User michael lacey - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:53:42Z http://mathoverflow.net/feeds/user/1158 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62672/random-trigonometric-polynomial/62700#62700 Answer by michael lacey for Random Trigonometric Polynomial michael lacey 2011-04-23T03:38:34Z 2011-04-23T03:38:34Z <p>The case of $a_k=1/k$ and Gaussian coefficents is easily seen that the series is a.s. convergent to a continuous function. (Corollary to Salem-Zygmund, already mentioned) So, in the case you are considering the distribution $m$ will have subgaussian tails. </p> <p>Under weaker conditions on the iid random coefficents, and $a_k=1/k$, this is discussed in an article of Michel Talagrand, <a href="http://www.jstor.org/pss/2245006" rel="nofollow"> A borderline Fourier Series" Ann Prob 1995. <a href="http://www.jstor.org/pss/2245006" rel="nofollow">http://www.jstor.org/pss/2245006</a> </a> </p> http://mathoverflow.net/questions/20225/why-do-littlewood-paley-projections-behave-like-iid-random-variables/20808#20808 Answer by michael lacey for Why do Littlewood-Paley projections behave like iid random variables michael lacey 2010-04-09T02:33:08Z 2010-04-09T02:33:08Z <p>It is much better to replace 'iid random variables' above by 'martingale differences.' </p> <p>The usual Littlewood-Paley square function is closely related to the Haar square function.<br> And the Haar square function is exactly a martingale square function, namely a sum of squares of martingale differences. </p> <p>One can pass back and forth, from martingale to continuous analogs. A striking method to do this was found by Stefanie Petermichl, when she found a simple way to obtain the Hilbert transform from a modification of a martingale multiplier. </p> http://mathoverflow.net/questions/3494/lp-multiplier-sets/3505#3505 Answer by michael lacey for L^{p} multiplier sets michael lacey 2009-10-31T02:32:44Z 2010-02-01T05:59:20Z <p>Let M_p be the class of L^p muliplier sets, as considered in this equation. It is known: </p> <ol> <li><p>This is an algebra of sets, but not a sigma-algebra. [Not sure of the reference.] </p></li> <li><p>The inclusion M_p \subsetneq M_q is strict for 2\le p </li> </ol> <p>MR1728363 (2001g:42013) Mockenhaupt, Gerd(5-NSW-SM); Ricker, Werner J.(5-NSW-SM) Idempotent multipliers for $L^p(\bf R)$. </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/3559/colloquial-catchy-statements-encoding-serious-mathematics/3619#3619 Answer by michael lacey for Colloquial catchy statements encoding serious mathematics michael lacey 2009-10-31T23:18:06Z 2009-10-31T23:18:06Z <p><strong> Time Average is the Space Average </strong></p> <p>which is the the Ergodic Theorem. </p> http://mathoverflow.net/questions/2409/strong-law-of-large-numbers-for-weakly-dependent-random-variables/2605#2605 Answer by michael lacey for Strong Law of Large Numbers for weakly dependent random variables michael lacey 2009-10-26T12:54:02Z 2009-10-31T02:50:57Z <p>This question sounds like an exercise: Split the sequence into I sequences of iid random variables. Apply the classical SLLN to each sequence. Recombine. </p> <p>Tom: Of course it is true with exponential decay of the corellation function, but it is not easy. </p> <p>The essential difficulty is that one wants to reduce the SLLN to an exponentially growing subsequence of N's. In the classical case, this is done by a martingale inequality. (Prob of a supremum of a martingale is dominated by the probability at end of the martingale.) </p> <p>Once one moves away from an implicit martingale structure, then tricks have to be employed---of which the most obvious is that if the sequence of random variables is bounded, then obviously you can reduce to an exponentially growing subsequence. This point is much of the content of the paper of Lyons cited already. </p> <p>Not sure that this would appear in a text book however. My sense is that these considerations are well-known. </p> http://mathoverflow.net/questions/723/is-the-fourier-transform-of-exp-x-non-negative/3506#3506 Answer by michael lacey for Is the Fourier transform of exp(-||x||) non-negative? michael lacey 2009-10-31T02:40:20Z 2009-10-31T02:40:20Z <p>These questions are closely related to the so-called stable distributions. In particular, the cauchy distribution on the real line has the characteristic function e^{-|x|}. </p> <p>Go to the wikipedia page, and in the definition section set: mu=0 (this is the drift parameter) alpha=0 (this is the skewness parameter) </p> <p>To get the same thing in higher dimensions, take independent copies in each coordinate. </p> <p>Take note: These distributions are not square integrable--otherwise the 'universal' Central Limit Theorem would hold. The cauchy distribution is only weakly integrable. </p>