Domain of attraction of normal law and zeros of random polynomials

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