On random orthogonal polynomials. (2001)

• Record Type: Journal Article
• Authors: Farahmand, K.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 14:Number 3(2001)
• Page Start: 265
• Page End: 274
• Publication Date: 2001
• Subjects: sharp crossings -- number of real roots -- Kac-Rice formula -- normal density -- Legendre polynomial; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953301000223
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : Let T 0 ∗ ( x ), T 1 ∗ ( x ), …, T n ∗ ( x ) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval ( − 1, 1 ) . The asymptotic estimate of the expected number of real zeros of the random polynomial g 0 T 0 ∗ ( x ) + g 1 T 1 ∗ ( x ) + … + g n T n ∗ ( x ) where g j, j = 1, 2, …, n are independent identically and normally distributed random variables is known. In this paper, we first present the asymptotic value for the above expected number when coefficients are dependent random variables. Further, for the case of independent coefficients, we define the expected number of zero up-crossings with slope greater than u or zero down-crossings with slope less than − u . Promoted by the graphical interpretation, we define these crossings as u -sharp. For the above polynomial, we provide the expected number of such crossings.