Asymptotic shape of the concave majorant of a L\'evy process

We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at $T$) of any L\'evy process on $[0,T]$ as $T\to\infty$. The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the L\'evy measure. The key tool in the proofs is the recent representation of the concave majorant for all L\'evy processes using a stick-breaking representation.