Introduction

In analysis and probability theory, one studies various sort of convergences (of random variables) for various reasons. In this post we study vague convergence, which is responsible for the convergence in distribution.

Vaguely speaking, vague convergence is the weakest kind of convergence one can expect (whilst still caring about continuity whenever possible). We do not consider any dependence relation between the sequence of random variables.

Throughout, fix a probability space $(\Omega,\mathscr{F},\mathscr{P})$, where $\Omega$ is the sample space, $\mathscr{F}$ the event space and $\mathscr{P}$ the probability function. Let $(X_n)$ be a sequence of random variables on this space. Each random variable $X_n$ canonically induces a probability space $(\mathbb{R},\mathscr{B},\mu_n)$ where $\mathscr{B}$ is the Borel $\sigma$-measure. To avoid notation hell we only consider the correspondence $X_n \leftrightarrow \mu_n$ where

$$\mu_n(B)=\mathscr{P}(X^{-1}(B))=\mathscr{P}\{X \in B\}.$$

Here comes the question: if $X_n$ tends to a limit, then we would expect that $\mu_n$ converges to a limit (say $\mu$) in some sense (at least on some intervals). But is that always the case? Even if the sequence converges, can we even have $\mu(\mathbb{R})=1$? We will see through some examples that this is really not the case.

Examples: Failure of Convergence on Intervals

Let $X_n\equiv\frac{(-1)^n}{n}$, then $X_n \to...