Limiting Spectral Distribution of Random k-Circulants

作者：Bose Arup^*; Mitra Joydip; Sen Arnab

来源：Journal of Theoretical Probability, 2012, 25(3): 771-797.

DOI：10.1007/s10959-010-0312-9

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

Consider random k-circulants A (k,n) with n -> a,k=k(n) and whose input sequence {a (l) } (la parts per thousand yen0) is independent with mean zero and variance one and for some delta > 0. Under suitable restrictions on the sequence {k(n)} (na parts per thousand yen1), 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 ga parts per thousand yen1 is fixed and p (1) is the smallest prime divisor of g. Suppose where {E (j) }(1a parts per thousand currency signja parts per thousand currency signg) are i.i.d. exponential random variables with mean one.
(i) If k (g) =-1+sn where s=1 if g=1 and if g > 1, then the empirical spectral distribution of n (-1/2) A (k,n) converges weakly in probability to where U (1) is uniformly distributed over the (2g)th roots of unity, independent of P (g) .
(ii) If ga parts per thousand yen2 and k (g) =1+sn with , then the empirical spectral distribution of n (-1/2) A (k,n) converges weakly in probability to where U (2) is uniformly distributed over the unit circle in a"e(2), independent of P (g) .
On the other hand, if ka parts per thousand yen2, k=n (o(1)) with gcd (n,k)=1, and the input is i.i.d. standard normal variables, then converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius .

• 出版日期2012-9