Consider a symmetric random walk of IID $X_i$ where $X_i = \pm 1$ and $Pr(\pm 1) = 0.5$. Define $S_n = X_1 + \ldots + X_n$, therefore $E[S_n] = 0$.
In the notes that I am looking at, it shows the following:
The way $S_{n+1}$ seems to be weird, and I am not sure if it is a typo. Shouldn't the 2 cases for $S_{n+1}$ be lower case $s_n \pm 1$ instead of upper case $S_n \pm 1$. Upper case is a random variable and lower case is a specific realization, I believe.
If this is correct, then should't the expectation $E[S_{n+1}^2 - (n+1)]$ be equal to $s_n^2 - n$ instead of $S_n^2 - n$?
The only typo here is that $E[S2n+1−(n+1)]$ should instead be $E[S2n+1−(n+1)\mid S_1,\ldots, S_n]$. Remember that the conditional expectation $E[X\mid Y]$ is a random variable that is a function of $Y$.