# Stable Law with Rates

If $X_{i}$ are a bunch of iid random variables with mean 0 and finite second moments, we know that $\sum_{i=1}^{n} \frac{X_{i}}{\sqrt{n}}$ converges in law to a Gaussian. Furthermore, by the Berry-Esseen theorem, we have some bounds on the rates of this convergence. Similar results hold, even if $X_{i}$ are only mixing in some sense, rather than actually iid.

If the $X_{i}$ are now instead regularly varying with exponent 1 (and satisfying some other conditions), and are 'very strongly mixing' but not iid, then I know that their suitably scaled partial sums converge in law to an appropriate Levy function with exponent 1.

My question is, what is a reasonable rate for this convergence? Of course, it will have something to do with the mixing rate, and with the constants associated with regular variation, and maybe other things. Are there any standard results in this direction?

I am not much of an expert, and have so far tried only some very standard tricks (e.g. concentration of regeneration times) which work in very specialized circumstances, and often not very well. Any pointers would be appreciated.

Edited: In response to Igor's very nice answer below, to state explicitly that I did not wish to assume the $X_{i}$ were iid.

## 1 Answer

Check out this paper:

Harmonic mean, random polynomials and stochastic matrices Natalia L. Komarova, , Igor Rivin

There are some results of the sort you are asking about there (and since the authors were not then and are not now probabilists, the proofs start from essentially nothing).

• Hello Igor, thank you for the response - the paper is quite interesting (and the application is even somewhat related to what I was thinking of studying). I think that you are studying the iid case, rather than the mixing case. On the other hand, the rates are much better than I would have expected. I may try to use the calculations anyway... – QAMS Nov 7 '11 at 4:44
• Also, I just realized I was being a little rude there with the edit - I think my initial question was quite unclear, with one mention of iid and then another mention of a mixing rate. Perhaps the new version is better. – QAMS Nov 7 '11 at 4:45
• No problem. I don't know much about the mixing case, but am guessing that some of the techniques work there too... – Igor Rivin Nov 7 '11 at 20:05