The Annals of Probability

Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law

Jean Bertoin

Full-text: Open access

Abstract

Let $\xi, \xi_1, \dots$ be i.i.d. real-valued random variables and $S_n = \xi_1 + \dots + \xi_n$. In the case when the distribution of $\xi$ is close to a stable $(\alpha)$ law for some $\alpha \epsilon (0, 1) \bigcup (1, 2)$, we investigate the asymptotic behavior in distribution of the maximum of normalized sums, $\max_{k=1,\dots,n} k^{-1/\alpha}S_k$. This completes the Darling-Erdös limit theorem for the case $\alpha = 2$.

Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 832-852.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855652

Digital Object Identifier
doi:10.1214/aop/1022855652

Mathematical Reviews number (MathSciNet)
MR1626527

Zentralblatt MATH identifier
0943.60041

Subjects
Primary: 60J30
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes

Keywords
Stable Lévy process normalized maximum Darling-Erdös theorem

Citation

Bertoin, Jean. Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law. Ann. Probab. 26 (1998), no. 2, 832--852. doi:10.1214/aop/1022855652. https://projecteuclid.org/euclid.aop/1022855652


Export citation

References

  • [1] Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Probab. 18 92-128.
  • [2] Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks Cole, Pacific Grove, CA.
  • [3] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705- 766.
  • [4] Blumenthal, R. M. (1992). Excursions of Markov Processes. Birkh¨auser, Boston.
  • [5] Breiman, L. (1968). A delicate law of the iterated logarithm for non-decreasing stable processes. Ann. Math. Statist. 39 1818-1824. [Correction: Ann. Math. Statist. 41 1126.]
  • [6] Darling, D. A. and Erd os, P. (1956). A limit theorem for the maximum of normalized sums of independent random variables. Duke Math. J. 23 143-154.
  • [7] Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992). Processus de Markov, compl´ements de calcul stochastique. Probabilit´es et Potentiel V. Hermann, Paris.
  • [8] Doney, R. A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 451-465.
  • [9] Einmahl, U. (1989). The Darling-Erd os theorem for sums of i.i.d. random variables. Probab. Theory Related Fields 82 241-257.
  • [10] Einmahl, U. and Mason, D. (1989). Darling-Erd os theorems for martingales. J. Theoret. Probab. 2 437-460.
  • [11] Feller, W. E. (1946). A limit theorem for random variables with infinite moments. Amer. J. Math. 68 257-262.
  • [12] Feller, W. E. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • [13] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745-789.
  • [14] Pruitt, W. E. (1981). The growth of random walks and L´evy processes. Ann. Probab. 9 948-956.
  • [15] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, Berlin.
  • [16] Rootz´en, H. (1978). Extremes of moving averages of stable processes. Ann. Probab. 6 847- 869.
  • [17] Rootz´en, H. (1986). Extreme value theory for moving average processes. Ann. Probab. 14 612-652.
  • [18] Rootz´en, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. in Appl. Probab. 20 371-390.
  • [19] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance. Chapman & Hall, New York.
  • [20] Shorack, G. R. (1979). Extension of the Darling and Erd os theorem on the maximum of normalized sums. Ann. Probab. 7 1092-1096.
  • [21] Tkachuk, S. G. (1977). Limit theorems for sums of independent random variables belonging to the domain of attraction of a stable law. Ph.D. thesis, Tashkent. (In Russian.)
  • [22] Vinogradov, V. (1994). Refined Large Deviation Limit Theorems. Longman, Essex.
  • [23] Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Amer. Math. Soc., Providence, RI.