Abstract
The purpose of the present paper is to establish a functional central limit theorem (FCLT) for partial-sum processes with random locations and indexed by Vapnik-Cervonenkis classes (VCC) of sets in arbitrary sample spaces. The context is as follows: Let X = (X,X) be an arbitrary measurable space, (ηj)j∈N be a sequence of independent and identically distributed (i.i.d.) random elements (r.e.) in X with distribution v on X (that is, the η; j ’s are asumed to be defined on some basic probability space (Ω, F, P) with values in X such that each ηj: (Ω, F) → (X,X) is measurable), and let (ξnj)1≤ j ≤ j (n),n∈ℕ be a triangular array of rowwise independent (but not necessarily identically distributed) real-valued random variables (r.v.) (also defined on (Ω, F, P)) such that the whole triangular arrray is independent of the sequence (ηj)j∈ℕ. Given a class C ⊂ X, define a partial-sum process (of sample size n ∈ IN) S n = (S n(C))C∈c by
where I C denotes the indicator function of C.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alexander, K.S. and Pyke, R. (1986). A uniform central limit theorem for setindexed partial-sum processes with finite variance. Ann. Probab. 14, 582–597.
Alexander, K.S. (1987). Central limit theorems for stochastic processes under random entropy conditions. Probab. The. Re. Fields 75, 351–378.
Araujo, A. and Giné (1980). The central limit theorem for real and Banach space valued random variables. Wiley, New York.
Assouad, P. (1981). Sur les classes de Vapnik-Cervonenkis. C. R. Acad. Sci. Paris 292 Sér. I 921–924.
Bass, R. F. and Pyke, R. (1984). Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets. Ann. Probab. 12, 13–34.
Dudley, R. M. (1978). Central limit theorems for empirical measures. Ann Probab. 6 899–929. (Correction (1979) ibid. 7 909-911)
Gaenssler, P. and Schlumprecht, Th. (1988). Maximal inequalities for stochastic processes which are given as sums of independent processses indexed by pseudometric parameter spaces (with applications to empirical processes). Preprint, Univ. of Munich.
Gaenssler, P. (1990). On weak convergence of certain processes indexed by pseudometric parameter spaces with applications to empirical processes. Invited paper for the 11th Prague Conference, August 27-31, 1990.
Hoffmann-Jørgensen, J. (1984). Convergence of stochastics processes on Polish spaces. Unpublised.
Marcus, M. B. and Pisier, G. (1981). Random Fourier series with applications to Harmonic analysis. Ann. Math. Studies 101, Princeton University Press, Princeton, New Jersey.
Pollard, D. (1984). Convergence of stochastic processes. Springer-Verlag, New York.
Pyke, R. (1983). A uniform central limit theorem for partial-sum processes indexed by sets. In Probability, Statistics and Analysis (J. F.C. Kingman and G. R. H. Reuter, eds.) 219–240. Cambridge University Press.
Stengle, G. and Yukich, J. E. (1989). Some new Vapnik-Červonenkis classes. Ann. Statist. 17, 1441–1446.
Vapnik, V.N. and Cervonenkis A. Ja. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Appl. 16, 264–280.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media New York
About this chapter
Cite this chapter
Arcones, M.A., Gaenssler, P., Ziegler, K. (1992). Partial-Sum Processes with Random Locations and Indexed by Vapnik-Červonenkis Classes of Sets in Arbitrary Sample Spaces. In: Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds) Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Progress in Probability, vol 30. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0367-4_26
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0367-4_26
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6728-7
Online ISBN: 978-1-4612-0367-4
eBook Packages: Springer Book Archive