Let X₁, X₂, …, Xₙ be independent identically distributed random variables with EX²₁ = ∞ but X₁ belonging to the domain of attraction of a stable law. It is known that the sample mean X̄ₙ appropriately normalized converges to a stable law. It is shown here that the bootstrap version of the normalized mean has a random distribution (given the sample) whose limit is also a random distribution implying that the naive bootstrap could fail in the heavy tailed case.
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Krishna B. Athreya (1987) studied this question.
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