## Electronic Communications in Probability

### A Law of the Iterated Logarithm for the Sample Covariance Matrix

Steven Sepanski

#### Abstract

For a sequence of independent identically distributed Euclidean random vectors, we prove a law of the iterated logarithm for the sample covariance matrix when $o(\log \log n)$ terms are omitted. The result is proved under the hypothesis that the random vectors belong to the generalized domain of attraction of the multivariate Gaussian law. As an application, we obtain a bounded law of the iterated logarithm for the multivariate t-statistic.

#### Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 7, 63-76.

Dates
Accepted: 20 May 2003
First available in Project Euclid: 18 May 2016

https://projecteuclid.org/euclid.ecp/1463608891

Digital Object Identifier
doi:10.1214/ECP.v8-1070

Mathematical Reviews number (MathSciNet)
MR1987095

Zentralblatt MATH identifier
1061.60028

Subjects
Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems

Rights

#### Citation

Sepanski, Steven. A Law of the Iterated Logarithm for the Sample Covariance Matrix. Electron. Commun. Probab. 8 (2003), paper no. 7, 63--76. doi:10.1214/ECP.v8-1070. https://projecteuclid.org/euclid.ecp/1463608891

#### References

• Araujo, A. and Gine, E. (1980) The Central Limit Theorem for Real and Banach Valued Random Variables. John Wiley and Sons. New York.
• Billingsley, P. (1966) Convergence of types in k-space. Z. Wahrsch. Verw. Gebiete. 5. 175-179.
• Dudley, R.M. (1989) Real Analysis and Probability. Chapman and Hall. New York.
•  Feller, W.(1968) An Introduction to Probability Theory and its Applications, Volume 1. Third Edition. John Wiley and Sons. New York.
• Feller, W.(1968) An extension of the law of the iterated logarithm to variables without variance. J. Math. Mechan. 18, 343-355.
• Gine, E. and Mason, D.M. (1998) On the LIL for self-normalized sums of iid random variables. J. Theor. Probab. v. 11 no. 2, 351-370.
• Griffin, P.S. and Kuelbs, J.D. (1989) Self normalized laws of the iterated logarithm. Ann. Probab. 17, 1571-1601.
• Hahn, M.G. and Klass, M.J. (1980) Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal. Ann. Probab. 8 (1980), no. 2, 262–280.
• Kuelbs, J. and Ledoux M. (1987) Extreme values and the law of the iterated logarithm. Prob. Theory and Rel. Fields. 74, 319-340.
•  Meerschaert, M. (1988) Regular variation in $\R^k$. Proc. Amer. Math. Soc. 102, 341-348.
•  Meerschaert, Mark M.and Scheffler, Hans-Peter (2001) Limit distributions for sums of independent random vectors. Heavy tails in theory and practice. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, 2001.
• Pruitt, W.E. (1981) General one sided laws of the iterated logarithm. Ann. Probab. 9, 1-48.
• Sepanski, S.J. (1994).Asymptotics for multivariate $t$-statistic and Hotelling's $T\sp 2$-statistic under infinite second moments via bootstrapping. J. Multivariate Anal. 49, no. 1, 41–54.
• Sepanski, Steven J. (1994) Necessary and sufficient conditions for the multivariate bootstrap of the mean. Statist. Probab. Lett. 19, no. 3, 205–216.
•  Sepanski, Steven J. (2002) Extreme values and the multivariate compact law of the iterated logarithm. J. Theoret. Probab. 14 no. 4, 989–1018.
• Sepanski, S. (2001) A law of the iterated logarithm for multivariate trimmed sums. Preprint.
• Sepanski, Steven J. (1996) Asymptotics for multivariate $t$-statistic for random vectors in the generalized domain of attraction of the multivariate normal law. Statist. Probab. Lett. 30, no. 2, 179–188.