Limit distributions of norms of vectors of positive i.i.d. random variables

Annals of Probability

Volumn 29, Issue 2, 2001, Pages 862-881

  Schlather, Martin ^a  

^a LANCASTER UNIVERSITY   (United Kingdom)

Author keywords

Central limit theorem; Extreme value theory; i.i.d. positive random variables; Limit theorems; lp norm; Normal distribution

Indexed keywords

EID: 0035537288     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1008956695     Document Type: Article

References (23)

1. ABRAMOWITZ, M. and STEGUN, I. A. (1984). Pocketbook of Mathematical Functions. Harri Deutsch, Frankfurt am Main.
2. ANDERSON, C. W. and TURKMAN, K. F. (1995). Sums and maxima of stationary sequences with heavy tailed distributions. Sankhyā Ser. A 57 1-10.
3. BILLINGSLEY, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
4. BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. (1987). Regular Variation. Cambridge Univ. Press.
5. CHOW, T. L. and TEUGELS, J. (1979). The sum and the maximum of i.i.d. random variables. In Proceedings of the Second Prague Symposium on Asymptotic Statistics (P. Mandl and M. Huskova, eds.) 81-92. North-Holland, Amsterdam.
6. EMBRECHTS, P., KLÜPPELBERG, C. and MIKOSCH, T. (1997). Modelling Extremal Events. Springer, Berlin.
7. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
8. GNEDENKO, B. W. (1963). The Theory of Probability, 2nd ed. Chelsea, New York.
9. GRADSHTEYN, I. S. and RYZHIK, I. M. (2000). Table of Integrals, Series, and Products, 6th ed. Academic, London.
10. GREENWOOD, P. E. and HOOGHIEMSTRA, G. (1991). On the domain of attraction of an operator between supremum and sum. Probab. Theory Related Fields 89 201-210.
11. GRIFFIN, P. and KUELBS, J. (1991). Some extensions of the LIL via self-normalisations. Ann. Probab. 19 380-395.
12. HAHN, M. G. and WEINER, D. C. (1992). Asymptotic behaviour of self-normalized trimmed sums: nonnormal limits. Ann. Probab. 20 455-482.
13. HO, H.-C. and HSING, T. (1996). On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. J. Appl. Probab. 33 138-145.
14. HOOGHIEMSTRA, G. and GREENWOOD, P. E. (1997). The domain of attraction of the α-sun operator for type II and type III distributions. Bernoulli 3 479-489.
15. HORVÁTH, L. and SHAO, Q.-M. (1996). Large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation. Ann. Probab. 24 1368-1387.
16. HSING, T. (1995). A note on the asymptotic independence of the sum and the maximum of strongly mixing stationary random variables. Ann. Probab. 23 938-947.
17. KALLENBERG, O. (1997). Foundations of Modern Probability. Springer, New York.
18. LEADBETTER, M. R., LINDGREN, G. and ROOTZÉN, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin.
19. 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.
20. RESNICK, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Appl. Probab. 4. Springer, New York.
21. SAMORODNITSKY, G. and TAQQU, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, Boca Raton, FL.
22. SATO, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
23. SHAD, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285-327.