The notions of statistical independence, conditional expectation and conditional probability are the cornerstones of probability theory.

1. 1.

   Criteria for percolation on thed-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. 2.

   Recall that the\(\sigma \)-field\(\mathcal{G}_i \) generated by\(\cup _{t\in \varLambda _i} \mathcal{F}_t\) is referred to as thejoin\(\sigma \)-field and denoted\(\bigvee _{t\in \varLambda _i}\mathcal{F}_t\).

3. 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. 4.

   Counterexamples have been constructed, see for example, Halmos (1950), p. 210.

5. 5.

   The Doob–Blackwell theorem provides the existence of a regular conditional distribution of a random mapY, 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.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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