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A Darling–Erdös type result for stationary ellipsoids

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  • Jirak, Moritz

Abstract

Let {Xk,k∈Z} be a zero mean d-dimensional stationary process, and let Sn,d=(Sn,1,Sn,2,…,Sn,d)T with Sn,h=1n∑k=1nXk,h, where Xk,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→∞, i.e., Xd2 converges to a Gumbel-type distribution exp(−e−z). We show that this result is valid as long as d→∞ and d=dn=O(nd) with d>0.

Suggested Citation

  • Jirak, Moritz, 2013. "A Darling–Erdös type result for stationary ellipsoids," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1922-1946.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:6:p:1922-1946
    DOI: 10.1016/j.spa.2013.01.018
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