An extension of almost sure central limit theorem for order statistics

作者：Tong Bin; Peng Zuoxiang; Nadarajah Saralees^*

来源：Extremes, 2009, 12(3): 201-209.

DOI：10.1007/s10687-008-0075-1

• Almost sure central limit theorems
• Extreme order statistics
• Summation methods

Let {X(n), n >= 1} be a sequence of i.i.d. random variables. Let M(n) and m(n) denote the first and the second largest maxima. Assume that there are normalizing sequences a(n) > 0, b(n) and a nondegenerate limit distribution G, such that a(n)(-1) (M(n) - b(n)) ->(d) G. Assume also that {d(k), k >= 1} are positive weights obeying some mild conditions. Then for x > y we have
lim 1/D(n) Sigma(n)(k=2) d(k)I {Mk-bk/ak <= x, m(k)-b(k)/a(k) <= y} = G(y) {log G(x)-log G(y)+1} a.s.
when G(y) > 0 (and to zero when G(y) = 0).

• 出版日期2009-9
• 单位西南大学