Summary
Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array (φ nj ) of stochastic processes, indexed by a set S, to a sample-continuous Gaussian process, when the array satisfies a “random entropy” condition, analogous to one used by Giné and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-Ĉervonenkis property and each φ nj (f)fdνnj is of the form νnjc for some reasonable random finite signed measure v nj. As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik-Ĉervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.
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Research supported by an NSF Postdoctoral Fellowship, grant no. MCS83-111686
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Alexander, K.S. Central limit theorems for stochastic processes under random entropy conditions. Probab. Th. Rel. Fields 75, 351–378 (1987). https://doi.org/10.1007/BF00318707
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DOI: https://doi.org/10.1007/BF00318707
Keywords
- Entropy
- Stochastic Process
- Probability Theory
- Limit Theorem
- Statistical Theory