Let $X_1,X_2,\ldots$ be <independent and identically distributed random variables>, let $S_n=\sum_{j=1}^nX_j$, and write $m=\mathbb E X_1$. <Cramér theorem> states that the empirical means $S_n/n$ obey a <large deviation principle> with good <rate function> $\psi^*$. In particular, for $x\geq m$,
$$
\lim_{n\to\infty}\frac1n\log\mathbb P(S_n/n\geq x)=-\psi^*(x),
$$
with the usual extended-real interpretation; the analogous lower-tail formula holds for $x\leq m$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 2 / b / Solution

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A rate function is a lower-semicontinuous map $I$ into $[0,\infty]$. It is good when every sublevel set $\{x:I(x)\leq c\}$ is compact.


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