Asymptotic Estimates K-Level Crossings of a Random Algebraic Polynomial
Keywords:
Independent, identically distributed random variables, random algebraic polynomial, random algebraic equation, real roots, domain of attraction of the normal law, slowly varying functionAbstract
This paper provides asymptotic estimates for the expected number of real zeros and klevel crossings of a random algebraic polynomial of the form a0(n-1 c 0)1/2 + a1(n-1 c 1)1/2 x + a2(n-1 c 2)1/2 x2 + …+an-1(n-1 c n-1)1/2xn-1, where a J (J=0, 1, 2, … , n-1 ) are independent standard normal random variables and k is constant independent of x . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form a0 + a1 x + a2 x2 + … + an-1xn-1 .
