Almost sure central limit theorem for partial sums and maxima

作者：Peng Zuoxiang; Wang Lili; Nadarajah Saralees^*

来源：Mathematische Nachrichten, 2009, 282(4): 632-636.

DOI：10.1002/mana.200610760

• Almost sure central limit theorem
• extreme value distribution
• partial sums and maxima
• stable distribution

Let X, X(1), X(2),... be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX(2) = 1. Let S(n) = Sigma(n)(i=1) X(i) and M(n) = max{X(i), 1 <= i <= n}. Suppose there exists constants a(n) > 0, b(n) is an element of R and a nondegenrate distribution G(y) such that
lim(n ->infinity) P (M(n) - b(n)/a(n) <= y) = G(y), -infinity < y < infinity.
Then, we have
lim(n ->infinity) 1/log n (n)Sigma(k=1) 1/k f (S(k)/root k, M(k) - b(k)/a(k)) = integral integral f(x,y)Phi(dx)G(dy) almost Surely. where f (x. y) denotes the bounded Lipschitz 1 function and Phi(x) is the standard normal distribution function.

• 出版日期2009-4
• 单位四川农业大学