Prove that $Z_n$ is independent of $(Y_{i,n})_i$

Suppose we have $$Y_{i,n}, i \ge 1, n \ge 1$$ iid with expectation $$\mu$$. And given $$Z_{n+1} = \sum_{i=1} ^{Z_n} Y_{i,n}$$ and $$Z_0 = 1$$.

In lecture it was stated that $$Y_{i,n}$$ is independent of $$Z_n$$. I do not see how one would prove that. In particular we therefore used Wald's equation to obtain:

$$E[Z_{n+1}] = E[Z_n] E[Y_{1,n}]$$.

I would be glad if someone could explain to me why $$Z_n$$ is independent of $$Y_{i,n}$$ or if there is a different argument justifying $$E[Z_{n+1}] = E[Z_n] E[Y_{1,n}]$$.

Edit: The context was that $$Z_n$$ could be interpreted as the number of individuals in a generation and $$Y_{i,n}$$ as the number of children that individual $$i$$ in generation $$n$$ has.

• Can you see that $Z_n$ is a function of $Y_{i,j}$ for $j < n$? (or more precisely, $Z_n$ is $\sigma(Y_{i,j} : i \geq 1, 1\leq j \leq n-1)$-measurable) Commented May 3 at 6:50
• Yes, I see that Commented May 3 at 6:52
• And can you see that $(Y_{i,n})_i$ is independent of $\sigma(Y_{i,j} : i \geq 1, 1 \leq j \leq n-1)$? Commented May 3 at 6:54
• Is it independent since all $Y_{i,n}$ are independent? Commented May 3 at 6:57
• I need to be a little careful with the subscripts, but essentially yes, since the entire 2d array $Y_{i,j}$ for $i \geq 1, j \geq 1$ are iid, the sigma algebras generated by any two disjoint subsets of this array will be independent. Commented May 3 at 7:01

First, note that $$Z_n$$ is $$\sigma(Z_{n-1}, Y_{1,n-1}, Y_{2,n-1}, \ldots)$$-measurable.
By induction on $$n$$, this shows that $$Z_n$$ is $$\sigma(Y_{i,j} : i\geq 1, 1\leq j < n)$$-measurable.
It now suffices to show that $$\sigma(Y_{i,n} : i\geq 1)$$ is independent of $$\sigma(Y_{i,j} : i\geq 1, 1\leq j < n)$$. However, this follows directly from the assumption that the 2D-array $$(Y_{i,j})_{i\geq 1, \, j\geq 1}$$ is an array of independent random variables, since the second subscript of the random variables generating $$\sigma(Y_{i,n} : i\geq 1)$$ is always $$n$$, which is larger than any of the second subscripts of the random variables generating $$\sigma(Y_{i,j} : i\geq 1, 1\leq j < n)$$.