论文标题
具有随机乘数系数的多项式的矩
Moments of polynomials with random multiplicative coefficients
论文作者
论文摘要
对于$ x（n）$ a rademacher或steinhaus随机乘法功能，我们考虑随机多项式$$ p_n（θ）= \ frac1 {\ sqrt {\ sqrt {n}} \ sum_ {n \ leq n \ leq n} p_n（θ）\ big |^{2k} \，dθ$$倾向于均值收敛的意义上的高斯瞬间，以$ k \ ll（\ log n / \ log \ log \ log \ log \ log n）^{1/3} $均匀地，但这与i.i.d的情况相反。系数，对于$ k $，这种行为不会持续得多。我们使用这些估计值（i）给出了几乎确定的salem -zygmund类型中心限制定理，$ p_n（θ）$，以前以不同的方法在未发表的哈珀作品中获得，并且（ii）表明，几乎毫无疑问地表明，几乎是$ $ $ $ $ $ n） \ ll \ exp（（（\ log n）^{1/2+\ varepsilon}），$$对于所有$ \ varepsilon> 0 $。
For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(θ) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(nθ), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1 \big| P_N(θ) \big|^{2k}\, dθ$$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k \ll (\log N / \log \log N)^{1/3}$, but that in contrast to the case of i.i.d. coefficients, this behavior does not persist for $k$ much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for $P_N(θ)$, previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely $$ (\log N)^{1/6 - \varepsilon} \ll \max_θ|P_N(θ)| \ll \exp((\log N)^{1/2+\varepsilon}), $$ for all $\varepsilon > 0$.
