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Let $t_{1},t_{2},\ldots, t_{n}$ be i.i.d. real Gaussian random variables of zero mean and variance one. Let $a_{1},a_{2},\ldots, a_{n}$ be positive and fixed real numbers and define the random polynomial $$ p(z):=\sum_{k=1}^{n}{a_{k}t_{k}z^{k}}. $$

Define the random variable $$ m=\max_{|z|=1}\Big\{\mathrm{Re}(p(z))\Big\}=\max_{\theta\in (0,2\pi]}\Bigg\{\sum_{k=1}^{n}{a_{k}t_{k}\cos(k\theta)}\Bigg\}. $$

How can one compute the probability distribution of $m$? Can we compute at least the first few moments $\mathbb{E}(m)$ and $\mathbb{E}(m^2)$?

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Have you taken a look at Marcus and Pisier's book? –  Mark Meckes Apr 23 '11 at 15:38
    
No, I'm not familiar with that book. Do you think it can help? –  ght Apr 23 '11 at 16:21
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Yes. The sharpest known results on what you ask are probably in there; the main difficulty might be extracting the rather classical special case you're interested in. You could also take a look at Kahane's book, Some Random Series of Functions. –  Mark Meckes Apr 24 '11 at 13:37

2 Answers 2

By coincidence I recently needed to consider essentially this same question.

By homogeneity one can fix the value of $\sum_{k=1}^{n} a_k^2$. Let us take this to be $n$. If all of the a_i's were equal to 1, then the max m will be asymptotically $\sqrt{n \ln(n)}$ (with an error term of $O\left( \sqrt{\frac{n}{\ln(n)}} \ln \ln(n) \right)$) with probability $1$. In the case of random +/- signs, the upper-bound is an old result of Salem and Zygmund (Some properties of trigonometric series whose terms have random signs) and the lower bound is a result of Halasz (On a result of Salem and Zygmund concerning random polynomials). The proof should easily modify to include the case of random Gaussians or more general i.i.d. random variables.

Its more difficult to state a result for general coefficients, since if the coefficients are concentrated on just a few terms (say just one term) the above result is obviously false. However, as long as not too much of the $\ell^2$ weight is concentrated on a single coefficient then the above result should still hold. Results in this direction are contained in the Salem and Zygmund paper.

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@Mark: Thanks. As a matter of fact I'm interested in the case $a_k=\frac{1}{k}$ what paper should I looked? –  ght Apr 22 '11 at 22:54
    
By homogeneity, the max in this case should asymptotically be $\frac{\ln(n)}{\sqrt{n}}$ almost surely. This is done in the paper Halász, G. On a result of Salem and Zygmund concerning random polynomials. Studia Sci. Math. Hungar. 8 (1973), 369–377, when the rv's are random signs. It should be straightforward to modify the argument for Gaussians. –  Mark Lewko Apr 22 '11 at 23:02
    
Of course I meant $\sqrt{\frac{\ln(n)}{n}}$. –  Mark Lewko Apr 22 '11 at 23:16
    
Did you mean to say $\sqrt{\frac{\ln(n)}{n}}$? It seems to me that the max cannot go to zero as $n\to\infty$. Other thing, I believe that these papers deal with the case $$ \max_{\theta\in[0,2\pi]}{|\sum_{k=1}^{n}{a_k t_k \cos(k\theta)}|} $$ which is different to what I'm asking. –  ght Apr 23 '11 at 1:06
    
Sorry, I had misread your statement $a_k =1/k$ as $a_k=1/n$ (so $\sqrt{\frac{ln(n)}{n}}$ won't be right in your case). –  Mark Lewko Apr 23 '11 at 1:08

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

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I'm familiar with the paper that you mentioned where it is essentially proved that the random series $$ \sum_{n=1}^{\infty}{\frac{t_k}{k}\cos(k\theta)} $$ is absolutely convergent. How this helps to solve my problem? I'm considering only a finite sum (trigonometric polynomial). –  ght Apr 23 '11 at 11:29

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