摘要:
{Xn, n≥1}是独立同分布随机变量序列, M(1)n, M(2)n分别表示{X1, X2, …, Xn}的第一个最大值与第二个最大值. 若存在 an＞0, bn 使得 P(Mn(1)≤anx+bn)w/→G(x) 成立(其中 G(x)为极值指数分布), 则对 x＞y 有limN→∞1/log N∑Nn=11/nI{M(1)n≤un, M(2)n≤vn}=G(y){log G(x)-log G(y)+1} a.s.其中un=anx+bn, vn=any+bn.
Abstract:
Let {Xn} be a sequence of i.i.d. random variables with distribution function F(x), M(1)n, M(2)n denote, respectively, the first and the second largest maximum of {X1, X2, …, Xn}, assume also that there are normalizing sequences an＞0, bn and a nondegenerate limit distribution G(x), such that P(Mn(1)≤anx+bn)w/→G(x), then for x＞y we have an almost sure central limit theorem for M(1)n and M(2)n, i.e.limN→∞(1)/(log N)∑Nn=11/nI{Mn(1)≤un, M(2)n≤vn}=G(y){log G(x)-log G(y)+1} a.s.where un=anx+bn, vn=any+bn.
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