Product of exponentials and spectral radius of random k-circulants

Arup Bose; Rajat Subhra Hazra; Koushik Saha

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 424-443
  • ISSN: 0246-0203

Abstract

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We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {al}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.

How to cite

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Bose, Arup, Hazra, Rajat Subhra, and Saha, Koushik. "Product of exponentials and spectral radius of random k-circulants." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 424-443. <http://eudml.org/doc/272018>.

@article{Bose2012,
abstract = {We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence \{al\}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.},
author = {Bose, Arup, Hazra, Rajat Subhra, Saha, Koushik},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {eigenvalues; Gumbel distribution; k-circulant matrix; Laplace asymptotics; large dimensional random matrix; linear process; normal approximation; spectral radius; spectral density; tail of product},
language = {eng},
number = {2},
pages = {424-443},
publisher = {Gauthier-Villars},
title = {Product of exponentials and spectral radius of random k-circulants},
url = {http://eudml.org/doc/272018},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Bose, Arup
AU - Hazra, Rajat Subhra
AU - Saha, Koushik
TI - Product of exponentials and spectral radius of random k-circulants
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 424
EP - 443
AB - We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {al}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.
LA - eng
KW - eigenvalues; Gumbel distribution; k-circulant matrix; Laplace asymptotics; large dimensional random matrix; linear process; normal approximation; spectral radius; spectral density; tail of product
UR - http://eudml.org/doc/272018
ER -

References

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