CLT for stationary sequences with infinte variance - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:14:34Z http://mathoverflow.net/feeds/question/5826 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5826/clt-for-stationary-sequences-with-infinte-variance CLT for stationary sequences with infinte variance Piotr Miłoś 2009-11-17T15:12:36Z 2009-11-17T20:45:14Z <p>There is a well-known central limit theorem for as a stationary sequences. </p> <p>If $( X_n )_n$ is a sationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\sum_{i=1}^n X_i$ converges weakly to a normal random variable. (This is very simplified version of Theorem 7.7.6 of Durrett's Probability Theory ...).</p> <p>This theorem is very nice but works only when $X_n$ have finite variance (the mixing conditions above require it). </p> <p>I am almost sure that there must be an analogue of this theorem for variables with infinite variance (of course the sequence will converge to a stable variable). But I couldn't find it in popular textbooks (I check Durrett - "Probability theory...", Kallemberg - "Fundations of probability" and Jacod, Shiryaev - "Limit theorems ..."). Does anybody know any good reference (e.g. a textbook)?</p> http://mathoverflow.net/questions/5826/clt-for-stationary-sequences-with-infinte-variance/5839#5839 Answer by Kristal Cantwell for CLT for stationary sequences with infinte variance Kristal Cantwell 2009-11-17T17:36:42Z 2009-11-17T20:45:14Z <p>I have found an article "A central limit theorem for independent summands with infinite variances" here:</p> <p><a href="http://www.springerlink.com/content/n462221853449467/" rel="nofollow">http://www.springerlink.com/content/n462221853449467/</a> </p> <p>Also see page 235 of <em>Financial modelling with jump processes</em> more information here:</p> <p><a href="http://en.wikipedia.org/wiki/Stable%5Fdistribution" rel="nofollow">http://books.google.com/books?id=3X2j2Gjv-oMC&amp;dq=central+limit+theorem+%22infinite+variance%22&amp;lr=&amp;source=gbs_navlinks_s</a></p> <p>There is a generalization of the central limit theorem involving stable distributions which involves infinite variance see the following:</p> <p><a href="http://en.wikipedia.org/wiki/Stable%5Fdistribution" rel="nofollow">http://en.wikipedia.org/wiki/Stable_distribution</a></p> <p>More on stable distributions:</p> <p><a href="http://academic2.american.edu/~jpnolan/stable/stable.html" rel="nofollow">http://academic2.american.edu/~jpnolan/stable/stable.html</a></p> http://mathoverflow.net/questions/5826/clt-for-stationary-sequences-with-infinte-variance/5840#5840 Answer by michael lacey for CLT for stationary sequences with infinte variance michael lacey 2009-11-17T17:49:46Z 2009-11-17T17:49:46Z <p>Feller vol 2 Chapter IX should do the trick. </p> <p>A more modern reference--which I have not looked at--is</p> <p>Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Stochastic Modeling) (Hardcover) ~ Gennady Samorodnitsky</p> <p>Not sure if this covers convergence issues or not. </p> http://mathoverflow.net/questions/5826/clt-for-stationary-sequences-with-infinte-variance/5842#5842 Answer by PeterR for CLT for stationary sequences with infinte variance PeterR 2009-11-17T17:51:22Z 2009-11-17T17:51:22Z <p>For IID rv's see Durrett's "Probability: theory &amp; Examples" Section 2.7 Stable Laws</p> <p>The more general (non-independent) case, is probably in "Stable non-Gaussian random processes: stochastic models with infinite variance" By Gennady Samorodnitsky, Murad S. Taqqu</p> http://mathoverflow.net/questions/5826/clt-for-stationary-sequences-with-infinte-variance/5843#5843 Answer by Jonathan Kariv for CLT for stationary sequences with infinte variance Jonathan Kariv 2009-11-17T17:57:00Z 2009-11-17T17:57:00Z <p>Just to be more explicit about what PeterR saud. The sum of n Cauchy random varibles (scaled by 1/n) is a cauchy. Maybe it would be helpful if you defined what nice properties you'd like your analog to have. </p>