Almost sure central limit theorem for the products of U-statistics

作者：Peng Zuoxiang; Tan Zhongquan; Nadarajah Saralees^*

来源：Metrika, 2011, 73(1): 61-76.

DOI：10.1007/s00184-009-0265-0

• Almost sure central limit theorem
• U-statistics
• Unbounded measurable function

Let (X(n)) be a sequence of i.i.d random variables and U(n) a U-statistic corresponding to a symmetric kernel function h, where h(1)(x(1)) = Eh(x(1), X(2), X(3),..., X(m)), mu = E(h(X(1), X(2),..., X(m))) and zeta(1) = Var(h(1)(X(1))). Denote gamma = root zeta 1/mu, the coefficient of variation. Assume that P(h(X(1), X(2),..., X(m)) > 0) = 1, zeta 1 > 0 and E|h(X(1), X(2),..., X(m))|(3) < infinity. We give herein the conditions under which
lim(N ->infinity) 1/log N (n=1)Sigma(N) 1/ng (((k=m)Pi(n) Uk/mu)(1/m gamma root n)) = (-infinity)integral(infinity) g(x)dF(x) a.s.
for a certain family of unbounded measurable functions g, where F(.) is the distribution function of the random variable exp(root 2 xi) and xi is a standard normal random variable.

• 出版日期2011-1
• 单位西南大学; 遵义师范学院