Abstract
Let {X, X n , n ≥ 1} be a sequence of independent and identically distributed non-degenerate random variables. Put \(S_{0} = 0,\ S_{n} =\sum _{ i=1}^{n}X_{i}\) and \(V _{n}^{2} =\sum _{ i=1}^{n}X_{i}^{2},\ n \geq 1.\) A weak convergence theorem is established for the self-normalized partial sums processes \(\{S_{[int]} /V _{n},0 \leq t \leq 1\}\) when X belongs to the domain of attraction of a stable law with index α ∈ (0, 2]. The respective limiting distributions of the random variables \(\max _{1\leq i\leq n}\vert X_{i}\vert /S_{n}\) and \(\max _{1\leq i\leq n}\vert X_{i}\vert /V _{n}\) are also obtained under the same condition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)
Chistyakov, G.P., Götze, F.: Limit distributions of Studentized means. Ann. Probab. 32, 28–77 (2004)
Csörgő, M., Horváth, L.: Asymptotic representations of self-normalized sums. Probab. Math. Stat. 9, 15–24 (1988)
Csörgő, M., Szyszkowicz, B., Wang, Q.: Donsker’s theorem for self-normalized partial sums processes. Ann. Probab. 31, 1228–1240 (2003)
Csörgő, M., Szyszkowicz, B., Wang, Q.: On weighted approximations and strong— limit theorems for self-normalized partial sums processes. In: Horváth, L., Szyszkowicz, B. (eds.) Asymptotic Methods in Stochastics. Fields Institute Communications, vol. 44, pp. 489–521. American Mathematical Society, Providence (2004)
Csörgő, M., Szyszkowicz, B., Wang, Q.: On weighted approximations in D[0, 1] with application to self-normalized partial sum processes. Acta Math. Hung. 121, 307–332 (2008)
Csörgő, S.: Notes on extreme and self-normalised sums from the domain of attraction of a stable law. J. Lond. Math. Soc. 39, 369–384 (1989)
Darling, D.A.: The influence of the maximum term in the addition of independent random variables. Trans. Am. Math. Soc. 73, 95–107 (1952)
de la Peña, V.H., Lai, T.L., Shao, Q.-M.: Self-normalized Processes: Limit Theory and Statistical Applications. Probability and Its Applications (New York). Springer, Berlin (2009)
Efron, B.: Student’s t-test under symmetry conditions. J. Am. Stat. Assoc. 64, 1278–1302 (1969)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1971)
Giné, E., Götze, F., Mason D.: When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25, 1514–1531 (1997)
Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Addison-Wesley, Cambridge (1968)
Griffin, P.S., Mason, D.M.: On the asymptotic normality of self-normalized sums. Proc. Camb. Philos. Soc. 109, 597–610 (1991)
Horváth, L., Shao, Q.-M.: Large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation. Ann. Probab. 24, 1368–1387 (1996)
Jing, B.-Y., Shao, Q.-M., Zhou, W.: Towards a universal self-normalized moderate deviation. Trans. Am. Math. Soc. 360, 4263–4285 (2008)
Kallenberg, O.: Canonical representations and convergence criteria for processes with interchangeable increments. Z. Wahrsch. Verw. Geb. 27, 23–36 (1973)
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)
Kallenberg, O.: Probabilistic Symmetries and Invariance Principles. Springer, New York (2005)
Kesten, H., Maller, R.A.: Infinite limits and infinite limit points for random walks and trimmed sums. Ann. Probab. 22, 1475–1513 (1994)
Logan, B.F., Mallows, C.L., Rice, S.O., Shepp, L.A.: Limit distributions of self-normalized sums. Ann. Probab. 1, 788–809 (1973)
Martsynyuk, Yu.V.: Functional asymptotic confidence intervals for the slope in linear error-in-variables models. Acta Math. Hung. 123, 133–168 (2009a)
Martsynyuk, Yu.V.: Functional asymptotic confidence intervals for a common mean of independent random variables. Electron. J. Stat. 3, 25–40 (2009b)
O’Brien, G.L.: A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Probab. 17, 539–545 (1980)
Petrov, V.V.: Limit Theorems of Probability Theory, Sequences of Independent Random Variables. Clarendon, Oxford (1995)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Shao, Q.-M.: Self-normalized large deviations. Ann. Probab. 25, 285–328 (1997)
Shao, Q.-M.: Recent developments on self-normalized limit theorems. In: Szyszkowicz, B. (ed.) Asymptotic Methods in Probability and Statistics. A Volume in Honour of M. Csörgő, pp. 467–480. North-Holland, Amsterdam (1998)
Shao, Q.-M.: Recent progress on self-normalized limit theorems. In: Lai, T.L., Yang, H., Yung, S.P. (eds.) Probability, Finance and Insurance. World Scientific, Singapore (2004)
Shao, Q.-M.: Stein’s method, self-normalized limit theory and applications. In: Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), Hyderabad, pp. 2325–2350 (2010)
Acknowledgements
We wish to thank two referees for their careful reading of our manuscript. The present version reflects their much appreciated remarks and suggestions. In particular, we thank them for calling our attention to the newly added reference Kallenberg [19], and for advising us that the proof of our Theorem 2.1 needs to be done more carefully, taking into account the remarks made in this regard. The present revised version of the proof of our Theorem 2.1 is done accordingly, with our sincere thanks attached herewith.
This research was supported by an NSERC Canada Discovery Grant of Miklós Csörgő at Carleton University and, partially, also by NSFC(No.10801122), the Fundamental Research Funds for the Central Universities, obtained by Zhishui Hu.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Csörgő, M., Hu, Z. (2015). Weak Convergence of Self-normalized Partial Sums Processes. In: Dawson, D., Kulik, R., Ould Haye, M., Szyszkowicz, B., Zhao, Y. (eds) Asymptotic Laws and Methods in Stochastics. Fields Institute Communications, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3076-0_1
Download citation
DOI: https://doi.org/10.1007/978-1-4939-3076-0_1
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-3075-3
Online ISBN: 978-1-4939-3076-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)