Rezakhah, S., Soltani, A. (2015). Domain of attraction of normal law and zeros of random polynomials. Bulletin of the Iranian Mathematical Society, 41(4), 1019-1030.

S. Rezakhah; A. Soltani. "Domain of attraction of normal law and zeros of random polynomials". Bulletin of the Iranian Mathematical Society, 41, 4, 2015, 1019-1030.

Rezakhah, S., Soltani, A. (2015). 'Domain of attraction of normal law and zeros of random polynomials', Bulletin of the Iranian Mathematical Society, 41(4), pp. 1019-1030.

Rezakhah, S., Soltani, A. Domain of attraction of normal law and zeros of random polynomials. Bulletin of the Iranian Mathematical Society, 2015; 41(4): 1019-1030.

Domain of attraction of normal law and zeros of random polynomials

^{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.

Abstract

Let $ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic polynomial, 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 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 $Oleft( (log n)^s log (log n)right)$.