The Annals of Probability

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

Kambiz Farahmand

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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 $K = o(\sqrt{n})$.

Article information

Source
Ann. Probab., Volume 18, Number 3 (1990), 1403-1409.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990751

Digital Object Identifier
doi:10.1214/aop/1176990751

Mathematical Reviews number (MathSciNet)
MR1062074

Zentralblatt MATH identifier
0708.60064

JSTOR
links.jstor.org

Keywords
60H 42 Number of real roots Kac-Rice formula random equation

Citation

Farahmand, Kambiz. On the Average Number of Level Crossings of a Random Trigonometric Polynomial. Ann. Probab. 18 (1990), no. 3, 1403--1409. doi:10.1214/aop/1176990751. https://projecteuclid.org/euclid.aop/1176990751


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