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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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