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On random algebraic polynomials. (English) Zbl 0933.60081
The author considers a random algebraic polynomial of the form $P(x)= P_n(x,\omega)= \sum^{n-1}_{j= 0} a_j(\omega){n-1\choose j}^{1/2}x^j,$ where $$a_j$$ are independent standard normal random variables, and obtains asymptotic estimates for the expected number of real zeros and $$k$$-level crossings of $$P(x)$$ where $$k$$ is a constant. He shows that these asymptotic estimates are much greater than those for algebraic polynomials of the form $Q(x)= Q_n(x,\omega)= \sum^{n-1}_{j= 0} a_j(\omega) x^j.$ The author has been motivated to study $$P(x)$$ by the fact that the $$j$$th coefficient of $$Q(x)$$ has variance $$\left(\begin{smallmatrix} n-1\\ j\end{smallmatrix}\right)$$.

##### MSC:
 60H99 Stochastic analysis
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