convergence in mean square

Question

$$\frac{(-1)^n}{n\sqrt{\omega}}$$ when w belongs to (0,1].

Does it converge a.s or m.s? I was able to proof that in converges in probability to zero (therefore converging in distribution).

Answer 1

For every $\omega \in (0,1]$ the limit is $0$. Hence your sequence converges almost surely (which implies it converges in probability and in distribution) (taking Lebesgue measure as the base measure). Convergence in mean square to $0$ means $\int_0^{1} |\frac {(-1)^{n}} {n\sqrt {\omega}}|^{2} \to 0$. But this integral is $\infty$ for very $n$ so mean square convergence is not true.