International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 579-586
doi:10.1155/S0161171299225793
Level crossings and turning points of random hyperbolic polynomials
Department of Mathematics, University of Ulster, Jordastown, Co. Antrim, BT37 0QB, UK
Received 1 October 1997
Copyright © 1999 K. Farahmand and P. Hannigan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x+⋯+ansinhnx, where aj(j=1,2,…,n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true for all K independent of x, provided K≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx+a2cosh2x+⋯+ancoshnx, with aj(j=1,2,…,n) as before, is also (1/π)logn.