 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-1/25j/d/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Split the overlapping products into two nonoverlapping families:

U_{n} = \sum_{j = 1}^{n} X_{2 j - 1} X_{2 j} + \sum_{j = 1}^{n} X_{2 j} X_{2 j + 1} .

Within each sum the summands depend on disjoint independent pairs and are independent identically distributed Bernoulli variables of mean

1/4

. The two families need not be independent of each other. The strong law of large numbers says that averages of independent identically distributed integrable random variables converge almost surely to their expectation. Applying it separately to each family and intersecting the two probability-one events gives

\frac{U _{n}}{n} ⟶ \frac{1}{4} + \frac{1}{4} = \frac{1}{2} almost surely .

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