On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients
نویسندگان
چکیده
Consider a random polynomial Gn(z) = ξnz+ · · ·+ξ1z+ξ0 with i.i.d. complex-valued coefficients. Suppose that the distribution of log(1 + log(1 + |ξ0|)) has a slowly varying tail. Then the distribution of the complex roots of Gn concentrates in probability, as n → ∞, to two centered circles and is uniform in the argument as n → ∞. The radii of the circles are |ξ0/ξτ |1/τ and |ξτ/ξn|, where ξτ denotes the coefficient with the maximum modulus.
منابع مشابه
On Classifications of Random Polynomials
Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...
متن کاملOn Roots of Random Polynomials
We study the distribution of the complex roots of random polynomials of degree n with i.i.d. coefficients. Using techniques related to Rice’s treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we ...
متن کاملAsymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables
Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy...
متن کاملTime-reversal symmetry and random polynomials
We analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system...
متن کاملOn the Distribution of Roots of Random Polynomials
In the study of algebraic and numerical properties of polynomials one occasionally introduces the notion of a random polynomial. For example, this chapter was originally motivated by investigations, including [Ki, Re, SS, Sml, Sm2], into the complexity of root-finding algorithms for polynomials. We restrict our attention to central normal measures on spaces of polynomials, i.e., we assume that ...
متن کامل