Using (5.26) or otherwise, show that in the univariate case, where X is a standardized sum with mean zero, unit variance and so on, then Y

∗ = X − ρ3X2/6 − ρ4X3/24 + ρ

2 3X3/9 has mean − ρ3/6 and standard deviation 1 − ρ4/8 + 7ρ

2 3/36 when terms of order O(n −3/2

) are ignored. Show also that Y

*+ρ3/6 1−ρ4/8+7ρ

2 3/36

~N (0, 1) + O (n

−3/2)