Sequence of random variables that converges in distribution but whose variances does not converge to limit's variance

Question

I am self-studying the Lehmann's amazing book "Elements of Large Sample Theory". There is a problem that I cannot figure out:

3.7 (page 122): Give an example in which $k_n(Y_n-c)$ tends in law to random variable $Y$ with cdf, but where $Var[k_n(Y_n-c)]$ does not tend to $\nu^2=Var(Y)$.

I thought maybe a good strategy would be an example where $Y_n$ has infinite variance for any finite $n$ but asymptotically it has finite variance. But (1) I cannot think of such a sequence and (2) I'm not even sure this is the right approach.

Answer 1

Let $k_n = 1, c=0$,

Let $Y_n = \begin{cases}-n, &\text{ with probability} \frac1{2n} \\ 0 &\text{ with probability} 1-\frac1n\\ n, &\text{ with probability} \frac1{2n}\end{cases}$

then we have

$$Pr(Y_n \le m) = \begin{cases} 0 &, \text{if } m < -n \\ \frac1{2n} &, \text{if } -n \le m < 0 \\ 1-\frac1{2n} &, \text{if } 0 \le m < n \\ 1&, \text{if } m \ge n\end{cases}$$ but $$Var(Y_n)=E(Y_n^2)=n^2\left( \frac1n \right)=n \to \infty$$