Local maxima of a random trigonometric polynomial

Abstract

In this paper we obtain a formula for the expected number of maxima of a normal process ζ(t) which occur below a levelu. The main condition assumed in the derivation is that ζ(t) and its first and second derivative have, with probability one, continuous one-dimensional distributions. The expected number of such maxima of a trigonometric polynomial with random coefficients follow from this result. It is shown that, wich probability one, all the maxima of this type of polynomial occur below a levelu if $$u/\sqrt n \to \infty $$ , and a sizeable number of maxima exist below zero level.