Advertisement

Probability Theory and Related Fields

, Volume 75, Issue 2, pp 261–277 | Cite as

A central limit theorem for generalized quadratic forms

  • Peter de Jong
Article

Summary

Random variables of the form \(W(n) = \mathop \sum \limits_{1 \leqq i \leqq n} \mathop \sum \limits_{{\text{ }}1 \leqq j \leqq n} w_{ijn} (X_i ,X_j )\) are considered with X i independent (not necessarily identically distributed), and w ijn (·, ·) Borel functions, such that w ijn (X i , X j ) is square integrable and has vanishing conditional expectations:
$$E(w_{ijn} (X_i ,X_j )|X_i = E(w_{ijn} (X_i ,X_j )|X_j ) = 0,{\text{ a}}{\text{.s}}{\text{.}}$$
A central limit theorem is proved under the condition that the normed fourth moment tends to 3. Under some restrictions the condition is also necessary. Finally conditions on the individual tails of w ijn (X i , X j ) and an eigenvalue condition are given that ensure asymptotic normality of W(n).

Keywords

Stochastic Process Probability Theory Quadratic Form Limit Theorem Statistical Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bloemena, A.R.: Sampling from a graph. Mathematical Centre Tract 2, Amsterdam (1964)Google Scholar
  2. Barbour, A.D., Eagleson, G.K.: Multiple comparisons and sums of dissociated random variables. Adv. Appl. Prob. 17, 147–162 (1985)MathSciNetCrossRefMATHGoogle Scholar
  3. Beran, R.J.: Rank spectral processes and tests for serial dependence. Ann. Math. Statist. 43, 1749–1766 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. Brown, B.M., Kildea, D.G.: Reduced U-statistics and the Hodge-Lehmann estimator. Ann. Statist. 6, 828–835 (1978)MathSciNetCrossRefMATHGoogle Scholar
  5. Chung, K.L.: A course in probability theory, 2nd edn. New York: Academic Press 1974MATHGoogle Scholar
  6. Feller, W.: An introduction to probability theory and its applications II. New York: Wiley 1971MATHGoogle Scholar
  7. Heyde, C.C., Brown, B.M.: On the departure from normality of a certain class of martingales. Ann. Math. Statist. 41, 2165–2165 (1970)MathSciNetMATHGoogle Scholar
  8. Hall, P.: Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivar. Anal. 14, 1–16 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. Jammalamadaka, R.S., Janson, S.: Limit theorems for a triangular scheme of U-statistics with applications to interpoint distances. Ann. Probab. 14, 1347–1358 (1986)MathSciNetCrossRefMATHGoogle Scholar
  10. Karlin, S., Rinott, Y.: Applications of ANOVA type decompositions of conditional variance statistics including Jackknife estimates. Ann. Statist. 10, 485–501 (1982)MathSciNetCrossRefMATHGoogle Scholar
  11. Kester, A.: Asymptotic normality of the number of small distances between random points in a cube. Stochastic Process. Appl. 3, 45–54 (1975)MathSciNetCrossRefMATHGoogle Scholar
  12. McGinley, W.G., Sibson, R.: Dissociated random variables. Math. Proc. Cambridge. Phil. Soc. 77, 185–188 (1975)MathSciNetCrossRefMATHGoogle Scholar
  13. Noether, G.E.: A central limit theorem with non-parametric applications. Ann. Math. Statist. 41, 1753–1755 (1970)MathSciNetCrossRefMATHGoogle Scholar
  14. Robinson, J.: Limit theorems for standardized partial sums of exchangeable and weakly exchangeable arrays. (Preprint) (1985)Google Scholar
  15. Rotar', V.I.: Some limit theorems for polynomials of second degree. Theor. Probab. Appl. 18, 499–507 (1973)CrossRefMATHGoogle Scholar
  16. Sevast'yanov, B.A.: A class of limit distributions for quadratic forms of normal stochastic variables. Theor. Probab. Appl. 6, 337–340 (1961)MATHGoogle Scholar
  17. Shapiro, C.P., Hubert, L.: Asymptotic normality of permutation statistics derived from weighted sums of bivariate functions. Ann. Statist. 1, 788–794 (1979)MathSciNetCrossRefMATHGoogle Scholar
  18. Weber, N.C.: Central limit theorems for a class of symmetric statistics. Math. Proc. Cambridge Philos. Soc. 94, 307–313 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. Whittle, P.: On the convergence to normality of quadratic forms in independent variables. Theor. Probab. Appl. 9, 113–118 (1964)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Peter de Jong
    • 1
  1. 1.Mathematical Institute of the University of AmsterdamAmsterdamThe Netherlands

Personalised recommendations