Blackwell–Girshick equation? We have the following theorem:
Let $N$ be a random variable assuming positive integer values $1, 2, 3,\dots\,$. Let $(X_i)$ be a sequence of independent random variables which are also independent of $N$ with $V(X_i)$ the same for $i$, and $E(X_i) = E(X)$ the same for $i$. We denote the random walk as $S_N=\sum_{i=1}^{N}X_i$. Then
$$V(S_N)=E(N)V(X)+E(X)^2V(N)$$
Is there a variant for the case where $N$ is not independent of $X_i$ ? This is not clear for me, what are the examples for which we have $N$ dependent of $X_i$ ?
Thank you
 A: Think about it this way: if $N$ is independent of the $(X_i)$, then when you stop has nothing to do with the current value of the random walk. This is why Wald's equation makes sense, for instance.
You bring up the case of the random walk $(S_n)$ being bounded.  If $(S_n)$ is almost surely bounded, what must that tell us about $X$? I bet that $\mathbb{E}[X]=0$; so, in this case, we would get that $\mathbb{E}[S_N]=0$ and 
$$
\text{Var}[S_N]=\mathbb{E}[N]\cdot\text{Var}[X],
$$
which makes sense if you think about it: conditioned on $N$, we're adding up $N$ independent random variables with the same variance.
To answer your other question: the most basic form of Wald's equation does require that $N$ be independent of $(X_i)$; however, there is a more general form where that is not necessarily required.
A: Suppose $X$ and each $X_i$ has the same i.i. distribution of being $0$ or $1$ with equal probability, and that $N$ is the first $i$ for which $X_i=1$.
Then $S_N$ is almost surely $1$ with zero variance, even though all the terms on your right-hand side are positive.
