$n$ random variables $X_1,\ldots,X_n$ are an i.i.d. sample. $\bar X_n$ is the sample mean. $\mu$ is the expectation of distribution. Doesn't guarantee a finite variance.
Does this always hold?
$$E[(\bar X_n-\mu)^2]\rightarrow0 \qquad (n\rightarrow \infty)$$
If yes, in which sense does this hold? (ie. almost surely / in probability / in distribution)
If not, under what condition does this hold? What if we add that $\sigma^2<\infty$ is the variance?