Buchmann, BorisFan, YuguangMaller, Ross A.2016-12-162016-12-161350-7265http://hdl.handle.net/1885/111420Distributional 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<s≤t Xs, or by sup0<s≤t |Xs|. Thus, we obtain necessary and sufficient conditions for Xt / sup0<s≤t Xs and Xt / sup0<s≤t |Xs| to converge in probability to 1, or to ∞, as t ↓ 0, so that X is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of X is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of X at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as t → ∞) versions of the results can also be obtained.R. Maller’s research was partially supported by ARC grant DP1092502.application/pdf© 2016 ISI/BS. http://www.sherpa.ac.uk/romeo/issn/1350-7265/..."author can archive publisher's version/PDF" from SHERPA/RoMEO site (as at 16/12/16).distributional representationdomain of attraction to normalitydominanceLévy processmaximal jump processrelative stability2016-1110.3150/15-BEJ731