Vol. 103, No. 2, 1982

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Strong result for real zeros of random polynomials

M. N. Mishra, N. N. Nayak and Swadeenananda Pattanayak

Vol. 103 (1982), No. 2, 509–522
Abstract

Let Nn be the number of real zeros of Σr=0narXrxr = 0 where Xr’s are independent random variables identically distributed belonging to the domain of attraction of normal law; a0,a1,a2an are nonzero real numbers such that (kn∕tn) = o(log n) where kn = max0rn|ar| and tn = min0rn|ar|. Further we suppose that the coefficients have zero means and P{Xr0} > 0. Then there exists a positive integer n0 such that

P {sup(Nn ∕Dn) < μ} > 1 − μ′{log((kn0∕tn0)loglogn)∕logn0}1−𝜀∕2
n>n0

for n > n0 and 1 > 𝜀 > 0 where Dn = (log n∕log(kn∕tn)log log n)(1𝜀)2.

Mathematical Subject Classification 2000
Primary: 60G99
Milestones
Received: 19 August 1980
Revised: 16 September 1981
Published: 1 December 1982
Authors
M. N. Mishra
N. N. Nayak
Swadeenananda Pattanayak