Expected Number of Real Zeros of a Random Polynomial With Independent Identically Distributed Symmetric Long-Tailed Coefficients

Shepp, Larry AFarahmand, K.2023-05-232023-05-232011-01-012016-07-27https://repository.upenn.edu/handle/20.500.14332/47787We show that the expected number of real zeros of the nth degree polynomial with real independent identically distributed coefficients with common characteristic function φ(z) = e-A(ln|1/z|)^-a for 0 < |z| < 1 and φ(0) = 1, φ(z) ≡ 0 for 1 ≦ |z| < ∞, with 1 < a and A ≧ a(a-1), is (a-1)/(a-1/2) log(n) asymptotically as n → ∞.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.random polynomialsnumber of real zerosreal rootsKac-Rice formulacharacteristic functionProbabilityStatistics and ProbabilityExpected Number of Real Zeros of a Random Polynomial With Independent Identically Distributed Symmetric Long-Tailed CoefficientsArticle