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 languageEnglish (US)
Pages (from-to)771-797
Number of pages27
JournalJournal of Theoretical Probability
Volume25
Issue number3
DOIs
StatePublished - Sep 1 2012

Keywords

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

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