Asymptotic variance of the number of real roots of random polynomial systems

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of thi...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2018-12, Vol.146 (12), p.5437-5449
Main Authors: Armentano, D., Azaïs, J-M., Dalmao, F., León, J.
Format: Article
Language:English
Subjects:
F. STATISTICS AND PROBABILITY
Mathematics
Probability
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Summary:We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of this random variable.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/14215