Suppose I have a blackbox generating values $x_n$. If I take mean($x_n^2$) it flies all over the place as new values are added -- a supermassive 48-digit value could suddenly ramp up a previously stable-looking mean by a factor of 100. However mean(log($x_n$)) converges to L. how can I estimate mean($x_n^2$) from observing log($x_n$)?
I imagine I need to extract more properties of the distribution of log($x_n$), maybe the variance?
A simple analogy might be squares of numbers: if I take the numbers [1,3,4,2,2,12,5,1,8,2], the mean is 4. but if I now square each number: [1,9,16,4,4,144,25,1,64,4], the mean is now 209. So what transformation on 4 was required to achieve 209?
Can anyone recommend a direction of research?
PS The blackbox generator is as follows: take a random permutation of 1000 elements & return the order (i.e. number of cycles required before the original order is restored).