Let Y,Y₁,Y₂, ... be i.i.d. r.v.’s, E{Y} = 0, Var{Y} = 1. Let the random vector (X_1n, ...,X_nn) = (Y₁, ...,Y_n) with probability 1−1, and (X_1n, ...,X_nn) = (Y, ...,Y) with probability 1. Let S_n = X_1n+...+X_nn. Show that

while Var{S_n} = 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.