Building a CLT counter example $E(X_n)=0, Var(X_n)=1$ but not C.L.T

Question

I want to build an example of independent random variables: $X_n:\mathbb R\rightarrow \mathbb R$, $E(X_n)=0$, $Var(X_n)=1$ But $Y_n = n^{-1/2}\sum_{k\leq n} X_k$ does not converge in distribution to $\mathcal N (0,1)$.

My progress:

Building X_n with mean 0 is easy - if $\{t_1,\dots,t_m\}\in \mathbb R_{0<}$ then $P(X_n=t_k)=P(X_n=-t_k)=p_k$ has mean $0, k\in \{1,\dots,m\}$

Building X_n with var 1 - possible if I divide $X'_n=1/var(X_n) *X_n$

But, for now all the examples I have built did converge to $\mathcal N(0,1)$.

Any ideas?

Answer 1

With the @Clement C comment in mind (meaning you will violate the "identically distributed" assumption), you can attempt to violate the assumptions in the "Lyapunov CLT" statement given here: https://en.wikipedia.org/wiki/Central_limit_theorem

Here is a simple way to do it: Consider $\{X_i\}_{i=1}^{\infty}$ defined as a sequence of mutually independent random variables with: $$X_i = \left\{ \begin{array}{ll} -i &\mbox{ with prob $\frac{1}{2i^2}$} \\ i & \mbox{ with prob $\frac{1}{2i^2}$}\\ 0 & \mbox{ else} \end{array}\right. $$ Then $E[X_i]=0$, $Var(X_i)=1$ for all $i$, and $\lim_{n\rightarrow\infty} \frac{1}{\sqrt{n}}\sum_{i=1}^n X_i = 0$ with prob 1.