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

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 $$n\rightarrow \infty $$ .