## The Annals of Probability

### Limit Distributions Of Norms Of Vectors Of Positive i.i.d.Random Variables

Martin Schlather

#### Abstract

This paper aims to combine the central limit theorem with the limit theorems in extreme value theory through a parametrized class of limit theorems where the former ones appear as special cases. To this end the limit distributions of suitably centered and normalized $l_{cp(n)}$-norms of $n$-vectors of positive i.i.d. random variables are investigated. Here, $c$ is a positive constant and $p(n)$ is a sequence of positive numbers that is given intrinsically by the form of the upper tail behavior of the random variables. A family of limit distributions is obtained if $c$ runs over the positive real axis. The normal distribution and the extreme value distributions appear as the endpoints of these families, namely, for $c =0 +$ and $c = \infty$, respectively.

#### Article information

Source
Ann. Probab., Volume 29, Number 2 (2001), 862-881.

Dates
First available in Project Euclid: 21 December 2001

https://projecteuclid.org/euclid.aop/1008956695

Digital Object Identifier
doi:10.1214/aop/1008956695

Mathematical Reviews number (MathSciNet)
MR1849180

Zentralblatt MATH identifier
1014.60015

#### Citation

Schlather, Martin. Limit Distributions Of Norms Of Vectors Of Positive i.i.d.Random Variables. Ann. Probab. 29 (2001), no. 2, 862--881. doi:10.1214/aop/1008956695. https://projecteuclid.org/euclid.aop/1008956695

#### References

• Abramowitz, M. and Stegun, I. A. (1984). Pocketbook of Mathematical Functions. Harri Deutsch, Frankfurt am Main.
• Anderson, C. W. and Turkman, K. F. (1995). Sums and maxima of stationary sequences with heavy tailed distributions. Sankhy¯a Ser. A 57 1-10.
• Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
• 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.
• Embrechts, P., Kl ¨uppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
• Gnedenko, B. W. (1963). The Theory of Probability, 2nd ed. Chelsea, New York.
• Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products, 6th ed. Academic, London.
• 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.
• Griffin, P. and Kuelbs, J. (1991). Some extensions of the LIL via self-normalisations. Ann. Probab. 19 380-395.
• Hahn, M. G. and Weiner, D. C. (1992). Asymptotic behaviour of self-normalized trimmed sums: nonnormal limits. Ann. Probab. 20 455-482.
• 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.
• Hooghiemstra, G. and Greenwood, P. E. (1997). The domain of attraction of the -sun operator for type II and type III distributions. Bernoulli 3 479-489.
• Horv´ath, 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.
• Hsing, T. (1995). A note on the asymptotic independence of the sum and the maximum of strongly mixingstationary random variables. Ann. Probab. 23 938-947.
• Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
• Leadbetter, M. R., Lindgren, G. and Rootz´en, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin.
• Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of selfnormalized sums. Ann. Probab. 1 788-809.
• Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Appl. Probab. 4. Springer, New York.
• Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, Boca Raton, FL.
• Sato, K.-I. (1999). L´evy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
• Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285-327.