Weak Drifts of Infinitely Divisible Distributions and Their Applications

Ken-iti Sato () and Yohei Ueda
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Yohei Ueda: Keio University

Journal of Theoretical Probability, 2013, vol. 26, issue 3, 885-898

Abstract: Abstract Weak drift of an infinitely divisible distribution μ on ℝ d is defined by analogy with weak mean; properties and applications of weak drift are given. When μ has no Gaussian part, the weak drift of μ equals the minus of the weak mean of the inversion μ′ of μ. Applying the concepts of having weak drift 0 and of having weak drift 0 absolutely, the ranges, the absolute ranges, and the limit of the ranges of iterations are described for some stochastic integral mappings. For Lévy processes, the concepts of weak mean and weak drift are helpful in giving necessary and sufficient conditions for the weak law of large numbers and for the weak version of Shtatland’s theorem on the behavior near t=0; those conditions are obtained from each other through inversion.

Keywords: Infinitely divisible distribution; Weak mean; Weak drift; Inversion; Stochastic integral mapping; Weak law of large numbers; Shtatland’s theorem; 60E07; 60G51; 60H05 (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-012-0419-2

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