Solution

The conclusion printed as convergence in probability is already an assumption; the substantive conclusion is convergence in

L^{p}

, and we prove that stronger statement.

First, the almost-sure subsequence result from part (a) and Fatou's lemma show that

X \in L^{r}

E ∣ X ∣^{r} \leq lim inf_{k} E ∣ X_{n_{k}} ∣^{r} < \infty.

Consequently

sup_{n} E ∣ X_{n} - X ∣^{r} < \infty.

Fix

1 \leq p < r

and

δ > 0

. Splitting according to

∣ X_{n} - X ∣ \leq δ

and applying Hölder's inequality on the complement gives

E ∣ X_{n} - X ∣^{p} \leq δ^{p} + E [∣ X_{n} - X ∣^{p} 1_{{∣ X_{n} - X ∣ > δ}}] \leq δ^{p} + (E ∣ X_{n} - X ∣^{r})^{p / r} P (∣ X_{n} - X ∣ > δ)^{1 - p / r} .

The second term tends to zero because of convergence in probability and the uniform

L^{r}

bound. Taking the upper limit and then

δ ↓ 0

proves

∥ X_{n} - X ∥_{p} ⟶ 0 .

Solved by gpt-5.6-sol high.

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