Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 3 / c

Table of contents
- Solution c

Solution

For

K > 0

, the Cauchy-Schwarz inequality and part a give

E [Y_{n}^{-} 1_{{Y_{n}^{-} > K}}] \leq E [(Y_{n}^{-})^{2}] P (Y_{n}^{-} > K) \leq \frac{1}{K} .

Thus

(Y_{n}^{-})

is uniformly integrable. Combining this with the weak convergence of random variables from part b yields convergence of the first moments:

E [Y_{n}^{-}] ⟶ E [Y^{-}] .

By symmetry of the standard normal density,

E [Y^{-}] = \int_{0}^{\infty} x \frac{e ^{- x^{2} /2}}{2 π} d x = \frac{1}{2 π} .

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