A example of a CDF tends in law to a variable Y but its variance does not tends to var(Y)

Question

I was thinking in a example of Kn(Yn-c) tends in law to a ramdom variable Y with CDF(cumulative distribution function), but where Variance of Kn(Yn-c) does not tend to v^2=Var(Y). Do you think that example can exist?

Answer 1

On $(0,1)$ with Lebesgue measure let $K_n=1, c=0, Y_n=nI_{(0,\frac 1 n)}$. Then $Y_n \to 0$ a.s. but $var(Y_n) \to \infty$.