expansion for outcome's estimate of a single draw from a probability distribution

Question

Limit theorems, like CLT, are useful to obtain an estimate of the sum (or the mean) of a big number of draw from a given probability distribution $P_x(x)$.
I have a fat tail probability distribution (second moment is not defined); can I estimate, with a similar expansion approach, the outcome of a single draw from $P_x(x)$?
Of course the difference (and the hurdle), with respect to the limit theorems case, is that, in this case, I'm not considering a quantity (like the mean) whose distribution narrows increasing the number of draws.