Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 26J / d

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- Solution d

Solution

m_{N} = E Y_{N} = \frac{1}{N} \sum_{k = 1}^{N} E X_{k} .

By variance additivity for independent random variables,

∥ Y_{N} - m_{N} ∥_{2}^{2} = Var (Y_{N}) = \frac{1}{N ^{2}} \sum_{k = 1}^{N} Var (X_{k}) \leq \frac{1}{N} ⟶ 0.

Thus

Y_{N}

differs in

L^{2}

by

o (1)

from the deterministic scalar

m_{N}

. The L2 convergence of averages of independent variables with bounded second moments gives the necessary and sufficient condition

Y_{N} converges in L^{2} ⟺ \frac{1}{N} k = 1 \sum N E X_{k} converges .

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