Sharp moderate maximal inequalities for upward skip-free Markov chains

2018/03/03 by Jia, Chen
#60E15 #60J27 #60J80 #60K25 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1803.01120

Abstract

The 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 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.

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