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 ?



Which type of convergence for this sequence of random variables?
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).


