# Does convergence in probability w.r.t. a topology make sense?

Let $(S,d)$ be a separable metric space. A sequence of random variables $X_n$ is said to converge in probability to $a \in S$ if and only if for all $\varepsilon > 0$ $$P(d(X_n,a) > \varepsilon) \to 0,$$ as $n \to \infty$.

Would this make sense in a topological space? Let $S$ be a topological space instead and, instead of the condition above. require that for any open $U \ni a$ $$P(X_n \in U) \to 1.$$

Does this type of concept exist? Would it make any sense? What I have in mind is a probabilistic version of convergence of operators on Hilbert space w.r.t. the weak operator topology.

## 1 Answer

I'm not sure I entirely understand all the parts of this question, but I've been interested in what's in the literature on convergence of random variables in topological spaces.

Your proposed definition of convergence in probability that $\text{plim}_{n\to\infty}X_n=x$ if $P(X_n\in U)\rightarrow 1$ for all open neighbourhoods, $U$ of $x$, makes sense, and can be found in the literature. Two examples I've seen recently are Liese and Vajda (1995), or in Heijmans and Magnus (1986). In both cases the definition is given in the context where the random variable is assumed to take values in a metric space, but that assumption is not needed for the definition to make sense.

That said I haven't found anyone explicitly defining convergence in probability for random variables taking values in more generic topological spaces and I think the reason is that convergence in probability is usually defined as convergence to a random variable rather than a point and that can't be so easily generalised to topological spaces.

The most general definition I can find is these lectore notes by Pierre-Loïc Méliot, which gives the standard definition of convergence in probability for metric spaces but applies it to all metrizable topological spaces since which sequences converge on the definition is independent of metric consistent with a given topological space. The reason for the independence is because of the well-known result that convergence in probability is equivalent to every subsequence containing an almost surely convergent subsequence, and almost sure convergence of random variables does depend only on the topology. This suggests taking the characterisation in terms of almost sure convergence as the definition of convergence in probability so it can apply to arbitrary topological spaces, but I have been unable to find anyone doing it.