Multivariate Convergence in Distribution

Definition

Let

• $\{{\mathbf{X}}_n\}$ : Sequence of $p$-dimensional vectors, with
  • $F_n(\mathbf{x})$ : cdf of ${\mathbf{X}}_n$
• $\mathbf{X}$ : Random vector, with
  • $F(\mathbf{x})$ : cdf of $\mathbf{X}$
• $C(F)$ denote the set of all points where $F$ is continuous

If

$$\underset{n\to \infty }{\lim }{F}_n(\mathbf{x})=F(\mathbf{x}),\forall \mathbf{x}\in C(F)$$

Then

• We say $\{{\mathbf{X}}_n\}$ converges in distribution to $\mathbf{X}$
• We write ${\mathbf{X}}_n\stackrel{D}{\to }\mathbf{X}$