> [!definition] > > Let $\seqi{X} \subset L^1$ be a family of $\real^n$-valued [[Random Variable|random variables]] and $\seqi{\mu}$ be their [[Probability Distribution|probability distributions]]. The family $\seqi{\mu}$ is **tight** if for all $\varepsilon > 0$, there exists a [[Compactness|compact]] set $K \subset \real^n$ such that > $ > \sup_{i \in I}\mu_i(\real^n \setminus K) < \varepsilon > $ > [!theorem] > > Let $\seq{\mu_i}$ be a tight family of [[Probability|probability measures]] on $\real$. Then there exists a subsequence $\seq{n_k}$ such that $\mu_{n_k}$ converges [[Convergence in Distribution|in distribution]]. > > *Proof*. Stochastically bounded + pull a [[Tychonoff's Theorem|Tychonoff]].