Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 3 / c

Table of contents
- Solution c

Solution

Two applications of the Cauchy-Schwarz inequality give

\frac{1}{n} \sum_{i} F_{i} G_{i} ε_{i} ξ_{i} \leq (\frac{1}{n} \sum_{i} F_{i}^{2} G_{i}^{2})^{1/2} (\frac{1}{n} \sum_{i} ε_{i}^{2} ξ_{i}^{2})^{1/2} p 0

and

\frac{1}{n} \sum_{i} ∣ ε_{i} F_{i} ∣ G_{i}^{2} \leq (\frac{1}{n} \sum_{i} F_{i}^{2} G_{i}^{2})^{1/2} (\frac{1}{n} \sum_{i} ε_{i}^{2} G_{i}^{2})^{1/2} p 0.

Here the empirical residual second moment is again

O_{p} (1)

by the weak law of large numbers, while the assumed product error and the conclusion of part b are

o_{p} (1)

.

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