Stopped energy estimates for supremum occupations

Finite-grid summation by parts and dominated convergence give the bounded stopped energy estimate. Doob and Fatou transfer it to the finite auxiliary inverse clock. The terminal mean and the full positive-filter stopped convergence statements follow from these moments.

References

• J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapters IV and VI.
• O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, 2002, Chapter 7 (Doob's inequality).