Probability theory
Asymptotic variance and CLT for the number of zeros of Kostlan Shub Smale random polynomials
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1141-1145.

In this note, we find the asymptotic main term of the variance of the number of roots of Kostlan–Shub–Smale random polynomials and prove a central limit theorem for this number of roots as the degree goes to infinity.

Dans cette note, nous calculons un equivalent de la variance asymptotique du nombre de racines réelles des polynômes aléatoires de Kostlan–Shub–Smale et nous démontrons un théorème de la limite centrale pour ce même nombre quand le degré tend vers l'infini.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.016
Dalmao, Federico 1, 2

1 Departamento de Matemática y Estadística del Litoral, Universidad de la República, CP 50000, Salto, Uruguay
2 Unité de Recherche en Mathématiques, Université du Luxembourg, campus Kirchberg, L-1359, Luxembourg
@article{CRMATH_2015__353_12_1141_0,
     author = {Dalmao, Federico},
     title = {Asymptotic variance and {CLT} for the number of zeros of {Kostlan} {Shub} {Smale} random polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1141--1145},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.016},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.09.016/}
}
TY  - JOUR
AU  - Dalmao, Federico
TI  - Asymptotic variance and CLT for the number of zeros of Kostlan Shub Smale random polynomials
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1141
EP  - 1145
VL  - 353
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.09.016/
DO  - 10.1016/j.crma.2015.09.016
LA  - en
ID  - CRMATH_2015__353_12_1141_0
ER  - 
%0 Journal Article
%A Dalmao, Federico
%T Asymptotic variance and CLT for the number of zeros of Kostlan Shub Smale random polynomials
%J Comptes Rendus. Mathématique
%D 2015
%P 1141-1145
%V 353
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.09.016/
%R 10.1016/j.crma.2015.09.016
%G en
%F CRMATH_2015__353_12_1141_0
Dalmao, Federico. Asymptotic variance and CLT for the number of zeros of Kostlan Shub Smale random polynomials. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1141-1145. doi : 10.1016/j.crma.2015.09.016. http://www.numdam.org/articles/10.1016/j.crma.2015.09.016/

[1] Azaïs, J.-M.; León, J. CLT for crossings of random trigonometric polynomials, Electron. J. Probab., Volume 18 (2013) no. 68, p. 68 (17 pp.)

[2] Azaïs, J.-M.; Wschebor, M. Level Sets and Extrema of Random Processes and Fields, John Wiley & Sons Inc., Hoboken, NJ, 2009 (xi+393 pp.) (ISBN: 978-0-470-40933-6)

[3] Azaïs, J.-M.; Dalmao, F.; León, J. CLT for the zeros of classical trigonometric polynomials, Ann. Inst. Henri Poincaré (2015) (in press)

[4] Bharucha-Reid, A.; Sambandham, M. Random Polynomials. Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, USA, 1986 (xvi+206 pp.) (ISBN: 0-12-095710-8)

[5] Bloch, A.; Pólya, G. On the roots of certain algebraic equations, Proc. Lond. Math. Soc., Volume 33 (1932) no. 1, pp. 102-114

[6] Bogomolny, E.; Bohigas, O.; Lebœuf, P. Distribution of roots of random polynomials, Phys. Rev. Lett., Volume 68 (1992) no. 18, pp. 2726-2729

[7] Cramér, H.; Leadbetter, M.R. Stationary and Related Stochastic Processes, Sample Function Properties and Their Applications, Dover Publications, Inc., Mineola, NY, USA, 2004 (reprint of the 1967 original xiv+348 pp.) (ISBN: 0-486-43827-9)

[8] Kostlan, E. On the distribution of roots of random polynomials, Berkeley, CA, 1990, Springer, New York (1993), pp. 419-431

[9] Kostlan, E. On the expected number of real roots of a system of random polynomial equations, Hong Kong, 2000, World Science Publishing, River Edge, NJ, USA (2002), pp. 149-188

[10] Kratz, M.; León, J. Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes, Stoch. Process. Appl., Volume 66 (1997) no. 2, pp. 237-252

[11] Kratz, M.; León, J. Central limit theorems for level functionals of stationary Gaussian processes and fields, J. Theor. Probab., Volume 14 (2001) no. 3, pp. 639-672

[12] Maslova, N. The distribution of the number of real roots of random polynomials, Teor. Veroâtn. Primen., Volume 19 (1974), pp. 488-500

[13] Peccati, G.; Tudor, C. Gaussian limits for vector-valued multiple stochastic integrals, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., vol. 1857, Springer, Berlin, 2005, pp. 247-262

[14] Schehr, G.; Majumdar, S. Real roots of random polynomials and zero crossing properties of diffusion equation, J. Stat. Phys., Volume 132 (2008) no. 2, pp. 235-273

[15] Shub, M.; Smale, S. Complexity of Bézout's theorem. II. Volumes and probabilities, Nice, 1992 (Progress in Mathematics), Volume vol. 109, Birkhäuser, Boston, MA, USA (1993), pp. 267-285

[16] Wschebor, M. On the Kostlan–Shub–Smale model for random polynomial systems. Variance of the number of roots, J. Complex., Volume 21 (2005) no. 6, pp. 773-789

Cited by Sources: