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

For an integrable real random variable $X$, its cumulant-generating function is the extended-real convex functionwhere $\psi (\theta )=+\infty$ if the exponential moment diverges. Its Legendre transform of a cumulant-generating function is

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Let $X_1,X_2,\dots$ be independent and identically distributed random variables, let $S_n={\sum }_{j=1}^n{X}_j$, 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 ${\psi }^{\ast }$. In particular, for $x\ge m$,with the usual extended-real interpretation; the analogous lower-tail formula holds for $x\le m$.

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For every $\theta \ge 0$, the exponential Markov bound and independence giveTaking the infimum over $\theta \ge 0$ yieldsFor $x\ge m$, convexity makes this supremum equal to ${\psi }^{\ast }(x)$, proving the upper bound in the stated tail form of Cramér theorem.

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Fix $a>m$ and then $x>a$. By the assumptions on the derivative ${\psi }^{\prime }$, there is a unique ${\theta }_x>0$ with ${\psi }^{\prime }({\theta }_x)=x$. Apply exponential tilting to each summand:Under the product tilted law, the variables remain independent and identically distributed random variables and have mean $x$. Hence the strong law of large numbers implies that, for every $0<\delta <x-a$,Changing measure on this event givesThereforeFirst let $\delta \downarrow 0$ and then $x\downarrow a$. The continuity of a convex function gives the lower bound $-{\psi }^{\ast }(a)$. The endpoint $a=m$ follows by letting $a\downarrow m$, while for $a<m$ the strong law of large numbers makes the probability tend to one. This proves the required lower bound.

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