# Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has.

Consider the following infinite triangular matrix: \begin{align} \begin{bmatrix} v_{11} \\ v_{21} & v_{22} \\ . & . & . \\ . & . & . & . \\ v_{N1} & v_{N2} & . & . & v_{NN} \\ . & . & . & . & . & . \end{bmatrix}, \end{align} whose entries are all real elements. Each row in the infinite triangular matrix is a finite population, which we can denote by $$\Pi_1, \Pi_2, \ldots$$, where $$\Pi_1$$ consists of a single element with the value $$v_{11}$$, $$\Pi_2$$ consists of two elements with the values $$v_{21}, v_{22}$$ and so on and so forth. The population $$\Pi_N$$ refers to row $$N = 1, 2, \ldots$$ of the infinite triangular matrix that consists of elements with values $$v_{N1}, v_{N2}, \ldots, v_{NN}$$.

Focusing on the sample sum (call it $$S_N$$), Theorem 6 of Lehmann (1975, Appendix 4) shows that the standardized sample sum converges in distribution to $$\mathcal{N}(0, 1)$$ as $$N \to \infty$$ when the following condition holds: $$\begin{eqnarray} \label{eq: regularity conds} n, N - n \to \infty \text{ and } \dfrac{\max \limits_{1 \leq i \leq N} \left(v_{Ni} - \bar{v}_N\right)^2}{\sum \limits_{i = 1}^N\left(v_{Ni} - \bar{v}_N\right)^2} \max\left(\dfrac{N - n}{n}, \dfrac{n}{N - n}\right) & \to & 0 \text{ as } N \to \infty \end{eqnarray}$$ or, equivalently, $$\dfrac{n}{N} \to p \in \left(0, 1\right)$$ and $$\dfrac{\max \limits_{1 \leq i \leq N} \left(v_{Ni} - \bar{v}_N\right)^2}{\sum \limits_{i = 1}^N\left(v_{Ni} - \bar{v}_N\right)^2}$$ is bounded as $$N \to \infty$$, where $$n$$ is the sample size and $$\bar{v}_N$$ is the population mean.

However, in David Freedman's 2008 article in The Annals of Applied Statistics, he shows that $$\sqrt{N}(S_N - \mathbb{E}[S_N])$$ converges in distribution to $$\mathcal{N}(0, \nu)$$, where $$\mathbb{E}[\cdot]$$ is the expected value operator, $$\nu$$ is $$\lim \limits_{N \to \infty} \mathbb{V}[\sqrt{N}(S_N - \mathbb{E}[S_N])]$$ and $$\mathbb{V}[\cdot]$$ is the variance operator, when the following three conditions hold:

(1) For all $$N = 1, 2, \ldots$$, $$\mathbf{v}_N$$ has a bounded fourth central moment, i.e., $$\dfrac{1}{N} \sum \limits_{i = 1}^N \left\lvert v_{Ni} \right\rvert^4 < L < \infty$$.

(2) As $$N \to \infty$$, $$\dfrac{n}{N} \to p \in \left(0, 1\right)$$.

(3) The population quantities $$\dfrac{1}{N} \sum \limits_{i = 1}^N v_{Ni}$$ and $$\dfrac{1}{N} \sum \limits_{i = 1}^N v_{Ni}^2$$ tend to finite limits as $$N \to \infty$$.

(Technically Freedman proves this result for the difference-in-means estimator in the context of finite population causal inference with two finite populations of fixed potential outcomes, but I am just drawing on the fact that the difference-in-means estimator can be represented as a sample sum via scale and shift factors that do not depend on treatment assignment.)

I think I see why Freedman needs condition (3). Is it to ensure that the variance of the sample sum converges to a positive, finite value? Presumably Lehmann doesn't need this assumption because standardizing the sample sum ensures that the variance is 1 for all $$N = 1, 2, \ldots$$. I can also see that Freedman's condition (2) is equivalent to Lehmann's assumption that $$\dfrac{n}{N} \to p \in \left(0, 1\right)$$.

However, I am having trouble understanding the role of Freedman's assumption (1) and whether it implies Lehmann's assumption that $$\dfrac{\max \limits_{1 \leq i \leq N} \left(v_{Ni} - \bar{v}_N\right)^2}{\sum \limits_{i = 1}^N\left(v_{Ni} - \bar{v}_N\right)^2}$$ is bounded as $$N \to \infty$$.

Does Freedman's condition (1) of a bounded fourth central moment imply Lehmann's condition that $$\dfrac{\max \limits_{1 \leq i \leq N} \left(v_{Ni} - \bar{v}_N\right)^2}{\sum \limits_{i = 1}^N\left(v_{Ni} - \bar{v}_N\right)^2}$$ is bounded as $$N \to \infty$$? And if so, does anyone have a proof of this? Or am I missing something altogether about the relationship between the regularity conditions in these two version of the finite population CLT?