 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/27h/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

X = lim inf_{n \to \infty} X_{n}

almost surely. Since the random variables are nonnegative, Fatou lemma applies and gives

E X \leq lim inf_{n \to \infty} E X_{n} \leq sup_{n \in N} E X_{n} .

The inequalities remain valid in the extended nonnegative reals, so no integrability assumption beyond nonnegativity is required.

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