On the quadratic moment of self-normalized sums

Let an integer n ≥ 2 and a vector of independent, identically distributed random variables X = (X1, . . . , Xn) be given with P(X = 0) = 0 and define the self-normalized sum Zn = ( Pn i=1 Xi)/( Pn i=1 X 2 i ) . We derive a formula for EZ n which enables us to prove that EZ 2 n ≥ 1 and that EZ n = 1 if and only if the summands are symmetrically distributed. The formula moreover suggests nonparametric estimators of EZ n given X which we comment upon. We also construct examples where Zn converges to the standard normal distribution as n tends to infinity while EZ n tends to infinity (the distribution of the summands varies with n).