## 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 _{n}n ^{-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=1}E _{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 ^{-1/2}A _{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 ^{-1/2}A _{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) |
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Pages (from-to) | 771-797 |

Number of pages | 27 |

Journal | Journal of Theoretical Probability |

Volume | 25 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2012 |

### Bibliographical note

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

Copyright:

Copyright 2012 Elsevier B.V., All rights reserved.

## Keywords

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