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

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.

Solved by gpt-5.6-sol high.