I begin with a simpler example than the one with which my question is concerned:
Let $X_1 , X_2, \ldots$ be a sequence of i.i.d. random variables with $\Bbb E [X_1^4] < \infty $ and $\mu := \Bbb E [X_1]$. With the strong law of large numbers it clear that, with an additional sequence $Z_1 , Z_2 , \ldots $ of i.i.d. random variables and $0\leq Z_1 \leq 1$ it holds $$\frac 1 n \sum_{i=1}^n Z_i X_i \to p \mu \tag{1}\label{eq1}$$ where $p = \Bbb E[Z_1]$. On the other hand, intuitively, one should get the same when doing the following: Let $M_n\subset \{1, \ldots, n\}$ be a random subset with $\vert M_n \vert = \lfloor pn\rfloor$ (We draw $\lfloor pn\rfloor$ times from the set $\{1, \ldots , n\}$ without replacement). Let $$\hat Z_i := 1_{\{i \in M_n\}} = \begin{cases} 1 : i\in M_n\\ 0 : i\notin M_n \end{cases}$$ Then one could guess that also $$\frac 1 n \sum_{i=1}^n \hat Z_i X_i \overset{?}\to p \mu \tag{2}\label{eq2}$$ To apply the strong law of large numbers here as in \eqref{eq1} one had to apply a statement for triangular arrays with weakly correlated random variables, because the random variables $Y_i = \hat Z_i X_i$ are not independent anymore and the distribution varies with $n$.
Question 1: Is there such a statement? For example, a version for triangular arrays of Strong laws of large numbers for weakly correlated random variables. (1988, Russell Lyons)
But in some sense for large $n$ we have that the distributions of $Z_i$ and $\hat Z_i$ are very similar and the correlation of two $\hat Z_i$ should vanish in the limit.
Question 2: Is it possible to derive \eqref{eq2} from \eqref{eq1} ?
The original problem I am concerned with is more of the following form: Let $G_1 , G_2 ,\ldots $ be independent random variables with identical distribution and $G_1 \in \Bbb N$ almost surely and $\Bbb E [G_1^k] < \infty$ for all $k\in\Bbb N$. Furthermore, for every $i\in\Bbb N$ let $X_1^i , X_2^i, \ldots$ a sequence of bounded random variables with identical distribution, independent from the other randomness. For every $n$, depending on $N_n := G_1 + \ldots + G_n$ let $M_n \subset \{ (i,j) : 1\leq i\leq n , 1\leq j \leq G_i\}$ with $\vert M_n \vert = \lfloor p N_n \rfloor$ be a randomly chosen set and $Z_j^i := 1_{\{ (i,j) \in M_n\}}$.
With the law of the large numbers and Wald's identity we then have
$$\frac 1 n \sum_{i=1}^n \sum_{j=1}^{G_i} X_j^i \to \Bbb E [G_1] \Bbb E [X_1^1]$$
Question 3: Is it also possible that $$\frac 1 n \sum_{i=1}^n \sum_{j=1}^{G_i} Z_j^i X_j^i \overset ? \longrightarrow p\Bbb E [G_1] \Bbb E [X_1^1]$$