Discussion
Suppose \(\{X_{n}, n \ge 1\}\) is a sequence of random variables such that :
\[ \begin{aligned} &P[X_{n} = 0] = 1 - \frac{1}{n^{3}} \text{ and} \\ &P[X_{n} = n] = \frac{1}{n^{3}},\ n \ge 1 \end{aligned} \]
Which of the following is not true ?
Answer: C
The statement that is not true is Xn does not converge to 0 almost surely.
Since \(\sum P(X_{n} \neq 0) = \sum \frac{1}{n^{3}} \lt \infty\), the Borel-Cantelli lemma says that with probability 1 only finitely many \(X_{n}\) are non-zero. So \(X_{n} \to 0\) almost surely, and the claim that it does not is false.
The other modes hold. In quadratic mean:
\[ \begin{aligned} E(X_{n}^{2}) &= n^{2} \cdot \frac{1}{n^{3}} \\ &= \frac{1}{n} \\ &\to 0 \end{aligned} \]
and convergence in quadratic mean (or almost surely) implies convergence in probability and in law.
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