On the Average Number of Level Crossings of a Random Trigonometric Polynomial

Abstract

There are many known asymptotic estimates of the number of zeros of the polynomial $T(\theta) = g_1 \cos \theta + g_2 \cos 2\theta + \cdots + g_n \cos n \theta$ for $n \rightarrow \infty$, where $g_i (i = 1, 2,\ldots, n)$ is a sequence of independent normally distributed random variables with mathematical expectation 0 and variance 1. The present paper provides an estimate of the expected number of times that such a polynomial assumes the real value $K$. It is shown that the results for $K = 0$ are valid when $K = o(\sqrt{n})$.