We generalize the notion of the submartingale property and Doob’s inequality. Furthermore, we show how the latter leads to new inequalities for several stochastic processes: certain time series, Lévy processes, random walks, processes with independent increments, branching processes and continuous state branching processes, branching diffusions and superdiffusions, as well as some Markov processes, including geometric Brownian motion.
a-achieving process Doob’s inequality Maximal inequality Time series Lévy process Processes with independent increments Submartingale Random walk Branching process Branching diffusion Superprocess Continuous state branching process Markov process Geometric Brownian motion Approximate convexity
Mathematics Subject Classification (2010)
60E15 60G45 60G48 60G51 60J80
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The author is grateful to an anonymous referee for his/her close reading of the manuscript and for pointing out some glitches.
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