Supplementary Material

Before we proceed with proving the lemma, we first establish some shorthand notation. We are now in a position to prove (A.2). Combining this with (A.4) shows that max Rearranging the resulting expression yields the desired identity. Fact 1 is a special case of the following lemma. Moreover, we can express the variance d V AR ( g ( X) | X 2 t) in a similar form, viz., d V AR ( g ( X) | X 2 t) = 1 N (t) With these specifications fixed, the program (A.19) becomes min Hence by (A.12), we obtain the desired (A.8).