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

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.