Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 2 / a

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- Solution a

Solution

Let

X_{1}, X_{2}, \dots

be i.i.d. with mass function

Q

on a finite alphabet, and let

P_{n}

be their empirical distribution. Sanov theorem states that for every set

Γ

of probability mass functions,

- in f_{R \in Γ^{\circ}} D_{e} (R ∥ Q) \leq lim inf_{n \to \infty} \frac{1}{n} lo g P (P_{n} \in Γ)

and

lim sup_{n \to \infty} \frac{1}{n} lo g P (P_{n} \in Γ) \leq - in f_{R \in \overline{Γ}} D_{e} (R ∥ Q),

where interior and closure use the probability-simplex topology. Thus the empirical distributions satisfy a large deviation principle with rate function

D_{e} (\cdot ∥ Q)

.

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