Summary
Let |x j ,j∈ℕq} be independent identically formed random elements with values in a Banach spaceB, indexed byq-tuples of positive integers. We obtain the almost sure approximation as well as the approximation in probability of the partial sum process\(\{ \sum\limits_{j \in nA} {x_j ,A \in A\} } \) by a partial sum process\(\{ \sum\limits_{j \in nA} {y_j ,A \in A\} } \) asn→∞ uniformly over all setsA in a certain class\(A\) of subsets of theq-dimensional unit cube. Here |y j ,j∈ℕq} are independent identically distributed Gaussian random variables. These results are then applied to obtain the approximation of empirical processes over sets and indexed by sets by Gaussian partial sum processes indexed by sets.
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Research supported in part by grants from the National Science Foundation
This work was done while the first author was visiting Texas A & M University
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Morrow, G.J., Philipp, W. Invariance principles for partial sum processes and empirical processes indexed by sets. Probab. Th. Rel. Fields 73, 11–42 (1986). https://doi.org/10.1007/BF01845991
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DOI: https://doi.org/10.1007/BF01845991
Keywords
- Positive Integer
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Unit Cube