Abstract
The notions of statistical independence, conditional expectation and conditional probability are the cornerstones of probability theory.
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Notes
- 1.
Criteria for percolation on the d-dimensional integer lattice is a much deeper and technically challenging problem. In the case \(d=2\) the precise identification of the critical probability for (bond) percolation as \(p_c = {1\over 2}\) is a highly regarded mathematical achievement of Harry Kesten, see Kesten, H. (1982). For \(d\ge 3\) the best known results for \(p_c\) are expressed in terms of bounds.
- 2.
Recall that the \(\sigma \)-field \(\mathcal{G}_i \) generated by \(\cup _{t\in \varLambda _i} \mathcal{F}_t\) is referred to as the join \(\sigma \)-field and denoted \(\bigvee _{t\in \varLambda _i}\mathcal{F}_t\).
- 3.
This inequality appears in J. Neveu (1988): Multiplicative martingales for spatial branching processes, Seminar on Stochastic Processes, 223–242, with attribution to joint work with Brigitte Chauvin.
- 4.
Counterexamples have been constructed, see for example, Halmos (1950), p. 210.
- 5.
The Doob–Blackwell theorem provides the existence of a regular conditional distribution of a random map Y, given a \(\sigma \)-field \(\mathcal{G}\), taking values in a Polish space equipped with its Borel \(\sigma \)-field \(\mathcal{B}(S)\). For a proof, see Breiman (1968), pp. 77–80.
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Bhattacharya, R., Waymire, E.C. (2016). Independence, Conditional Expectation. In: A Basic Course in Probability Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-47974-3_2
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DOI: https://doi.org/10.1007/978-3-319-47974-3_2
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