Bounded law of the iterated logarithm for sums of independent random vectors normalized by matrices

Author:
V. O. Koval’

Translated by:
Oleg Klesov

Journal:
Theor. Probability and Math. Statist. **72** (2006), 69-73

MSC (2000):
Primary 60F15

DOI:
https://doi.org/10.1090/S0094-9000-06-00665-X

Published electronically:
August 18, 2006

MathSciNet review:
2168137

Full-text PDF Free Access

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Abstract: Let $(X_n,n\geq 1)$ be a sequence of independent centered random vectors in $\mathbf R^d$ with finite moments of order $p\in (2,3]$ and let $(A_n,n\geq 1)$ be a sequence of $m\times d$ matrices. We find explicit conditions under which \[ \limsup _{n\to \infty } c_n \left \|A_n\sum _{i=1}^n X_i\right \|<\infty \] almost surely, where $(c_n,n\geq 1)$ is some sequence of positive numbers.

References
- Valery Koval,
*A new law of the iterated logarithm in ${\bf R}^d$ with application to matrix-normalized sums of random vectors*, J. Theoret. Probab. **15** (2002), no. 1, 249–257. MR **1883931**, DOI https://doi.org/10.1023/A%3A1013851720494
- V. V. Buldygin and V. A. Koval,
*Convergence to zero and boundedness of operator-normed sums of random vectors with application to autoregression processes*, Georgian Math. J. **8** (2001), no. 2, 221–230. Dedicated to Professor Nicholas Vakhania on the occasion of his 70th birthday. MR **1851031**

References
- V. Koval,
*A new law of the iterated logarithm in $R^d$ with application to matrix-normalized sums of random vectors*, J. Theoret. Probab. **15** (2002), no. 1, 249–257. MR **1883931 (2003a:60049)**
- V. V. Buldygin and V. A. Koval,
*Convergence to zero and boundedness of operator-normed sums of random vectors with application to autoregression processes*, Georgian Math. J. **8** (2001), no. 2, 221–230. MR **1851031 (2003d:60067)**

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Additional Information

**V. O. Koval’**

Affiliation:
Department of Higher Mathematics, Zhitomir State University for Technology, Chernyakhovskiĭ Street 103, 10005 Zhitomir, Ukraine

Email:
vkoval@com.zt.ua

Keywords:
Law of the iterated logarithm,
sums of independent random vectors,
matrix normalizations

Received by editor(s):
August 31, 2004

Published electronically:
August 18, 2006

Article copyright:
© Copyright 2006
American Mathematical Society