Conditioning and martingales
The concept of conditioning, or equivalently conditional expectation, is one of the most fundamental notions in Probability. After some motivation for the concept, we discuss some immediate properties of conditional expectations (and probabilities). Then the regularity of conditional probabilities and some related results are discussed. Next the concept of a martingale, the basic inequalities, and the decompositions of Riesz, Doob (in the discrete case) and Jordan type [for (sub) martingales] are presented. The discrete parameter convergences of martingales, including the Andersen-Jessen theorems, are discussed. Related Complements with some details, are contained as exercises in the last section.
KeywordsProbability Space Conditional Expectation Banach Lattice Vector Measure Orlicz Space
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