I'm trying to read Lehmann's "Elements of Large Sample Theory" and I have the following question about the text.
The classical Central Limit Theorem is stated as:
Now, the author goes on to provide the following counter-example:
My question is, what assumption of the CLT does this counter-example violate? Could it be that in this case $\{X_i\}$ are not identically distributed, since they depend on $n$? Perhaps we do not have finite variance? But I checked and variance does look finite.
The author talks about the fact that the distribution F of $\{X_1, X_2, X_3, \ldots, X_n\}$ depends on $n$ but that CLT assumes that this distribution $F$ (of any sequence where CLT applies) to be fixed. Where exactly does CLT make this assumption as stated? Or is the author just simply not giving a precise enough statement of CLT?
Please shed light, ye wise math stackexchange folk.