Abstract
In Chapter 11 we investigate infinitely divisible processes in a far more general setting than what mainstream probability theory has yet considered: we make no assumption of stationarity of increments of any kind and our processes are actually indexed by an abstract set. These processes are to Lévy processes what a general Gaussian process is to Brownian motion. Our main tool is a representation theorem due to J. Rosinski, which makes these processes appear as conditionally Bernoulli processes. For a large class of such processes we are able to prove lower bounds that extend those given in Chapter 8 for p-stable process. These lower bounds are not upper bounds in general, but we succeed in showing in a precise sense that they are upper bounds for “the part of boundness of the process which is due to cancellation”. Thus, whatever bound might be true for the “remainder of the process” owes nothing to cancellation.
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Talagrand, M. (2014). Infinitely Divisible Processes. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54075-2_11
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