Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 26K / b / iii

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- Solution iii

Solution

A finite linear combination of independent Gaussian random variables is Gaussian, so

\sum_{k = 1}^{n} a_{k} Y_{k} \sim N (0, v_{n}), v_{n} = \sum_{k = 1}^{n} a_{k}^{2} .

Since

v_{n} \to v := \sum_{k \geq 1} a_{k}^{2} < \infty

, the characteristic functions converge pointwise:

exp (- \frac{1}{2} v_{n} t^{2}) ⟶ exp (- \frac{1}{2} v t^{2}) .

The Lévy continuity theorem yields convergence in distribution to

N (0, k \geq 1 \sum a_{k}^{2}) .

On the given common probability space, the same variance calculation actually proves convergence in

L^{2}

as well.

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