Probabilities not bounded away from zero

We have a population or cohort of $N$ people divided into $H$ sampling strata, with a sample of size $n_h$ taken from the population $N_h$ in stratum $h$. Let ${\pi }_{ij}$ be the sampling probability for person $i$ in stratum $h$. When we do asymptotics we usually assume ${\pi }_{ih}$ are bounded away from zero. That’s not ideal for, say, case-control studies of rare diseases, where we might want asymptotic approximations based on the case incidence being small (ie, converging to zero). 

In the situations where I’m interested in ${\pi }_{ih}$ being small, it’s usually small for a whole stratum. Since sampling is independent between strata, there should be a central limit theorem separately for each stratum, and we should be able to add up the limiting Normal approximations for the stratum totals to get a Normal limit for the population total estimate and the population mean estimate. 

To formalise this,  suppose $n_h\to \infty$ for every stratum (so that asymptotics makes sense), and that ${\pi }_{ih}{N}_h/n_h$ is bounded above and below, so that within each stratum the sampling probability has a finite (relative) range. As a simple example, we might have a case stratum with ${\pi }_i\approx 1$ and a control stratum with very small ${\pi }_i$. 

[Update: As Stas Kolenikov points out, I’m assuming the same strata are small large along the infinite sequence, so I need something like $n_{h_1}/(n_{h_1}+n_{h_2})\to c_{h_1,h_2}\in [0,1]$ for each pair of strata.  This isn’t a meaningful loss of generality since (a) the infinite sequence is an analytic fiction and we might as well set it up for our maximum convenience; and (b) even without assuming anything, every subsequence will have a subsubsequence along which the condition holds]

By standard results, $n_h^{1/2}({\overline{X}}_{.h}-{\mu }_h)\stackrel{d}{\to }N(0,{\sigma }_h^2)$ for each stratum $h$ , and by the Skorohod representation theorem we can find an $H$-variate normal vector $⟨Z_h{⟩}_{h=1}^H$ with
$$n_h^{1/2}({\overline{X}}_{.h}-{\mu }_h)\stackrel{p}{\to }{Z}_h$$
(possibly on a different probability space), to get
$${\overline{X}}_{.h}={\mu }_h+n_h^{-1/2}{Z}_h+o_p\left(n_h^{-1/2}\right)$$
The $Z_h$ will be independent, with mean zero; write ${\sigma }_h^2$ for the variances. 

[Update: Note that ${\sigma }_h^2$ is just $var\left[Z_h\right]$, nothing more fundamental. Under stratified random sampling, ${\sigma }_h^2$ will be $var\left[X\right]$ in stratum $h$ multiplied by the ‘finite population correction” $(N_h-n_h)/N_h$, but under other sampling schemes it will be something else]

Now,
$${\overline{X}}_{..}=\frac{1}{N}\sum _{h=1}^H{N}_h{\overline{X}}_{.h}$$
giving
$$\begin{array}{cc}{\overline{X}}_{..}& =\sum _{h=1}^H\frac{N_h}{N}{\mu }_h+\frac{N_h{n}_h^{-1/2}}{N}{Z}_h+o_p\left(\frac{N_h{n}_h^{-1/2}}{N}\right)\\ & =\mu +\left(\sum _{h=1}^H\frac{N_h{n}_h^{-1/2}}{N}{Z}_h\right)+o_p\left(\sum _{h=1}^H\frac{N_h}{N\sqrt{n_h}}\right)\end{array}$$

First, suppose $ N_h/N$ converges to a non-zero constant for each $h$. Let $n_{\ast }={min}_h{n}_h$ and define $H=\{h:lim{n}_{\ast }/n_h>0\}$
$$\begin{array}{ccc}{\overline{X}}_{..}& =& \mu +\left(\sum _{h=1}^H\frac{N_h{n}_h^{-1/2}}{N}{Z}_h\right)+o_p\left(\frac{{max}_h{N}_h}{N{min}_h\sqrt{n_h}}\right)\\ & =& \mu +\left(\sum _{h\in H}\frac{N_h{n}_{\ast }^{-1/2}}{N}{Z}_h\right)+\sum _{h\notin H}{o}_p\left(n_{\ast }^{-1/2}\right)+o_p\left(\frac{{max}_h{N}_h}{N\sqrt{n_{\ast }}}\right)\\ & =& \mu +n_{\ast }^{-1/2}Z+o_p\left(n_{\ast }^{-1/2}\right)\end{array}$$

where $Z\sim N(0,{\sigma }^2)$ with
$${\sigma }^2=\underset{n_{\ast }\to \infty }{lim}\sum _{h\in H}\frac{N_h^2{n}_{\ast }{\sigma }_h^2}{N^2{n}_h}$$

Alternatively, for case–control sampling we may have $N_h/N\to 0$ in the case stratum, but we would have $n_h$ all of the same order, and so of the same order as their total, $n$. The limiting distribution is dominated by the largest strata: define $H^{\prime }=\{h:lim{N}_h/N>0\}$ (which is non-empty as $H$ is finite)

$$\begin{array}{ccc}{\overline{X}}_{..}& =& \mu +\left(\sum _{h=1}^H\frac{N_h{n}_h^{-1/2}}{N}{Z}_h\right)+o_p\left(\sum _{h=1}^H\frac{N_h}{N\sqrt{n_h}}\right)\\ & =& \mu +\left(\sum _{h\in H^{\prime }}\frac{N_h{n}^{-1/2}}{N}{Z}_h\right)+\sum _{h\notin H^{\prime }}{o}_p\left(n^{-1/2}\right)+o_p\left(n^{-1/2}\right)\\ & =& \mu +n^{-1/2}Z+o_p\left(n^{-1/2}\right)\end{array}$$
where $Z\sim N(0,{\sigma }^2)$ with
$${\sigma }^2=\underset{n\to \infty }{lim}\sum _{h\in H}\frac{N_h^{2}n{\sigma }_h^2}{N^2{n}_h}$$

Weaker conditions on $N_h$ and $n_h$ are clearly possible: it is only necessary to identify which terms dominate the limiting distribution of ${\overline{X}}_{..}$, since the limiting distribution of estimated stratum totals is always independent $H$-variate Normal under appropriate scaling.