What exactly does $\dot\sim$ notation mean? What does $\dot\sim$ notation (dot over tilde) mean, in the context like $x \mathrel{\dot\sim} \mathcal N(0,1)$?
Turns out it is easier to find how to typeset it correctly: tex.SE explains that one should type \mathrel{\dot\sim} instead of simply \dot\sim to fix the spacing issue -- than to find what it actually means. It has only been used 4 times on CV until now; is it standard?
 A: Unless there was some other clue as to the intended meaning, I'd interpret that as "is approximately distributed as". 
It's fairly standard. Note that some of the other usual ways of indicating "approximation" by modifying a symbol don't really work with $\sim$.
Note that $\sim$ can be read as "is distributed as" and that adding the dot over a symbol at least sometimes indicates approximation -- compare $=$ with $\mathrel{\dot =}$.
So "$x \mathrel{\dot\sim} \mathcal N(0,1)$" could be read something like "$x$ is approximately distributed as standard normal".  Personally, I don't mind the closer spacing in \dot\sim ($\dot\sim$) for that use.
A: "$\dot \sim$" means "approximately distributed as". It is often used as short hand for something like 
$\sqrt n (\bar x - \mu)/\sigma \rightarrow_d  N(0,1)$ as $n \rightarrow \infty$
i.e. convergence in distribution, but you are too lazy to write out the necessary $n \rightarrow \infty$ to make the statement actually mathematically rigorous.  
(Of course, in the above statement, this is exactly distributed if the $x_i \sim_{iid} N(\mu, \sigma)$. But if $x_i$ are not normal, it would only converge in distribution to $N(0,1)$.)
During grad school, one my professors went on a technical, but justified, rampage about how this notation is often used in an abusive manner. For example, if you were to write 
$ \hat p \mathrel{\dot\sim} N(p, \sqrt{p(1-p)/n})$
where $\hat p$ is the standard MLE for a binomial distribution, this seems to imply that $\hat p$ is approximately normal for any n, which is of course not true. We were not allowed to use $\dot \sim$ notation in his class, but rather wrote everything out in the proper "converges in distribution" notation. 
None of my other professors cared. 
