Example of a stochastic process $X_t$ such that given condition holds

Question

any thoughts on this?

Find a stochastic process $X_t$ such that it satisfies the following two conditions:

1. ms-$\lim_{t \to \infty} {X_t} = 0$
2. ms-$\lim_{t \to \infty} {X_t}^2 \neq 0$

Thanks!

Answer 1

On $(0,1)$ with Lebesgue measure define $X_t(\omega)=t^{\frac 1 4} I_{(0,\frac 1 t)}(\omega)$. Then $EX_t^{2} \to 0$ but $EX_t^{4}=1$ for all $t$.