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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| Published in: | International journal of stochastic analysis Vol. 2008; no. 1 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Hindawi Publishing Corporation
2008
Hindawi Limited |
| Subjects: | |
| ISSN: | 1048-9533, 2090-3332, 1687-2177, 2090-3340 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1048-9533 2090-3332 1687-2177 2090-3340 |
| DOI: | 10.1155/2008/189675 |