Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 28K / b

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- Solution b

Solution

Conditionally on

Y_{1}, \dots, Y_{m}

, the cumulative distribution function of

X^{*}

is the weighted empirical distribution

F_{m}^{*} (t) = P (X^{*} \leq t ∣ Y_{1}, \dots, Y_{m}) = \frac{m ^{- 1} \sum _{i = 1}^{m} w ( Y _{i} ) 1 _{{Y_{i} \leq t}}}{m ^{- 1} \sum _{i = 1}^{m} w ( Y _{i} )}, w (y) = \frac{f ( y )}{h ( y )} .

For fixed

t

, the strong law of large numbers gives, almost surely,

\frac{1}{m} \sum_{i = 1}^{m} w (Y_{i}) 1_{{Y_{i} \leq t}} ⟶ \int_{(- \infty, t]} f (y) d y = F (t),

while

\frac{1}{m} \sum_{i = 1}^{m} w (Y_{i}) ⟶ \int f (y) d y = 1.

The ratio therefore converges almost surely to

F (t)

. This is precisely the claimed convergence in distribution of the self-normalized importance sampling resample.

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