The Expected Number of Local Maxima of a Random Algebraic Polynomial

K. Farahmand and P. Hannigan

Journal of Theoretical Probability, 1997, vol. 10, issue 4, 991-1002

Abstract: Abstract In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial $$a_0 + a_1 x + a_2 x^2 + \cdots + a_{n - 1} x^{n - 1} $$ , where a j (j=0, 1,...,n−1) are independent, normally distributed random variables with mean μ and variance one. It is shown that for nonzero μ, the expected number of maxima is asymptotic to $$((\sqrt 3 + 1)/4\pi )$$ log n, when n is large.

Keywords: Gaussian process; number of real roots; Kac-Rice formula; normal density; covariance matrix; random algebraic polynomial; local maxima (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1023/A:1022618801587

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