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July, 1990 On the Average Number of Level Crossings of a Random Trigonometric Polynomial
Kambiz Farahmand
Ann. Probab. 18(3): 1403-1409 (July, 1990). DOI: 10.1214/aop/1176990751

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

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Kambiz Farahmand. "On the Average Number of Level Crossings of a Random Trigonometric Polynomial." Ann. Probab. 18 (3) 1403 - 1409, July, 1990. https://doi.org/10.1214/aop/1176990751

Information

Published: July, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0708.60064
MathSciNet: MR1062074
Digital Object Identifier: 10.1214/aop/1176990751

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

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 3 • July, 1990
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