Almost sure central limit theorem on maxima and minima

Abstract

We prove an almost sure central limit theorem on maxima and minima: limn→∞ 1/log n Σⁿk=1 1/k I (Vk < mk ≤ Mk ≤ μk)=e^{− T+η} a.s. as the Stationary Gaussian sequence with covariance rn under the condition 1/n Σk=1|rk|log k exp{γ|rk|log k} ≪ (log log n)^−(1+ε)