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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-209/2/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 209 / 2 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

The empirical characteristic function replaces the expected value in a characteristic function by the sample average:

φ_{n} (u) = \frac{1}{n} j = 1 \sum n e^{i u X_{j}}, u \in R .

Its centered fluctuation at the central limit theorem scale is the empirical characteristic process:

C_{n} (u) = n (φ_{n} (u) - φ (u)), φ (u) = E e^{i u X_{1}} .

In particular,

φ_{n} (0) = 1

and

C_{n} (0) = 0

.

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