Let Y,Y 1 ,Y 2 , ... be i.i.d. r.v.s, E{Y} = 0, Var{Y} = 1. Let

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Let Y,Y1,Y2, ... be i.i.d. r.v.’s, E{Y} = 0, Var{Y} = 1. Let the random vector (X1n, ...,Xnn) = (Y1, ...,Yn) with probability 1−1, and (X1n, ...,Xnn) = (Y, ...,Y) with probability 1. Let Sn = X1n+...+Xnn. Show thatimage

while Var{Sn} = 2n−1 ∼ 2n, which says that the standard deviation of the sum of r.v.’s is not always a good normalization even in the case of normal convergence.

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