Proofs of Results of Chapter II

Abstract

We use similar techniques as in the proof of Proposition I.2.2. We first observe the obvious equality

$$ \Sigma_{{i = 1}}^n\;\left( {{W_{{1,n}}} - {{1} \left/ {n} \right.}} \right)\;h\left( {{X_i}} \right) $$

$$ = {n^{{ - 1}}}\Sigma_{{i = 1}}^n\left( {{Y_{{i,n}}} - 1} \right)h\left( {{X_i}} \right) + \quad {\overline {{Y_n}}^{{ - 1}}}\left( {1 - \overline {{Y_n}} } \right)\quad {n^{{ - 1}}}\Sigma_{{i = 1}}^n{Y_{{i,n}}}h\left( {{X_i}} \right) $$

.