Level crossings and turning points of random hyperbolic polynomials

• Main Authors: K. Farahmand, P. Hannigan
• Format: Article
• Language: English
• Published: Hindawi Limited 1999-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: Gaussian process; number of real roots; Kac-Rice formula; normal density; covariance matrix.
• Online Access: http://dx.doi.org/10.1155/S0161171299225793

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.