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Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

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  • R. A. Doney

    (University of Manchester)

  • R. A. Maller

    (University of Western Australia)

Abstract

We prove some limiting results for a Lévy process X t as t↓0 or t→∞, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t↓0; for example, we may have X t /t → P +∞ (t↓0) (weak divergence to +∞), whereas X t /t→∞ a.s. (t↓0) is impossible (both are possible when t→∞), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. “Almost sure stability” of X t , i.e., X t tending to a nonzero constant a.s. as t→∞ or as t↓0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to “strong law” behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.

Suggested Citation

  • R. A. Doney & R. A. Maller, 2002. "Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity," Journal of Theoretical Probability, Springer, vol. 15(3), pages 751-792, July.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:3:d:10.1023_a:1016228101053
    DOI: 10.1023/A:1016228101053
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    References listed on IDEAS

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    1. Harry Kesten & R. A. Maller, 1998. "Random Walks Crossing High Level Curved Boundaries," Journal of Theoretical Probability, Springer, vol. 11(4), pages 1019-1074, October.
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    Cited by:

    1. Søren Asmussen, 2022. "On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance," Finance and Stochastics, Springer, vol. 26(3), pages 383-416, July.
    2. David M. Mason, 2021. "Self-Standardized Central Limit Theorems for Trimmed Lévy Processes," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2117-2144, December.
    3. Yuguang Fan, 2017. "Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times," Journal of Theoretical Probability, Springer, vol. 30(2), pages 675-699, June.
    4. Grahovac, Danijel, 2022. "Intermittency in the small-time behavior of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 187(C).

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