Discussion
Suppose \(\{X_{n}\}\) is a sequence of random variables and X is a random variable such that \(P\left[\lim_{n\to\infty} X_{n} = X\right] = 1\).
Then which of the following may not hold ?
Answer: C
The one that may not hold is \(\lim E[|X_{n} - X|^{2}] = 0\).
Almost sure convergence implies convergence in probability (A) and in distribution (D), and a continuous function such as exp preserves almost sure convergence (B). It does not imply convergence in mean square.
Example: on (0, 1) with Lebesgue measure, \(X_{n} = n\,1_{(0, 1/n)}\) tends to \(X = 0\) at every point, but \(E[X_{n}^{2}] = n^{2} \cdot \tfrac{1}{n} = n \to \infty\). Without a uniform integrability condition, the mean square need not converge.
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