Abstract
In this chapter, we consider some extensions of the Central Limit Theorem to classes of sums (or averages) of dependent random variables. Many further extensions are possible, but we focus on ones that will be useful in the sequel. Section 12.2 considers sampling without replacement from a finite population. As an application, the potential outcomes framework is introduced in order to study treatment effects. The class of U-statistics is studied in Section 12.3, with applications to the classical one-sample signed-rank statistic and the two-sample Wilcoxon rank-sum statistic.
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Notes
- 1.
Alternatively, the celebrated Kolmogorov Three-Series Theorem may be used to easily show that the series (12.51) converges with probability one; see Billingsley (1995), Theorem 22.8. In addition, if \(Var ( \epsilon _j ) < \infty \), we may write, \(X_j = \lim _{m \rightarrow \infty } X_{m,j}\), where \(X_{m,j} = \sum _{i=0}^{m-1} \rho ^i \epsilon _{j-i}\), and the limit can be interpreted in the mean-squared sense; see Problem 11.65.
- 2.
The arithmetic-geometric mean inequality says that, for \(y_i \ge 0\), \((y_1 + \cdots + y_k )/k \ge (y_1 \cdots y_k)^{1/k}\).
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Lehmann, E.L., Romano, J.P. (2022). Extensions of the CLT to Sums of Dependent Random Variables. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_12
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DOI: https://doi.org/10.1007/978-3-030-70578-7_12
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