
Matrices--Problems, exercises, etc. | Eigenvalues--Problems, exercises, etc.
Classification: LCC QA196.5 .B6725 2018 | DDC 512.9/434--dc23
LC record available at https://lccn.loc.gov/2018028758
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To the memory of my teacher J.C. Gupta
A.B.
To my parents and teachers
K.S.

Contents
Preface xi
About the Authors xiii
Introduction xv
1 Circulants 1
1.1 Circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Reverse circulant . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 k-circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Symmetric and reverse circulant 9
2.1 Spectral distribution . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Moment method . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Input and link . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Trace formula and circuits . . . . . . . . . . . . . . . . 13
2.2.4 Words and vertices . . . . . . . . . . . . . . . . . . . . 14
2.2.5 (M1) and Riesz’s condition . . . . . . . . . . . . . . . 16
2.2.6 (M4) condition . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Reverse circulant . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Related matrices . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Reduced moment . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.1 A metric. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.2 Minimal condition . . . . . . . . . . . . . . . . . . . . 24
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 LSD: normal approximation 27
3.1 Method of normal approximation . . . . . . . . . . . . . . . 27
3.2 Circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 k-circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vii
viii Contents
4 LSD: dependent input 37
4.1 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Reverse circulant . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 k-circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Spectral radius: light tail 67
5.1 Circulant and reverse circulant . . . . . . . . . . . . . . . . . 67
5.2 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Spectral radius: k-circulant 79
6.1 Tail of product . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Additional properties of the k-circulant . . . . . . . . . . . . 83
6.3 Truncation and normal approximation . . . . . . . . . . . . . 86
6.4 Spectral radius of the k-circulant . . . . . . . . . . . . . . . . 88
6.4.1 k-circulant for sn=kg+1 . . . . . . . . . . . . . . . 97
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Maximum of scaled eigenvalues: dependent input 99
7.1 Dependent input with light tail . . . . . . . . . . . . . . . . . 99
7.2 Reverse circulant and circulant . . . . . . . . . . . . . . . . . 100
7.3 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . . 104
7.4 k-circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4.1 k-circulant for n=k2+1 . . . . . . . . . . . . . . . . 115
7.4.2 k-circulant for n=kg+1, g >2 . . . . . . . . . . . . 117
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 Poisson convergence 119
8.1 Point process . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Reverse circulant . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.3 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . 126
8.4 k-circulant, n=k2+1 . . . . . . . . . . . . . . . . . . . . . 128
8.5 Reverse circulant: dependent input . . . . . . . . . . . . . . . 135
8.6 Symmetric circulant: dependent input . . . . . . . . . . . . 137
8.7 k-circulant, n=k2+1: dependent input . . . . . . . . . . . 137
8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9 Heavy-tailed input: LSD 139
9.1 Stable distribution and input sequence . . . . . . . . . . . . 139
9.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.3 Reverse circulant and symmetric circulant . . . . . . . . . . 144
9.4 k-circulant: n=kg+1 . . . . . . . . . . . . . . . . . . . . . 145
9.4.1 Proof of Theorem 9.4.2 . . . . . . . . . . . . . . . . . 149
Contents ix
9.5 k-circulant: n=kg−1 . . . . . . . . . . . . . . . . . . . . . 154
9.6 Tail of the LSD . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10 Heavy-tailed input: spectral radius 159
10.1 Input sequence and scaling . . . . . . . . . . . . . . . . . . . 159
10.2 Reverse circulant and circulant . . . . . . . . . . . . . . . . . 160
10.3 Symmetric circulant . . . . . . . . . . . . . . . . . . . . . . . 164
10.4 Heavy-tailed: dependent input . . . . . . . . . . . . . . . . . 166
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
11 Appendix 173
11.1 Proof of Theorem 1.4.1 . . . . . . . . . . . . . . . . . . . . . 173
11.2 Standard notions and results . . . . . . . . . . . . . . . . . . 177
11.3 Three auxiliary results . . . . . . . . . . . . . . . . . . . . . 182
Bibliography 185
Index 189