Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains

Chen Jia ()
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Chen Jia: University of Texas at Dallas

Journal of Theoretical Probability, 2019, vol. 32, issue 3, 1382-1398

Abstract: Abstract The $$L^p$$ L p maximal inequalities for martingales are one of the classical results in probability theory. Here we establish the sharp moderate maximal inequalities for upward skip-free Markov chains, which include the $$L^p$$ L p maximal inequalities as special cases. Furthermore, we apply our theory to two specific examples and obtain their moderate maximal inequalities: the first one is the M/M/1 queue and the second one is an upward skip-free Markov chain with large death jumps. These two examples have the same total birth and death rates. However, the former exhibits a phase transition phenomenon while the latter does not.

Keywords: Birth–death process; Single-birth process; M/M/1 queue; Moderate function; Lenglart domination principle; Good $$\lambda $$ λ inequality; 60J27; 60E15; 60K25; 60J80 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-018-0820-6

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