On the small-time behavior of subordinators

Shaul K. Bar-Lev; Andreas Löpker; Wolfgang Stadje

doi:10.3150/11-BEJ363

On the small-time behavior of subordinators

Shaul K. Bar-Lev
, Andreas Löpker
, Wolfgang Stadje

Department of Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove several results on the behavior near t = 0 of Y^-t _t for certain (0,∞)-valued stochastic processes (Y _t)_t>0. In particular, we show for Lévy subordinators that the Pareto law on [1,∞) is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of tL(Y_t) as t →0 for a decreasing function L that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.

Original language	English
Pages (from-to)	823-835
Number of pages	13
Journal	Bernoulli
Volume	18
Issue number	3
DOIs	https://doi.org/10.3150/11-BEJ363
State	Published - Aug 2012

Keywords

Pareto law
Regular variation
Subordinator
Weak limit theorem

ASJC Scopus subject areas

Statistics and Probability

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10.3150/11-BEJ363

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title = "On the small-time behavior of subordinators",

abstract = "We prove several results on the behavior near t = 0 of Y-t t for certain (0,∞)-valued stochastic processes (Y t)t>0. In particular, we show for L{\'e}vy subordinators that the Pareto law on [1,∞) is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of tL(Yt) as t →0 for a decreasing function L that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.",

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T1 - On the small-time behavior of subordinators

AU - Bar-Lev, Shaul K.

AU - Löpker, Andreas

AU - Stadje, Wolfgang

PY - 2012/8

Y1 - 2012/8

N2 - We prove several results on the behavior near t = 0 of Y-t t for certain (0,∞)-valued stochastic processes (Y t)t>0. In particular, we show for Lévy subordinators that the Pareto law on [1,∞) is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of tL(Yt) as t →0 for a decreasing function L that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.

AB - We prove several results on the behavior near t = 0 of Y-t t for certain (0,∞)-valued stochastic processes (Y t)t>0. In particular, we show for Lévy subordinators that the Pareto law on [1,∞) is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of tL(Yt) as t →0 for a decreasing function L that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.

KW - Pareto law

KW - Regular variation

KW - Subordinator

KW - Weak limit theorem

UR - https://www.scopus.com/pages/publications/84871462288

U2 - 10.3150/11-BEJ363

DO - 10.3150/11-BEJ363

M3 - Article

AN - SCOPUS:84871462288

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VL - 18

SP - 823

EP - 835

JO - Bernoulli

JF - Bernoulli

IS - 3

ER -

On the small-time behavior of subordinators

Abstract

Keywords

ASJC Scopus subject areas

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