Asymptotic properties for the loglog laws under positive association

作者：Yang, Xiao-Rong^*; Fu, Ke-Ang

来源：Mathematica Slovaca, 2012, 62(5): 979-994.

DOI：10.2478/s12175-012-0059-0

• the law of the iterated logarithm
• Chung-type law of the iterated logarithm
• positive association
• moment convergence
• tail probability

Let {X-n : n >= 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set S-n = n Sigma(k=1) X-k, M-n = max(k <= n) vertical bar S-k vertical bar, n >= 1. Suppose that 0 < sigma(2) = EX12 + 2 Sigma(infinity)(k=2) EX1Xk < infinity. In this paper, we prove that if E vertical bar X-1 vertical bar(2+delta) < infinity for some delta is an element of (0, 1], and Sigma(infinity)(j=n+1) Cov(X-1, X-j) = O(n(-alpha)) for some alpha > 1, then for any b > -1/2
lim(epsilon SE arrow 0) epsilon(2b+1) Sigma(infinity)(n=1) (log log n)(b-1/2)/n(3/2) log n E{M-n - sigma epsilon root 2n log log n}+
= 2(-1/2-b)E vertical bar N vertical bar(2(b+1))/(b+1)(2b+1) Sigma(infinity)(k=0) (-1)(k)/(2k + 1)(2(b+1))
and
lim(epsilon SE arrow 0) epsilon(-2(b+1)) Sigma(infinity)(n=1) (log log n)(b)/n(3/2) log n E{sigma epsilon root pi(2)n/8 log log n - M-n}+
=Gamma(b + 1/2)/root 2(b + 1) Sigma(infinity)(k=0) (-1)(k)/(2k + 1)(2b+2), where x(+) = max{x, 0}, N is a standard normal random variable, and Gamma(.) is a Gamma function.

• 出版日期2012-10
• 单位浙江工商大学