Scaling limit results for the sum of many inverse Lévy subordinators

2012/03/30 by Ingemar Kaj, Kaj, Ingemar, Anders Martin‐Löf +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1203.6831

openalex publication_date 2012/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The first passage time process of a Lévy subordinator with heavy-tailed Lévy measure has long-range dependent paths. The random fluctuations that appear under two natural schemes of summation and time scaling of such stochastic processes are shown to converge weakly. The limit process is fractional Brownian motion in one case and a non-Gaussian and non-stable process in the other case. The latter appears to be of independent interest as a random process that arises under the influence of coexisting Gaussian and stable domains of attraction and is known from other applications to provide a bridge between fractional Brownian motion and stable Lévy motion.

Related

• Scaling Limit of Dependent Random Walk
• Trimmed Lévy Processes and their Extremal Components
• Lévy processes with jumps governed by lower incomplete gamma subordinator and its variations
• Lévy processes linked to the lower-incomplete gamma function
• Convergence to stable limits for ratios of trimmed Levy processes and\n their jumps