Perturbation Approach Applied to the Asymptotic Study of Random Operators

We prove that, for the main kind of limit theorems (laws of large numbers, central limit theorems, large deviations principles, laws of the iterated logarithm) asymptotic results for selfadjoint random operators yield equivalent results for their eigenvalues and associated projections.

[1]  Denis Bosq,et al.  Linear Processes in Function Spaces , 2000 .

[2]  D. Bosq.  Linear Processes in Function Spaces: Theory And Applications , 2000 .

[3]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[4]  Jeanne Fine.  On the validity of the perturbation method in asymptotic theory , 1987 .

[5]  Jacob T. Schwartz,et al.  Linear operators. Part II. Spectral theory , 2003 .

[6]  F. Smithies.  Linear Operators , 2019, Nature.

[7]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[8]  I. Gohberg.  Classes of Linear Operators, Vol.I, Operator Theory , 1990 .

[9]  M. A. Kaashoek,et al.  Classes of Linear Operators Vol. I , 1990 .

[10]  André Mas,et al.  Large and moderate deviations for infinite-dimensional autoregressive processes , 2003 .

[11]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[12]  Frits H. Ruymgaart,et al.  Some Applications of Watson's Perturbation Approach to Random Matrices , 1997 .

[13]  J. Dauxois,et al.  Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference , 1982 .

[14]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[15]  David E. Tyler.  Asymptotic Inference for Eigenvectors , 1981 .

[16]  J. Diestel,et al.  On vector measures , 1974 .

[17]  Ludovic Menneteau,et al.  Some laws of the iterated logarithm in Hilbertian autoregressive models , 2005 .

[18]  André Mas,et al.  Weak convergence for the covariance operators of a Hilbertian linear process , 2002 .