Suppose that $X_1,X_2...$ is a sequence of non-negative real random variables. I have that $\mathbb{E}(X_i^2) \to 0$ as $i \to +\infty$, therefore my sequence converges at least in distribution to the random vairbale that is identically zero.
Does it converge in proba ? almost surely ?
Yes in probability: The definitions of convergence in distribution to a constant random variable and convergence in probability to a constant random variable are the same.
No almost surely: If we let $\alpha$ be randomly distributed between $[0,1]$ and we let $X_n$ be $1$ if $$\alpha + \sum_{i=1}^{n-1} (1/i) \mod 1 > 1 -1/n $$ and $0$ otherwise then $X_n$ has a probability $1/n$ of being $1$ and a probability $1-1/n$ of being $0$, but with probability $1$, $X_n$ is $1$ infinitely often (at the points where $\alpha + \sum_{i=1}^n (1/i)$ passes an integer).
