Abstract
Doob’s intriguing and delicate upcrossing inequality is derived in this chapter. One of its consequences is the (sub) martingale convergence theorem, which in turn leads to a proof of the strong law of large numbers and a derivation of DeFinetti’s representation of exchangeable (symmetrically dependent) sequences of random variables. Other applications include regularity of sample paths of continuous parameter stochastic processes to be derived in Chapter 13.
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Bhattacharya, R., Waymire, E.C. (2021). The Upcrossings Inequality and (Sub)Martingale Convergence. In: Random Walk, Brownian Motion, and Martingales. Graduate Texts in Mathematics, vol 292. Springer, Cham. https://doi.org/10.1007/978-3-030-78939-8_12
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DOI: https://doi.org/10.1007/978-3-030-78939-8_12
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