Convergence in Probability Implies Convergence in Distribution

A sequence of random variables $\left< X_n \right>$ is convergent in probability to a random variable $X$ if for any $\epsilon > 0$

$\lim_{n\rightarrow \infty} P[ |X_n - X| > \epsilon] = 0$

It is convergent in distribution if for all points $x$ for which $F$ is continuous at $x$ ,

$\lim_{n \rightarrow \infty} F_n(x) = F(x)$

where $F_n$ is the distribution function of $X_n$ and $F$ is the distribution function of $X$

Convergence in probability implies convergence in distribution.

Since

$X \le x - \epsilon \leftrightarrow (X \le x - \epsilon \wedge X_n \le x) \vee (X \le x - \epsilon \wedge X_n > x) \\ \rightarrow X_n \le x \vee X_n - X > \epsilon$

$X_n \le x \leftrightarrow (X_n \le x \wedge X \le x + \epsilon) \vee (X_n \le x \wedge X > x + \epsilon) \\ \rightarrow X \le x + \epsilon \vee X - X_n > \epsilon$

$F(x - \epsilon) \le F_n (x) + P(|X_n - X| > \epsilon)$

$F_n(x) \le F(x + \epsilon) + P(|X_n - X| > \epsilon)$

$F(x - \epsilon) \le \liminf_n F_n(x)$

$\limsup_n F_n(x) \le F(x + \epsilon)$

Let $F$ be continuous at $x$ . Then $\lim_{\epsilon \downarrow 0} F(x \pm \epsilon) = F(x)$ . Since $\liminf_n F_n(x) \le \limsup_n F_n(x)$ it follows that

$F(x) \le \liminf_n F_n(x) \le \limsup_n F_n(x) \le F(x)$

and so $\lim_{n\rightarrow \infty} F_n(x) = F(x)$ . That is, the sequence of random variables is convergent in probability.

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