ESTIMATES FOR LARGE DEVIATIONS IN RANDOM TRIGONOMETRIC POLYNOMIALS

Let F(t) = SIGMA(n=1)N a(n) exp(iX(n)t), where X1, X2,...,X(N) are ind ependent random variables and the coefficients a(n) are real or comple x constants. Probabilistic estimates of the form P[sup(t is-an-element -of K) \F(t) - E[F(t)]\ greater-than-or-equal-to C square-root NlogN] less-than-or-equal-to epsilon are obtained where K is an interval on t he 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 metho d includes trigonmetric interpolation and straightforward probabilisti c techniques to obtain explicit numerical bounds that are applicable i n 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.