TY - JOUR

T1 - Distributional representations and dominance of a Lévy process over its maximal jump processes

AU - Buchmann, Boris

AU - Fan, Yuguang

AU - Maller, Ross A.

N1 - Publisher Copyright:
© 2016 ISI/BS.

PY - 2016/11

Y1 - 2016/11

N2 - Distributional identities for a Levy 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 sup0s≤tδXs, or by sup0s≤t |δXs |. Thus, we obtain necessary and sufficient conditions for Xt / sup0s≤tδXs and Xt / sup0s≤t |δXs | 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 Levy 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.

AB - Distributional identities for a Levy 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 sup0s≤tδXs, or by sup0s≤t |δXs |. Thus, we obtain necessary and sufficient conditions for Xt / sup0s≤tδXs and Xt / sup0s≤t |δXs | 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 Levy 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.

KW - Distributional representation

KW - Domain of attraction to normality

KW - Dominance

KW - Lévy process

KW - Maximal jump process

KW - Relative stability

UR - http://www.scopus.com/inward/record.url?scp=84969590526&partnerID=8YFLogxK

U2 - 10.3150/15-BEJ731

DO - 10.3150/15-BEJ731

M3 - Article

SN - 1350-7265

VL - 22

SP - 2325

EP - 2371

JO - Bernoulli

JF - Bernoulli

IS - 4

ER -