Statistics Papers

Document Type

Journal Article

Date of this Version

11-1995

Publication Source

Transactions of the American Mathematical Society

Volume

347

Issue

11

Start Page

4365

Last Page

4384

Abstract

Mark Kac gave an explicit formula for the expectation of the number, νn(Ω), of zeros of a random polynomial,

Pn(z) = ∑n−1j=0 ηjzj ,

in any measurable subset Ω of the reals. Here, η0, . . . , ηn−1 are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which

n(Ω) = ∫ gn(x)dx.

Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω of the complex plane IC. Namely, we show that

n(Ω) = ∫ hn(x, y)dxdy + ∫Ω∩IR gn(x)dx,

where hn is an explicit intensity function. We also study the asymptotics of hn showing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.

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Date Posted: 27 November 2017