題組內容

3. Let \( X \) and \( Y \) be random variables and \( \{X_n, n \ge 1\} \) be a sequence of random variables. Define \[ \rho(X) = E\left( \frac{|X|}{1+|X|} \right). \] Show that

\( \lim_{n \to \infty} \rho(X_n - X) = 0 \) if and only if \( X_n \to X \) in probability. (15%)