Comptes Rendus
no. 12
Statistics
A sufficient condition for asymptotic normality of the normalized quadratic form Ψn(f,g)
[Une condition suffisante de normalité asymptotique de la forme quadratique standardisée Ψn(f,g)]

Présenté par : Paul Deheuvels

Valentin Solev1 ; Léo Gerville-Reache2
1 Steklov Mathematical Institute, St Petersbourg, Russia
2 Statistique mathématique, université Victor-Segalen, 146, rue Léo-Saignat, 33076 Bordeaux, France
Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 971-975.

Des conditions suffisantes de normalité asymptotique de la forme quadratique standardisée Ψn(f,g) se sont succédées depuis 1958. La moins restrictive fût proposée par L. Giraitis et D. Surgailis en 1990. En abordant le problème sous l'angle des opérateurs linéaires, il est possible de produire une condition suffisante encore moins restrictive sur le couple de fonctions (f,g).

Many sufficient conditions of asymptotic normality of the normalized quadratic form Ψn(f,g) have been proposed since 1958. The less restrictive was given in the paper of L. Giraitis and D. Surgailis (1990). Using a linear operator approach, it is possible to produce an even less restrictive sufficient condition on the couple of functions (f,g).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.04.022
Valentin Solev 1 ; Léo Gerville-Reache 2

1 Steklov Mathematical Institute, St Petersbourg, Russia
2 Statistique mathématique, université Victor-Segalen, 146, rue Léo-Saignat, 33076 Bordeaux, France 
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Valentin Solev; Léo Gerville-Reache. A sufficient condition for asymptotic normality of the normalized quadratic form $ {\Psi }_{n}(f,g)$. Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 971-975. doi : 10.1016/j.crma.2006.04.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.022/

[1] F. Avram On Bilinear Forms in Gaussian Random Variables and Toeplitz Matrices, University of California Press, Berkeley and Los Angeles, 1958

[2] M.S. Ginovyan On Toeplitz type quadratic forms in Gaussian stationary process, Probab. Theory Related Fields, Volume 100 (1994), pp. 395-406

[3] M.S. Ginovyan, A.A. Saakian, On central limit theorem for Toeplitz quadratic forms in Gaussian stationary sequences, Theory Probab. Appl. (2005) submitted for publication

[4] L. Giraitis; D. Surgailis A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate, Probab. Theory Related Fields, Volume 86 (1990), pp. 87-104

[5] U. Grenander; G. Szegö Toeplitz Form and Their Applications, University of California Press, Berkeley and Los Angeles, 1958

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[7] I.A. Ibragimov; Yu.A. Rozanov Gaussian Processes, Mir, Moscow, 1974

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