Domain of attraction of normal law and zeros of random polynomials

Abstract

Let<br />$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic<br />polynomial,<br /> where $A_{0},A_{1}, cdots $ is a sequence of independent<br /> random variables belong to the domain of attraction of the normal law.<br /> Thus<br /> $A_j$'s for $j=0,1cdots $<br /> possesses the characteristic functions $exp {<br />-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowly<br />varying functions.<br />Under the assumption that there exist a real positive slowly varying<br />function $H(cdot)$ and positive constants $t_{0}$, $ C_{ast}$ and<br />$C^{ast}$ that $C_{ast}<br />H(t) leq mbox{Re}[H_{j}(t)] leq C^{ast} H(t),;tleq<br />t_{0},;j=1,cdots,n$, we find that<br />while the variance of coefficients are bounded, real zeros are concentrated around $pm 1$, and <br /> the expected number of real<br />zeros of $P_n(x)$ round the origin at a distance $(log n)^(-s)$ of $pm 1$ are at most of order <br />$Oleft( (log n)^s log (log n)right)$.