 Solution

ID: past-exam-of-the-probability-measure-and-martingales-course-of-the-university-of-oxford/2023/2/b/i/solution

The floor function gives the deterministic bound

0 \leq X - X_{n} < 2^{- n}

for every outcome, including

X = 1

. Thus all three convergence assertions hold: there is pointwise, and hence almost sure convergence, as well as convergence in probability and convergence in L2. In detail, for

ε > 0

P (∣ X - X_{n} ∣ > ε)

is zero as soon as

2^{- n} \leq ε

; also

E [∣ X - X_{n} ∣^{2}] \leq 4^{- n} ⟶ 0.

The uniform distribution even gives the exact second moment of the error by integrating over the dyadic partition:

E [∣ X - X_{n} ∣^{2}] = 2^{n} \int_{0}^{2^{- n}} t^{2} d t = \frac{4 ^{- n}}{3} .

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