Multivariate Random Variables Probability Convergence 📂Mathematical Statistics

Multivariate Random Variables Probability Convergence

Definition ¹

$p$-dimensional random vector $\mathbf{X}$ and a sequence of random vectors $\left\{ \mathbf{X}_{n} \right\}$ are said to converge in probability to $n \to \infty$ as $\mathbf{X}_{n}$ if they satisfy the following. It is denoted by $\mathbf{X} _ {n} \overset{P}{\to} \mathbf{X}$. $$ \forall \varepsilon > 0 , \lim_{n \to \infty} P \left[ \left\| \mathbf{X}_{n} - \mathbf{X} \right\| < \varepsilon \right] = 1 $$

$\| \cdot \|$ is defined as the Euclidean norm, defined by $\left\| \left( x_{1} , \cdots , x_{n} \right) \right\| = \sqrt{ x_{1}^{2} + \cdots + x_{n}^{2}}$.

Theorem

Let us represent the $p$-dimensional random vector as $\mathbf{X} = \left( X_{1} , \cdots , X_{p} \right)$. Then $$ \mathbf{X}_{n} \overset{P}{\to} \mathbf{X} \iff X_{nk} \overset{P}{\to} X_{k} \qquad, \forall k = 1, \cdots, p $$

Proof

$(\Rightarrow)$

Let $\mathbf{X}_{n} \overset{P}{\to} \mathbf{X}$. According to the definition of the Euclidean norm for $\varepsilon > 0$, $$ \varepsilon \le \left| X_{nk} - X_{k} \right| \le \left\| \mathbf{X}_{nk} - \mathbf{X}_{k} \right\| $$ therefore, $$ \limsup_{n \to \infty} P \left[ \left| X_{nk} - X_{k} \right| \ge \varepsilon \right] \le \limsup_{n \to \infty} P \left[ \left\| \mathbf{X}_{nk} - \mathbf{X}_{k} \right\| \ge \varepsilon \right] = 0 $$

$(\Leftarrow)$

Let $X_{nk} \overset{P}{\to} X_{k} , \forall k = 1, \cdots, p$. According to the definition of the Euclidean norm for $\varepsilon > 0$, $$ \varepsilon \le \left\| \mathbf{X}_{n} - \mathbf{X} \right\| \le \sum_{k=1}^{p} \left| X_{nk} - X_{k} \right| $$ therefore, $$ \begin{align*} & \limsup_{n \to \infty} P \left[ \left\| \mathbf{X}_{n} - \mathbf{X} \right\| \ge \varepsilon \right] \\ \le & \limsup_{n \to \infty} P \left[ \left| X_{nk} - X_{k} \right| \ge \varepsilon \right] \\ \le & \sum_{k=1}^{p} \limsup_{n \to \infty} P \left[ \left| X_{nk} - X_{k} \right| \ge \varepsilon \right] \\ =& 0 \end{align*} $$

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