The expected $L_{p}$ norm of random polynomials

Abstract:

The results of this paper concern the expected $L_{p}$ norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval $[0, 2\pi ]$). Specifically, for a random polynomial \[ q_{n}(\theta ) = \sum _{0}^{n-1}X_{k}e^{ik\theta }\] let \[ ||q_{n}||_{p}^{p}= \int _{0}^{2\pi } |q_{n}(\theta )|^{p} d\theta /(2\pi ). \] Assume the random variables $X_{k},k\ge 0$, are independent and identically distributed, have mean 0, variance equal to 1 and, if $p>2$, a finite $p^{th}$ moment ${\mathrm E}(|X_{k}|^{p})$. Then \[ \frac {\text { E}(||q_{n}||_{p}^{p})}{n^{p/2}} \to \Gamma (1+p/2) \] and \[ \frac {\text {E}(||q_{n}^{(r)}||_{p}^{p})}{n^{(2r+ 1)p/2}} \to (2r+1)^{-p/2}\Gamma (1+p/2) \] as $n\to \infty$. In particular if the polynomials in question have coefficients in the set $\{+1,-1\}$ (a much studied class of polynomials), then we can compute the expected $L_{p}$ norms of the polynomials and their derivatives \[ \frac {\text { E}(||q_{n}||_{p})}{n^{1/2}} \to (\Gamma (1+p/2))^{1/p} \] and \[ \frac {\text { E}(||q_{n}^{(r)}||_{p})}{n^{(2r+1)/2}} \to (2r+1)^{-1/2}(\Gamma (1+p/2))^{1/p}. \] This complements results of Fielding in the $p:=0$ case, Newman and Byrnes in the $p:=4$ case, and Littlewood et al. in the $p=\infty$ case.