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 poin...
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| Veröffentlicht in: | Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2016-11, Vol.22 (4), p.2325-2371 |
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| container_title | Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability |
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| creator | BUCHMANN, BORIS FAN, YUGUANG MALLER, ROSS A. |
| description | 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 |
| doi_str_mv | 10.3150/15-BEJ731 |
| format | Article |
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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 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. 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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 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. 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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 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.</abstract><pub>International Statistical Institute and Bernoulli Society for Mathematical Statistics and Probability</pub><doi>10.3150/15-BEJ731</doi><tpages>47</tpages><oa>free_for_read</oa></addata></record> |
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| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; Project Euclid Complete |
| title | Distributional representations and dominance of a Lévy process over its maximal jump processes |
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