04:30 PM in MATH 175
Prem S. Puri Memorial Lecture
Yuan Shih Chow, Professor of Statistics, Department of Mathematical Statistics, Columbia University, New York, NY
Wald's Equation for A Class of Denormalized U-statistics
Abstract
For any integer k ≤ 1, define the de-normalized U-statistic
Sk,n= ∑1 ≤ i1 < ... < ik ≤ n Xi1...Xik where {X, Xn, n ≥1} is an i.i.d.
sequence with EX = 0. It is proved for k ≥ 2 that ESk,T = 0
whenever ElXlp < ∞, 1 < p ≤ 2 and T is a stopping time with
E T(h-1)/(p-1)< ∞. This Wald-type equation is used to elicit information
about the moments of Tk = inf{n ≥ k : Sk,n ≥ 0) and Wc = inf{n
≥ 2 : S1,n2 ≥ c ∑j=1nXj2}, c > 0. It is also noted that E|Sk, n/nk/p|p → 0
if E|X|p < ∞, O < p < 2 provided EX = 0 when 1 ≤ p < 2.
