An notation question about $\sim$ in probability

Question

This seems to be a elementary problem but I don't find any authoritative answer. In probability $\sim$ is understood as 'distributed as'. My question is: can we use $\sim$ in conditional distribution? For example, for a common random walk $X_k$ on $\mathbb{Z}$, we usually write $P(X_{k+1}=X_k\pm1)=\frac12$, but can we write $$X_{k+1}|X_k\sim \mathcal{U}(\{X_k+1,X_k-1\})?$$

Answer 1

Yes you can write it that way and it is clear. You probably ought to include for what values of $k$ it holds as well as the starting value so that the recursion is well defined