ON THE EXPECTED NUMBER OF LEVEL-CROSSINGS OF A RANDOM POLYNOMIAL

There are many known asymptotic estimates for the expected number of K -level crossings of an algebraic polynomial a(0) + a(1)x + a(2)x(2) ... + a(n)x(n) with normally distributed coefficients. The present pap er provides the estimate for the expected number of such level crossin gs when the coefficients are independent identically Cauchy distribute d random variables. Using a numerical approach we show that in the int erval (-1, 1) by increasing K the number of K-level crossings decrease s, while outside this interval this number is invariant, as long as K = o(root n). Since the Cauchy distribution does not belong to the wide class of distributions of domain of attraction of the normal law, the polynomials with Cauchy distributed coefficients are interesting as t hey indicate the behaviour of polynomials for distribution outside thi s class. (C) 1997 Academic Press.