The asymptotic distribution of self-normalized triangular arrays

Journal of Theoretical Probability

Volumn 18, Issue 4, 2005, Pages 853-870

  Mason, David M ^a  

^a UNIVERSITY OF DELAWARE   (United States)

Author keywords

Asymptotic distribution; Infinitely divisible laws; Self normalized sums; Stochastic compactness

Indexed keywords

EID: 26044476911     PISSN: 08949840     EISSN: 15729230     Source Type: Journal    
DOI: 10.1007/s10959-005-7529-z     Document Type: Article

References (19)

1. Araujo, A., and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-Chichester-Brisbane.
2. Berkes, I., and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7, 29-54.
3. Chistyakov, G. P., and Götze, F. (2004). Limit distributions of Studentized means. Ann. Probab. 32, 28-77.
4. Chow, Y. S., and Teicher, H. (1978). Probability Theory. Independence, Interchangeability, Martingales. Springer-Verlag, New York-Heidelberg.
5. Csörgo, S, Haeusler, E., and Mason, D. M. (1988). A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. Appl. Math. 9, 259-333.
6. Feller, W. (1966). An Introduction to Probability Theory and Its Applications, II., John Wiley & Sons, Inc., New York.
7. Giné, E., Götze, F., and Mason, D. M. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25, 1514-1531.
8. Giné, E., and Mason, D. M. (1998). On the LIL for self-normalized sums of IID random variables. J. Theoret. Probab. 11, 351-370.
9. Gnedenko, B. V., and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, MA.
10. Griffin, P. S. (2002). Tightness of the Student t-statistic. Electron. Comm. Probab. 7, 171-180.
11. de Haan, L. (1970). On regular variation and its application to the weak convergence of sample extremes, Mathematical Centre Tracts, 32 Mathematisch Centrum, Amsterdam.
12. Khintchine, A. Ya. (1937). Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze. Mat. Sbornik 44, 79-119.
13. Logan, B. F., Mallows, C. L., Rice, S. O., and Shepp, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab. 1, 788-809.
14. Maller, R. A. (1981a). A theorem on products of random variables, with application to regression. Austral. J. Statist. 23, 177-185.
15. Maller, R. A. (1981b). Some properties of stochastic compactness. J. Austral. Math. Soc. Ser. A 30, 264-277.
16. Mason, D. M., and Shao, Q-M. (2001). Bootstrapping the Student t-statistic. Ann. Probab. 29, 1435-1450.
17. Mason, D. M., and Zinn, J. (2005). When does a randomly weighted self-normalized sum converge in distribution? Electron. Comm. Probab. 10, 70-81.
18. Pruitt, W. E. (1983). The class of limit laws for stochastically compact normed sums. Ann. Probab. 11, 962-969.
19. Rosinski, J. (2001). Series representations of Levy processes from the perspective of point processes. In Barndorff-Nielsen, O. E., Mikosch T. and Resnick, S. I. (eds.), Lévy Processes Theory and Applications. Birkhäuser, Boston, pp. 401-415.