Convergence in Distribution Implies Probability Bound
Theorem
A sequence of random variables $\left\{ X_{n} \right\}$ is probabilistically bounded if it converges in distribution.
- $\overset{D}{\to}$ means convergence in distribution.
Explanation
Since we have shown that convergence almost surely implies convergence in distribution, by considering the contrapositive proposition, we can also obtain the common-sense corollary that ‘if it is not probabilistically bounded, it does not converge almost surely’.
Proof
Given $\epsilon>0$ and assuming that $X_{n}$ converges in distribution to random variable $X$ with the cumulative distribution function being $F_{X}$. Then, we can find $\eta_{1}, \eta_{2}$ satisfying $\displaystyle F_{X}(x) > 1- {\epsilon \over 2}$ from $\displaystyle x \ge \eta_{2}$ and $\displaystyle F_{X}(x) < {\epsilon \over 2}$ from $\displaystyle x \le \eta_{1}$. Now, if we set $$ \begin{align*} P[|X|\le \eta] =& F_X (\eta) - F_X (-\eta) \\ \ge& \left( 1 - {\epsilon \over 2} \right) - {\epsilon \over 2} \\ =& 1- \epsilon \end{align*} $$ Considering $X_{n}$ which converges in distribution to $X$, and taking both sides with $\displaystyle \lim_{n \to \infty}$ (that is, continuously choosing sufficiently large $N_{\epsilon}$), from the assumption of convergence in distribution, we get $$ \begin{align*} \lim_{n \to \infty} P[|X_{n}|\le \eta] =& \lim_{n \to \infty} F_{X_{n}} (\eta) - \lim_{n \to \infty} F_{X_{n}} (-\eta) \\ =& F_X (\eta) - F_X (-\eta) \\ \ge& 1 - \epsilon \end{align*} $$ By the definition of probabilistic boundedness, $\left\{ X_{n} \right\}$ is probabilistically bounded.
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