Domain of attraction of normal law and zeros of random polynomials

Document Type : Research Paper


1 Faculty of Mathematics and Computer Sciences‎, ‎Amirkabir University of Technology‎, ‎Tehran‎, ‎Iran.

2 Department of Statistics and Operational Research‎, ‎Faculty of Science‎, ‎Kuwait University‎, ‎State of Kuwait.


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


Main Subjects