Discussion
Let \(\{X_{n}, n \ge 1\}\) and X be random variables defined on the same probability space. Which of the following statements is not always true as \(n \to \infty\) ?
Answer: A
The statement not always true is the one claiming almost sure convergence of eXn from convergence in probability and distribution.
Convergence in probability (which already implies convergence in distribution) does not imply almost sure convergence. The 'typewriter' sequence of indicators of shrinking intervals sweeping across [0, 1] converges to 0 in probability but at no point almost surely; applying the continuous map \(e^{x}\) does not change that.
The other three are instances of the continuous mapping theorem: a continuous function preserves almost sure convergence (and hence convergence in probability), convergence in probability (hence in distribution), and convergence in distribution.
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