Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$.
I would like to calculate the probability generating function of $S_n,G(T)=E(T^{S_n})$.
Here's my attempt:
$$G_{S_n}(t)=E(t^{S_n})=E\left(\prod^n_{i=1}t^{X_i}\right)=G_X(t)^n$$
Where independence of $X_i$ was used, and
$$G_X(t)=E(t^X)=1+\frac{1}{9}t^{-1}+\frac{1}{9}+\frac{7}{9}t^2$$
Would this be correct?
Best Answer
Assuming your $S_n = 1 + \sum$ for $n\ge1$. Then I believe $$G_{S_n}(t) = E(t^1) [G_X(t)]^n = t[G_X(t)]^n, G_X(t) = \frac{1}{9}t^{-1}+\frac{1}{9}+\frac{7}{9}t^2$$