 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/26g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

lim_{K \to \infty} sup_{n} E [∣ X_{n} ∣ 1_{{∣ X_{n} ∣ > K}}] = 0.

sup_{n} E ∣ X_{n} ∣^{p} = C < \infty

for

p > 1

, then

E [∣ X_{n} ∣ 1_{∣ X_{n} ∣ > K}] \leq K^{1 - p} E ∣ X_{n} ∣^{p} \leq C K^{1 - p},

X_{n} = n

1/ n

and zero otherwise. Then

E ∣ X_{n} ∣ = 1

, but for every

K

and

n > K

the tail expectation is one.

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