# Wald's second equation

We have a random walk $S_N=\sum_{i=1}^{N}X_i$ where $X_i$ are i.i.d with $0<E(X_i)<\infty$ and $N$ is a stopping time. What is the "exact" second equation of Wald ?

I've seen different results and sometimes they are contradictory.

Thank you

• Could you state the different results you saw? – Davide Giraudo Jul 12 '13 at 11:04
• I've seen $E(S_N-E(X_1)N)^2=V(X_1)E(N)$ in this lecture note link. I've also seen $E(S_N^2)=V(X_1)E(N)$ when $E(X_1)=0$, the place of the square is very strange in the first case. – Diguer Romain Jul 12 '13 at 12:22
• I see again $E(S_N−E(X_1)N)2=V(X_1)E(_N)$ with the external power 2 at this link. – Diguer Romain Jul 12 '13 at 12:35
• I notice that sometimes a same scribe uses indifferently the parentheses either for the mean $E(.)$ or to precise the association of variables (and by omiting the parentheses for $E$)... – Diguer Romain Jul 12 '13 at 12:41
• @DavideGiraudo What do you think about the place of this square ? Maybe it's a variance and the square has to be applied on $(S_n-E(X_1)N)$ but I'm not completely sure. – Diguer Romain Jul 12 '13 at 12:54

All the results mean $$\mathbb E\left[\left(S_N-\mathbb E(X_1)\cdot N\right)^2\right]=\operatorname{Var}(X_1)\cdot\mathbb E(N).$$