On the expected number of level crossings of a random polynomial

[摘要] 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 paper provides the estimate for the expected number of such level crossings when the coefficients are independent identically Cauchy distributed random variables. Using a numerical approach we show that in the interval (-1, 1) by increasing K the number of K-level crossings decreases, 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 they indicate the behaviour of polynomials for distribution outside this class. (C) 1997 Academic Press.

[发布日期] 1997-04-01