convergence of random variable with parameter

Question

Consider the random Variables $X_{n} := n^{a}\mathbb{1}_{(0,\frac{1}{n})}$ on the probability space $(0,1)$ with borel-sigma algebra $\mathcal{B}((0,1))$ and the lebesgue measure. For which value $a > 0$ does $X_{n}$ converges in $\mathcal{L}^{2}$, in probability and almost surely?

Answer 1

For every $\omega\in\Omega=(0,1)$ we evidently have:$$\lim_{n\to\infty}X_n(\omega)=\lim_{n\to\infty}n^a1_{(0,\frac1n)}(\omega)=0$$

So $X_n$ converges to $0$ a.s. and consequently converges to $0$ in probability.

Further: $$\mathbb EX_n^2=n^{2a}P(X_n=n^a)=n^{2a}P(0,\frac1n)=n^{2a}\frac1n=n^{2a-1}$$

So $X_n$ converges also in mean square to $0$ if $0<a<\frac12$.