On the Asymptotic Behaviour of the Moving Block Bootstrap for Normalized Sums of Heavy-Tail Random Variables

This paper studies the performance of the moving block bootstrap procedure for normalized sums of dependent random variables. Suppose that X₁, X₂,… are stationary ρ-mixing random variables with ∑ ρ (2ⁱ) < ∞. Let Tₙ = (X₁ + ⋯ + Xₙ - bₙ)/aₙ, for some suitable constants aₙ and bₙ, and let Tᵐ,n denote the moving block bootstrap version of Tₙ based on a bootstrap sample of size m. Under certain regularity conditions, it is shown that, for Xₙ's lying in the domain of partial attraction of certain infinitely divisible distributions, the conditional distribution Ĥm,n of Tᵐ,n provides a valid approximation to the distribution of Tₙ along every weakly convergent subsequence, provided $m = o(n)$ as n → ∞. On the other hand, for the usual choice of the resample size m = n, Ĥn,n(x) is shown to converge to a nondegenerate random limit as given by Athreya (1987) when Tₙ has a stable limit of order α, 1 < α < 2.