Multivariate Convergence in Probability

Definition

Let

• $\mathbf{X}$ : Random vector
• $\{{\mathbf{X}}_n\}$ : Sequence of $p$-dimensional vectors
• $\{X_n\}$ and $\mathbf{X}$ defined on the same sample space

If

$$\underset{n\to \infty }{\lim }P\left[||{\mathbf{X}}_n-\mathbf{X}||\ge ϵ\right]=0,\forall ϵ>0$$

Then

• We say $\{{\mathbf{X}}_n\}$ converges in probability to $\mathbf{X}$
• We write ${\mathbf{X}}_n\stackrel{P}{\to }\mathbf{X}$