# Limiting Spectral Distribution of Random k-Circulants

Arup Bose, Joydip Mitra, Arnab Sen

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

## Abstract

Consider random k-circulants A k,n with n→∞,k=k(n) and whose input sequence {a l} l≥0 is independent with mean zero and variance one and sup nn -1 Σ n l=1 E{pipe}al{pipe} 2+δ < ∞ for some δ > 0. Under suitable restrictions on the sequence {k(n)} n≥1, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists, and we identify the limits. In particular, we prove the following: Suppose g≥1 is fixed and p 1 is the smallest prime divisor of g. Suppose P g = Π g j=1E j where {E j} 1≤j≤g are i. i. d. exponential random variables with mean one.(i) If k g=-1+sn where s=1 if g=1 and s=o(n p1-1) if g>1, then the empirical spectral distribution of n -1/2A k,n converges weakly in probability to U 1 P 1/(2g) g where U 1 is uniformly distributed over the (2g)th roots of unity, independent of P g.(ii) If g≥2 and k g=1+sn with s=o(n p1-1), then the empirical spectral distribution of n -1/2A k,n converges weakly in probability to U 1 P 1/(2g) g where U 2 is uniformly distributed over the unit circle in ℝ 2, independent of P g. On the other hand, if k≥2, k=n o(1) with gcd (n,k)=1, and the input is i. i. d. standard normal variables, then F n-1/2 Ak,n converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius r=exp(E[log√E 1]).

Original language English (US) 771-797 27 Journal of Theoretical Probability 25 3 https://doi.org/10.1007/s10959-010-0312-9 Published - Sep 2012

### Bibliographical note

Funding Information:
A. Bose was partially supported by J.C. Bose Fellowship, Government of India.

## Keywords

• Central limit theorem
• Circulant
• Eigenvalue
• Empirical spectral distribution
• Limiting spectral distribution
• Normal approximation
• k-circulant

## Fingerprint

Dive into the research topics of 'Limiting Spectral Distribution of Random k-Circulants'. Together they form a unique fingerprint.