 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009/a/1/b/ii/solution

n_{k} > n_{k - 1}

so that

P (∣ X_{n_{k}} - X ∣ > 2^{- k}) < 2^{- k} .

The corresponding events have a summable sequence of probabilities. The First Borel-Cantelli lemma, which requires no independence, therefore makes the probability of infinitely many of them zero. On the complementary event,

∣ X_{n_{k}} - X ∣ \leq 2^{- k}

for all sufficiently large

k

, proving

X_{n_{k}} ⟶ X almost surely .

The recursively chosen indices are strictly increasing and hence tend to infinity.

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