Journal of Applied Mathematics and Stochastic Analysis
Volume 14 (2001), Issue 3, Pages 265-274
doi:10.1155/S1048953301000223
On random orthogonal polynomials
University of Ulster at Jordanstown, Department of Mathematics, Co. Antrim, BT37 0QB, United Kingdom
Received 1 March 1999; Revised 1 September 2000
Copyright © 2001 K. Farahmand. 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
Let T0∗(x),T1∗(x),…,Tn∗(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
g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, 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.