The K-level crossings of a random algebraic polynomial with dependent coefficients

TL;DR

This paper investigates the number of level crossings in random algebraic polynomials with dependent Gaussian coefficients, showing that for certain covariance decay conditions, the behavior resembles the independent case.

Contribution

It extends previous results by analyzing the expected number of level crossings for polynomials with dependent coefficients having decaying covariance.

Findings

Expected number of crossings similar to independent case under certain covariance decay.

Behavior depends on the decay rate of the covariance function.

Results generalize previous independent coefficient models.

Abstract

For a random polynomial with standard normal coefficients, two cases of the K-level crossings have been considered by Farahmand. When the coefficients are independent, Farahmand was able to derive an asymptotic value for the expected number of level crossings, even if K is allowed to grow to infinity. Alternatively, it was shown that when the coefficients have a constant covariance, the expected number of level crossings is reduced by half. In this paper we are interested in studying the behavior for dependent standard normal coefficients where the covariance is decaying and no longer constant. Using techniques similar to those of Farahmand, we will be able to show that for a wide range of covariance functions behavior similar to the independent case can be expected.