Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems

Diego Armentano (1) , Jean-Marc Azaïs (2) , Federico Dalmao (1) , José Rafael León (1)

1 UDELAR - Universidad de la República [Montevideo]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We obtain a Central Limit Theorem for the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size as the degree goes to infinity. A study of the asymptotic variance of the number of roots is needed, this result was obtained in [2]. Afterwards we represent the number of roots as an explicit non linear functional belonging to the Itô-Wiener chaos. This representation provides a tool for applying the Fourth Moment Theorem and henceforth the asymptotic normality. MSC 2010 subject classifications: Primary 60F05, 30C15, ; secondary 60G60, 65H10.

Mots clés: 30C15; AMS Classification Primary 60F05; 65H10; Secondary 60G60; Kostlan-Shub-Smale ramdom polynomials; Central limit theorem; Kac-Rice formula; Hermite expansion

Domaines: Probabilités [math.PR]

Diego Armentano, Jean-Marc Azaïs, Federico Dalmao, José Rafael León. Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems. 2018. ⟨hal-01686277v2⟩