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The expected value of the number of real zeros
of a random sum of Legendre polynomials

Author: J. Ernest Wilkins Jr.
Journal: Proc. Amer. Math. Soc. 125 (1997), 1531-1536
MSC (1991): Primary 60G99; Secondary 41A60
MathSciNet review: 1377012
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Abstract: It is known that the expected number of zeros in the interval $(-1,1)$ of the sum $a_0\psi _0(t)+a_1\psi _1(t)+\dotsb +a_n\psi _n(t)$, in which $\psi _k(t)$ is the normalized Legendre polynomial of degree $k$ and the coefficients $a_k$ are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to $3^{-1/2}n$ for large $n$. We improve this result and show that this expected number is $3^{-1/2}n+o(n^\delta )$ for any positive $\delta $.

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Additional Information

J. Ernest Wilkins Jr.
Affiliation: Department of Mathematics, Clark Atlanta University, Atlanta, Georgia 30314

Keywords: Real zeros, random polynomials, Legendre polynomials
Received by editor(s): November 1, 1995
Communicated by: Richard T. Durrett
Article copyright: © Copyright 1997 American Mathematical Society