Abstract
For the tail probabilities of a general set-indexed empirical process in an arbitrary sample space a maximal inequality is derived. In the case that the class of sets by which the process is indexed possesses a total ordering, the application of our inequality yields an elementary proof for a functional central limit theorem without involving such advanced techniques as symmetrization, stratification, chaining or Gaussian domination. Analogously, the inequality leads to a weak uniform law of large numbers (including convergence rate).
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Ziegler, K. A Maximal Inequality and a Functional Central Limit Theorem for set-indexed empirical processes. Results. Math. 31, 189–194 (1997). https://doi.org/10.1007/BF03322161
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DOI: https://doi.org/10.1007/BF03322161
Key Words
- Maximal inequality
- set-indexed empircal process
- functional central limit theorem
- uniform law of large numbers