Question

My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in: $$\lim_{n\to\inf}||P(S_n\in\cdot)-P(S_n+k\in\cdot)||_{tv}=0$$ This is from coupling lectures in probability theory.

Answer 1

If $S_n$ is a real-valued random variable, then I would read $P(S_n \in \cdot)$ as denoting the probability measure $P \circ S_n^{-1}$ on $\mathbb{R}$, i.e. the set function $B \mapsto P(S_n \in B)$ where $B \in \mathcal{B}_{\mathbb{R}}$, the Borel $\sigma$-algebra on $\mathbb{R}$. The quantity in norm bars is then the total variation distance between these two measures, which means the given expression can be rewritten $$ \lim_{n \to \infty} \sup_{B \in \mathcal{B}_\mathbb{R}} |P(S_n \in B) - P(S_n + k \in B)| $$

Answer 2

I think the answer is that the $\cdot$ induces an anonymous function, e.g. the above is equal to:

$$\lim_{n\to\inf}||f||_{tv}=0$$ with f a function on the appropriate domain having $$f(x) := P(S_n=x)-P(S_n+k=x)$$