Estimates for large deviations in random trigonometric polynomials

Let $F(t) = \sum _{n = 1}^N a_n \exp (iX_n t)$, where $X_1 ,X_2 , \ldots ,X_N $ are independent random variables and the coefficients an are real or complex constants. Probabilistic estimates of the form \[ P\left[ {\mathop {\sup }\limits_{t \in K} | {F(t) - E[ {F(t)} ]} | \geq C\sqrt {N\log N} } \right] \geq \epsilon \] are obtained where K is an interval on the real line, C may be chosen more or less arbitrarily, and $\epsilon $ is an explicit function of C, K, N, and the random variables. This method includes trigonmetric interpolation and straightforward probabilistic techniques to obtain explicit numerical bounds that are applicable in a variety of engineering applications, particularly in the study of maximal sidelobe level for random arrays. Specific numerical examples are computed, and references to both the engineering and mathematical literature are provided.