A Darling–Erdös type result for stationary ellipsoids

Let {Xk,k∈Z} be a zero mean dd-dimensional stationary process, and let Sn,d=(Sn,1,Sn,2,…,Sn,d)T with Sn,h=1n∑k=1nXk,h, where Xk,hXk,h denotes the single components of {Xk,k∈Z}. Under a weak dependence condition, we show that the ellipsoid Xd2=max1≤k≤d(2k)−1/2|Sn,kTΓk−1Sn,k−k| follows a Darling–Erdös type law as d→∞d→∞, i.e., Xd2 converges to a Gumbel-type distribution exp(−e−z)exp(−e−z). We show that this result is valid as long as d→∞d→∞ and d=dn=O(nd)d=dn=O(nd) with d>0d>0.