The expected value of the number of real zeros of a random sum of Legendre polynomials
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- by J. Ernest Wilkins Jr. PDF
- Proc. Amer. Math. Soc. 125 (1997), 1531-1536 Request permission
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 $\delta$.References
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Additional Information
- J. Ernest Wilkins Jr.
- Affiliation: Department of Mathematics, Clark Atlanta University, Atlanta, Georgia 30314
- Received by editor(s): November 1, 1995
- Communicated by: Richard T. Durrett
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1531-1536
- MSC (1991): Primary 60G99; Secondary 41A60
- DOI: https://doi.org/10.1090/S0002-9939-97-03826-4
- MathSciNet review: 1377012