ON THE DISTRIBUTION OF COMPLEX ROOTS OF RANDOM POLYNOMIALS WITH HEAVY-TAILED COEFFICIENTS
Götze, Friedrich
Götze
Friedrich
Zaporozhets, D.
Zaporozhets
D.
Consider a random polynomial G(n)(z) = xi(n)z(n) + . . . + xi(1)z + xi(0) with independent identically distributed complex-valued coefficients. Suppose that the distribution of log(1 + log(1 + vertical bar xi(0)vertical bar)) has a slowly varying tail. Then the distribution of the complex roots of G(n) concentrates in probability, as n -> infinity, to two centered circles and is uniform in the argument as n -> infinity. The radii of the circles are vertical bar xi(0)/xi(tau)vertical bar(1/tau) and vertical bar xi(tau)/xi(n)vertical bar(1/(n-tau)), where xi(tau) denotes the coefficient with the maximum modulus.
56
4
696-U217
696-U217
Society For Industrial And Applied Mathematics
2012