Distributional representations and dominance of a Lévy process over its maximal jump processes

Distributional identities for a Lévy process Xt, its quadratic variation process Vt and its maximal jump processes, are derived, and used to make "small time" (as t ↓ 0) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of X. Apart from providing insight into the connections between X, V, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study "self-normalised" versions of Xt, that is, Xt after division by sup₀