An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Abstract

Let {ϕ i } ∞ i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure µ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say E n (µ), of random polynomials P n (z) := n i=0 Key words: random polynomials, orthogonal polynomials on the unit circle, expected number of real zeros, asymptotic expansion

An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials

“…It is known [27] that when m p |α m | is a bounded sequence for some p > 3/2, estimate (3) remains valid for random polynomials (6) with f m (z) = ϕ m (z) given by ( 8)- (9). Moreover, if the recurrence coefficients decay exponentially, it was shown by the authors in [1] that the expected number of real zeros has a full asymptotic expansion of the form (4) with the constant term still given by (5).…”

“…An interested reader can find a well referenced early history of the subject in the books by Bharucha-Reid and Sambandham [3], and by Farahmand [12]. In [15], Kac considered random polynomials (1) P n (z…”

“…where η i are certain real random variables and f m (z) are arbitrary functions on the complex plane that are real on the real line. Using a beautiful and simple geometrical argument they have shown 1 that if η 0 , . .…”

Real zeros of random sums with i.i.d. coefficients

“…(1,1) n (x, x), after algebraic simplification one sees (2.32) K n (x, x)K (1,1) n (x, x) − (K (0,1)…”

“…Asymptotics for the density function ρ n (x) in the case when the random variables {η k } are i.i.d. standard Gaussian has been well studied when the spanning functions are trigonometric functions [36], polynomials orthogonal on the real line ( [6], [7], [2], [28], [29], [31], [39]), and polynomials orthogonal on the unit circle ( [39], [1], [38]). As an application we consider the case…”