On the Average Number of Real Zeros of a Random Trigonometric Polynomial

Abstract

This work emphasizes the special role played by semi-stable distribution which is the generalization of the stable distribution. Here \(a_0,~ a _1,\ldots ,a_n\) be a sequence of mutually independent random variables following semi-stable distribution with characteristic function \(exp \left( - \left( C + \cos {\log |t|} \right) |t|^{\alpha } \right) \), \(1 \le \alpha \le 2\) and \(C>1\) and \(b_1,~ b_2,\ldots ,b_n\) be positive constants. We then obtain the average number of zeros in the interval \([0, 2\pi ]\) of random trigonometric polynomial of the form \(T_n(\theta )=\sum \nolimits _{k=1}^{n}\left( \frac{a_0}{n}+a_kb_k\sin {k\theta }\right) \) for large n. The case when \(b_k=k^{\sigma -\frac{1}{\alpha }}\), \(\sigma =-\frac{2}{3\alpha }\) is studied in detail. Here this average is asymptotically equal to \(2n+o(1)\) except for a set of measure zero as \(n\rightarrow \infty \).