Spatial homogeneity of simple random walk

Question

I'm reading the section on random walks (3.9) in "Probability and random process" by Grimmet and Stirzaker. It says that the simple random walk is spatially homogenous, that is
$$P(S_n=j|S_0=a)=P(S_n=j+b|S_0=a+b)$$
Where $S_n=S_0+\sum_1^n X_i$. As proof, it says that both sides of the equation are equal to
$P(\sum_1^n X_i=j-a)$, but it seems to me that
$$P(\sum_1^n X_i=j-a)=P(S_n=j\cap S_0=a) \quad \text{and}\quad P(S_n=j|S_0=a)=\frac{P(\sum_1^n X_i=j-a)}{P(S_0=a)}$$
What am I missing?

Answer 1

The event $\sum_1^n X_i=j-a$ is equivalent to the event $S_n - S_0 = j-a$

The latter includes $S_n=j\cap S_0=a$ but also $S_n=j+1\cap S_0=a+1$ , etc

Hence your assertion $P(\sum_1^n X_i=j-a)=P(S_n=j\cap S_0=a)$ is wrong.