Abstract
The paper unifies and extends a number of Hoffmann-j0rgensen-type inequalities known in the literature. Originally proved for sums of independent Banach-space valued random variables and commonly used in empirical process theory under much weaker measurability, the inequality is shown here to be still valid for arbitrary non-measurable mappings defined on the coordinates of a product probability space (thus replacing independence) and taking values in a real or complex vector space equipped with some (possibly infinite) seminorm. Additonally, our results are in “ψ-norm” (where ψ) is assumed to be a convex and nondecreasing function satisfying the Orlicz condition) generalizing the p-norms usually considered in the literature in this context. Applications of the inequality concern - among others - uniform laws of large numbers for triangular arrays of stochastic processes.
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References
Dudley, R.M. (1984): A course on empirical processes. Lecture Notes in Math. 1097, 1–142.
aenssler, P. (1992): On weak convergence of certain processes indexed by pseudo-metric parameter spaces with applications to empirical processes. Transactions of the 11 th Prague Conference, 49-78.
Giné, E. and Zinn, J. (1984): Some limit theorems for empirical processes. Ann. Probab. 12, 929–989.
Giné, E. and Zinn, J. (1986): Lectures on the central limit theorem for empirical processes. Lecture Notes in Math. 1221,50–113.
Giné, E. and Zinn, J. (1992): On Hoffmann-Jørgensen’s inequality for U-processes. In: Probability in Banach spaces, Vol. 8, Birkhäuser, Boston.
Hoffmann-ørgensen, J. (1974): Sums of independent Banach space valued random variables. Studia Math. 52, 159–189.
Hoffmann-Jørgensen, J. (1984): Stochastic processes on Polish spaces. Unpublished manuscript. Published in 1991 as Vol. 39 of the Various Publication Series, Aarhus Universitet.
Hoffmann-ørgensen, J. (1985): Necessary and sufficient conditions for the uniform law of large numbers. Lecture Notes in Math. 1153,258–272.
Ledoux, M. and Talagrand, M. (1991): Probability in Banach spaces. Springer, New York.
Marcus, M. (1981): Weak convergence of the empirical characteristic function. Ann. Probab. 9, 194–201.
Ossiander, M. (1987): A central limit theorem under metric entropy with L2-bracketing. Ann. Probab 15, 897–919.
Pollard, D. (1984): Convergence of stochastic processes. Springer, New York.
Strobl, F. (1994): Zur Theorie empirischer Prozesse unter besonderer Berücksichtigung des Bootstrap-Verfahrens und Fréchet-differenzierbarer statistischer Funktionale. Dissertation, Univ. of Munich.
Van der Vaart, A.W. and Wellner, J.A. (1996): Weak convergence and empirical processes (with Application to Statistics). Springer, New York.
Ziegler, K. (1994): On Functional Limit Theorems and Uniform Laws of Large Numbers for Sums of Independent Processes. Dissertation, Univ. of Munich.
Ziegler, K. (1995): Uniform Laws of Large Numbers for Triangular Arrays of Function-indexed Processes under Random Entropy Conditions. Submitted to Transactions of the American Mathematical Society.
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Ziegler, K. On Hoffmann-Jørgensen-type Inequalities for Outer Expectations with Applications. Results. Math. 32, 179–192 (1997). https://doi.org/10.1007/BF03322537
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DOI: https://doi.org/10.1007/BF03322537
Key words
- Hoffmann-Jørgensen inequality
- measurable cover function
- symmetrization
- Orlicz condition
- uniform law of large numbers
- asymptotic equicontinuity