A Darling–Erdös type result for stationary ellipsoids

Let { X k , k ∈ Z } be a zero mean d -dimensional stationary process, and let S n , d = ( S n , 1 , S n , 2 , … , S n , d ) T with S n , h = 1 n ∑ k = 1 n X k , h , where X k , h denotes the single components of { X k , k ∈ Z } . Under a weak dependence condition, we show that the ellipsoid X d 2 = max 1 ≤ k ≤ d ( 2 k ) − 1 / 2 | S n , k T Γ k − 1 S n , k − k | follows a Darling–Erdös type law as d → ∞ , i.e., X d 2 converges to a Gumbel-type distribution exp ( − e − z ) . We show that this result is valid as long as d → ∞ and d = d n = O ( n d ) with d > 0 .