Advertisement

Journal of Theoretical Probability

, Volume 18, Issue 4, pp 757–811 | Cite as

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

  • F. Götze
  • A. N. Tikhomirov
Article
We consider quadratic forms of the type
$$ Q(F,{\bf A})=\sum_{\mathop{1\le j,k \le N}\limits_{j\ne k}}a_{jk} X_j X_k, $$
where X j are i.i.d. random variables with common distribution F and finite fourth moment, \({\bf A}=\{a_{jk}\}_{j,k=1}^N\) denotes a symmetric matrix with eigenvalues λ1, ..., λ N ordered to be non-increasing in absolute value. We prove explicit bounds in terms of sums of 4th powers of entries of the matrix A and the size of the eigenvalue λ13 for the approximation of the distribution of Q(F,A) by the distribution of Q (φ, A) where φ is standard Gaussian distribution. In typical cases this error is of optimal order \({\cal {O}}(N^{-1})\)

Key words

Non-central limit theorems quadratic forms random vectors 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bentkus, V., Götze, F. 1996Optimal rates of convergence in the CLT for quadratic formsAnn. Probab24466490CrossRefGoogle Scholar
  2. 2.
    Bentkus, V., Götze, F. 1997aUniform rates of convergence in the CLT for quadratics forms in multidimensional spacesProbab. Theory Relat. Fields109367416CrossRefGoogle Scholar
  3. 3.
    Bentkus, V., Götze, F. 1997bOn the lattice point problem for the ellipsoidsActa ArithLXXX.2101125Google Scholar
  4. 4.
    Bentkus, V., Götze, F. 1999Optimal bounds in non-Gaussian limit theorems for U-statisticsAnn. Probab27454521CrossRefGoogle Scholar
  5. 5.
    Biggs, N.L. 1995Discrete MathematicsOxford Science PublOxford440Google Scholar
  6. 6.
    Chan, N.H., Wei, C.Z. 1987Asymptotic inference for nearly nonstationary AR(1) processesAnn. Statist1510501063Google Scholar
  7. 7.
    Gamkrelidze, N.G., Rotar’, V.I. 1977On the rate of convergence in the limit theorem for quadratic formsTheory Probab. Appl22394397CrossRefGoogle Scholar
  8. 8.
    Götze, F., Tikhomirov, A. 1999Asymptotic distribution of quadratic formsAnn. Probab5010721098Google Scholar
  9. 9.
    Götze, F., Tikhomirov, A. 2002Asymptotic distribution of quadratic formsJ. Theoret. Probab15423475CrossRefGoogle Scholar
  10. 10.
    Götze F., Ulyanov V.V. (2004). Sharp uniform approximations in the CLT for Balls in Hilbert spaces. Theory Probab. Appl. to appear.Google Scholar
  11. 11.
    Horn, R.A., Johnson, Ch.R. 1991Matrix AnalysisCambridge University PressNew York561Google Scholar
  12. 12.
    Rachkauskas, A. 1996Asymptotic accuracy of the least-squares estimates in nearly nonstationary autoregressive modelsLith. Math. J3692103Google Scholar
  13. 13.
    Rotar’, V.I. 1973Some limit theorems for polynomials of second degreeTheory Probab. Appl18499507CrossRefGoogle Scholar
  14. 14.
    Rotar’, V.I., Shervashidze, T.L. 1985Some estimates of distributions of quadratic formsTheory Probab. Appl30585591CrossRefGoogle Scholar
  15. 15.
    Sevast’yanov, B.A. 1961A class of limit distribution for quadratic forms of normal stochastic variablesTheory Probab. Appl6337340Google Scholar
  16. 16.
    Tikhomirov, A.N. 1991On the accuracy of the normal approximation of the probability of sums of weakly dependent Hilbert-space-valued random variables hitting a ball. ITheory Probab. Appl36738751CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeld 1Germany
  2. 2.Faculty of MathematicsSyktyvkarRussia
  3. 3.The Mathematical Department of IMM of the Ural Branch of Russian Academia of SciencesSyktyvkar State UniversityRussia

Personalised recommendations