On miminum modulus of trigonometric polynomials with random coefficients (Q1274120)

Let \(\xi_0,\dots, \xi_{n-1}\) be independent identically distributed random variables with finite third moment, nonzero second moment, and zero mean. Denote \[ P(u)= P_n(u):= \text{Pr}\Biggl[ \min_{x\in\mathbb{T}} \Biggl| \sum^{n-1}_{j= 0} \xi_j e^{ijx}\Biggr|> u\Biggr],\quad u\geq 0. \] The following theorem is proved. For any \(\varepsilon\in (0,1)\), \[ P(n^{-1/2+\varepsilon})\leq n^{-\varepsilon^2/180}\quad\text{for}\quad n> (16C_0(\xi))^{9936/\varepsilon^2}, \] where \(C_0(\xi)= 8(1/\gamma_0+ B+ 1/A+ 6E|\xi|^3+ 1/E\xi^2)\) and \(A\), \(B\), \(\gamma_0\) are positive numbers such that \(\text{Pr}[A< \xi_1- \xi_2< B]= \gamma_0\).