Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 27H / c

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- Solution c

Solution

Yes. Almost-sure convergence gives

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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