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I saw it mentioned here that Wald's Equation can still be used when the number of random variables being summed is not independent of the random variables themselves.

Consider finding the expected value of the stopping time $N$ defined below, where $U_i$ are iid and uniform in (0, 1).

$$N = \min \Big\{ n: \sum_{i = 1}^{n} U_i > 1 \Big\}$$

As mentioned here the expected value is $e$. Let $S_n = \sum_{i = 1}^{n} U_i$. Since, $S_n \geq 1$ due to the stopping rule, would it be valid to use Wald's equation for a crude lower bound as follows

$$ \mathbb{E}S_n = \mathbb{E}U_i \mathbb{E}N \geq 1\\ \mathbb{E}N \geq \frac{1}{\mathbb{E}U_i} = 2$$

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There seems to be some notational confusion – if I understand correctly, where you write $S_n$, except for the first time, you mean $S_N$.

Interpreted in this way, your derivation is correct. Wikipedia states the independence assumption under which Wald's equation holds:

$$ E[X_n1_{N\ge n}]=E[X_n]P(N\ge n)\;. $$

This is fulfilled if $N$ is a stopping time for the sequence $X_n$.

Your result also follows intuitively if you imagine starting a new “pile” to add the $X_n$ to as soon as the previous pile is $\ge1$. Then you use all the $X_n$ in these piles, and at any point in the process the sum of all the piles is the sum of all the $X_n$ up to that point, which couldn't work if you'd have less than $2$ of them per pile on average.

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