TITLE: Random Trigonometric Polynomial
QUESTION [10 upvotes]: 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 $\mathbb{E}(m^2)$?

REPLY [2 votes]: 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