Geodesic
On the distribution of the number of real roots of random polynomials
Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 488-500

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Let $\xi_0,\xi_1,\dots$ be a sequence of independent identically distributed random variables and $N$ be the number of real roots of the polynomial $$ Q(x)=\sum_{j=0}^n\xi_jx^j. $$ The main result is the following Theorem. {\it If $\mathbf P\{\xi_j=0\}=0$, $\mathbf E\xi_j=0$ and $\mathbf E|\xi_j|^{2+s}\infty$ for some positive number $s$, then, for any real} $t$, $$ \mathbf E\exp\{it(N-\mathbf EN)(\mathbf DN)^{-1/2}\}\underset{n\to\infty}\longrightarrow е^{-t^2/2}. $$ 
@article{TVP_1974_19_3_a2,
     author = {N. B. Maslova},
     title = {On the distribution of the number of real roots of random polynomials},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {488--500},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1974_19_3_a2/}
}
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N. B. Maslova. On the distribution of the number of real roots of random polynomials. Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 488-500. http://geodesic.mathdoc.fr/item/TVP_1974_19_3_a2/