Product of exponentials and spectral radius of random k-circulants

Abstract

We consider n x n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {a_l}_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 = k^g + 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.