User michael lacey - MathOverflow

Answer by michael lacey for Random Trigonometric Polynomial

2011-04-23T03:38:34Z

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.

Under weaker conditions on the iid random coefficents, and $ a_k=1/k$, this is discussed in an article of Michel Talagrand, A borderline Fourier Series" Ann Prob 1995. http://www.jstor.org/pss/2245006

Answer by michael lacey for Why do Littlewood-Paley projections behave like iid random variables

2010-04-09T02:33:08Z

It is much better to replace 'iid random variables' above by 'martingale differences.'

The usual Littlewood-Paley square function is closely related to the Haar square function.
And the Haar square function is exactly a martingale square function, namely a sum of squares of martingale differences.

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.

Answer by michael lacey for L^{p} multiplier sets

2009-10-31T02:32:44Z

Let M_p be the class of L^p muliplier sets, as considered in this equation. It is known:

This is an algebra of sets, but not a sigma-algebra. [Not sure of the reference.]
The inclusion M_p \subsetneq M_q is strict for 2\le p

MR1728363 (2001g:42013) Mockenhaupt, Gerd(5-NSW-SM); Ricker, Werner J.(5-NSW-SM) Idempotent multipliers for $L^p(\bf R)$.

Answer by michael lacey for CLT for stationary sequences with infinte variance

2009-11-17T17:49:46Z

Feller vol 2 Chapter IX should do the trick.

A more modern reference--which I have not looked at--is

Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Stochastic Modeling) (Hardcover) ~ Gennady Samorodnitsky

Not sure if this covers convergence issues or not.

Answer by michael lacey for Colloquial catchy statements encoding serious mathematics

2009-10-31T23:18:06Z

Time Average is the Space Average

which is the the Ergodic Theorem.

Answer by michael lacey for Strong Law of Large Numbers for weakly dependent random variables

2009-10-26T12:54:02Z

This question sounds like an exercise: Split the sequence into I sequences of iid random variables. Apply the classical SLLN to each sequence. Recombine.

Tom: Of course it is true with exponential decay of the corellation function, but it is not easy.

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.)

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.

Not sure that this would appear in a text book however. My sense is that these considerations are well-known.

Answer by michael lacey for Is the Fourier transform of exp(-||x||) non-negative?

2009-10-31T02:40:20Z

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|}.

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)

To get the same thing in higher dimensions, take independent copies in each coordinate.

Take note: These distributions are not square integrable--otherwise the 'universal' Central Limit Theorem would hold. The cauchy distribution is only weakly integrable.