Discussion
Suppose \(\{X_{n}\}\) and \(\{Y_{n}\}\) are sequences of random variables and X and Y are random variables such that as \(n \to \infty\), \(X_{n}\) converges in quadratic mean to X and \(Y_{n}\) converges in quadratic mean to Y.
Consider the following statements :
(I) The sequence \((X_{n} + Y_{n})\) converges in quadratic mean to (X + Y) as \(n \to \infty\)
(II) \(\lim_{n \to \infty} P[|X_{n}Y_{n} - XY| \gt \varepsilon] = 0\), \(\forall \varepsilon \gt 0\)
(III) \(\lim_{n \to \infty} E[e^{iX_{n}}] = E[e^{iX}]\)
Which of the above are true ?
Answer: A
All three statements are true.
(I) By Minkowski's inequality, \(\|(X_{n} + Y_{n}) - (X + Y)\|_{2}\) is at most \(\|X_{n} - X\|_{2} + \|Y_{n} - Y\|_{2} \to 0\), so the sum converges in quadratic mean.
(II) Convergence in quadratic mean implies convergence in probability, and products of sequences converging in probability converge in probability to the product (continuous mapping), so \(X_{n}Y_{n} \to XY\) in probability.
(III) Convergence in probability implies convergence in distribution, and \(e^{ix}\) is bounded and continuous, so \(E[e^{iX_{n}}] \to E[e^{iX}]\), the characteristic functions at \(t = 1\).
Each partial option leaves out a true statement.
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