Low-Order Inclusion Probabilities Do Not Determine Asymptotic Normality
Abstract
Central limit theorems in survey sampling are typically established under specific sampling designs rather than from generic characteristics of a design. Motivated by the search for more general conditions, we investigate whether low-order inclusion probabilities can provide such a basis. First- and second-order inclusion probabilities determine the first two moments of sampling estimators such as the Horvitz--Thompson estimator, but we show that even exact agreement of these probabilities with those of simple random sampling without replacement (SRSWOR) does not preserve asymptotic normality. We construct a sequence of fixed-size sampling designs having exactly the same first- and second-order inclusion probabilities as SRSWOR, together with a sequence of finite populations for which the sample total converges to N(0,1) under SRSWOR but to a centered standard exponential distribution under the alternative designs. Thus, the two sample totals have exactly the same mean and variance at every stage of the sequence, yet have different limiting distributions. The alternative designs are obtained from balanced incomplete block designs arising from finite projective planes. Their cyclic structure allows us to prescribe the distribution of the sample total under block sampling, while a probabilistic permutation argument, combined with an exponential inequality for sampling without replacement, yields finite populations satisfying the SRSWOR central limit theorem. The result shows that low-order inclusion probabilities do not control the global dependence structure relevant to asymptotic normality and therefore cannot, by themselves, provide general conditions for central limit theorems under complex sampling designs.
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BibTeX
@misc{chen2026loworder,
title = {{Low-Order Inclusion Probabilities Do Not Determine Asymptotic Normality}},
author = {Jiahua Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09168},
eprint = {2610.09168},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09168},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Low-Order Inclusion Probabilities Do Not Determine Asymptotic Normality
AU - Jiahua Chen
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09168
UR - https://arxiv.org/abs/2610.09168
AB - Central limit theorems in survey sampling are typically established under specific sampling designs rather than from generic characteristics of a design. Motivated by the search for more general conditions, we investigate whether low-order inclusion probabilities can provide such a basis. First- and second-order inclusion probabilities determine the first two moments of sampling estimators such as the Horvitz--Thompson estimator, but we show that even exact agreement of these probabilities with those of simple random sampling without replacement (SRSWOR) does not preserve asymptotic normality. We construct a sequence of fixed-size sampling designs having exactly the same first- and second-order inclusion probabilities as SRSWOR, together with a sequence of finite populations for which the sample total converges to N(0,1) under SRSWOR but to a centered standard exponential distribution under the alternative designs. Thus, the two sample totals have exactly the same mean and variance at every stage of the sequence, yet have different limiting distributions. The alternative designs are obtained from balanced incomplete block designs arising from finite projective planes. Their cyclic structure allows us to prescribe the distribution of the sample total under block sampling, while a probabilistic permutation argument, combined with an exponential inequality for sampling without replacement, yields finite populations satisfying the SRSWOR central limit theorem. The result shows that low-order inclusion probabilities do not control the global dependence structure relevant to asymptotic normality and therefore cannot, by themselves, provide general conditions for central limit theorems under complex sampling designs.
ER -