Abstract
It is proved that the central limit theorem for general bootstrap empirical process with random resample size indexed by a class of functions 𝓕 and based on a probability measure P holds a.s. if 𝓕 ∈ CLT(P), ∫F 2 dP < ∞, ∥P n − P∥∗ G → a.s. 0 and the random resample size {N n } satisfies \(\frac {N_{n}}n\to _{P}\nu \), where 𝓕 = sup f∈𝓕|f|, P n is the empirical measure, ν is a positive random variable, G = 𝓕 ∪ 𝓕2 ∪ 𝓕′2, 𝓕2, and 𝓕′2 denote the classes of squared functions and squared differences of functions from 𝓕, respectively. The bootstrap general empirical process with random resample size is also considered in the case where the resample size is independent of the original sample and of the bootstrap sample.
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Toan, N.V. On Weak Convergence of the Bootstrap General Empirical Process with Random Resample Size. Vietnam. J. Math. 42, 233–245 (2014). https://doi.org/10.1007/s10013-014-0074-2
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DOI: https://doi.org/10.1007/s10013-014-0074-2