Extreme value distribution for normalized sums from stationary Gaussian sequences

AbstractLet {Xi, i⩾0} be a sequence of independent identically distributed random variables with finite absolute third moment. Then Darling and Erdös have shown that for -∞<t<∞ where μn = max0⩽k⩽n k-12∑ki=0xi and Xn = (2 ln ln n)12. The result is extended to dependent sequences but assuming that {Xi} is a standard stationary Gaussian sequence with covariance function {ri}. When {Xi} is moderately dependent (e.g. when v(∑ni=1Xi) ∾ na, 0 < a < 2) we get where Ha is a constant. In the strongly dependent case (e.g. when v(∑ni=1Xi) ∾ n2r(n)) we get for-∞<t<∞