On Different Classes of Algebraic Polynomials with Random Coefficients

The expected number of real zeros of the polynomial of the form a0+a1x+a2x2+⋯+anxn, where a0,a1,a2,…,an is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in (−∞,∞) is asymptotic to (2/π)log n. In this paper, we show that this asymptotic...

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Veröffentlicht in:International journal of stochastic analysis Jg. 2008; H. 1
Hauptverfasser: Farahmand, K., Grigorash, A., McGuinness, B.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Hindawi Publishing Corporation 2008
Hindawi Limited
Schlagworte:
Probability
Random variables
Studies
ISSN:1048-9533, 2090-3332, 1687-2177, 2090-3340
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Zusammenfassung:The expected number of real zeros of the polynomial of the form a0+a1x+a2x2+⋯+anxn, where a0,a1,a2,…,an is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in (−∞,∞) is asymptotic to (2/π)log n. In this paper, we show that this asymptotic value increases significantly to n+1 when we consider a polynomial in the form a0(n0)1/2x/1+a1(n1)1/2x2/2+a2(n2)1/2x3/3+⋯+an(nn)1/2xn+1/n+1 instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
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ISSN:1048-9533
2090-3332
1687-2177
2090-3340
DOI:10.1155/2008/189675