 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/27l/a/solution

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

Put

S_{0} = 0, S_{n} = \sum_{i = 1}^{n} ξ_{i}, N_{t} = max {n \geq 0 : S_{n} \leq t} .

This is the renewal process generated by the interarrival times. Since the variables have a density and positive finite mean,

S_{n} \to \infty

N_{t}

is finite and tends to infinity with

t

. The strong law gives

\frac{S _{n}}{n} ⟶ \frac{1}{λ} almost surely.

S_{N_{t}} \leq t < S_{N_{t} + 1} .

Dividing by

N_{t}

and using

N_{t} \to \infty

gives

\frac{S _{N_{t}}}{N _{t}} \leq \frac{t}{N _{t}} < \frac{N _{t} + 1}{N _{t}} \frac{S _{N_{t} + 1}}{N _{t} + 1} .

Both outside expressions tend to

1/ λ

, so

\frac{N _{t}}{t} ⟶ λ almost surely.

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