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

Assume $\mu_2>\mu_1$, so the upper tail is governed by its boundary. The <large deviation principle> gives the exponentially equivalent estimate
$$
\boxed{\mathbb P(N_t/t\geq\mu_2)
\asymp e^{-tI(\mu_2)}},
$$
where now $I(x)=x\log(x/\mu_1)-x+\mu_1$. Increasing capacity to $\mu_3>\mu_2$ reduces this estimate by at least $10^6$ when
$$
\boxed{t[I(\mu_3)-I(\mu_2)]\geq6\log10}.
$$
Equivalently,
$$
\boxed{\mu_3\log\frac{\mu_3}{\mu_1}-\mu_3
-\mu_2\log\frac{\mu_2}{\mu_1}+\mu_2
\geq\frac{6\log10}{t}}.
$$

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / a / iii / Solution

{wiki}

A sequence $(Z_n)$ satisfies a large deviation principle with speed $n$ and rate function $I$ when probabilities of measurable sets $A$ decay exponentially between the bounds
$$
-\inf_{x\in A^\circ}I(x)
\leq\liminf_n\frac1n\log\mathbb P(Z_n\in A)
\leq\limsup_n\frac1n\log\mathbb P(Z_n\in A)
\leq-\inf_{x\in\overline A}I(x).
$$


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