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→General form of the theorem: Necessary that subset J be finite. |
The word "obvious" is unnecessary in the explanation. |
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The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process <math>X</math>. Then the probability <math>\mathbb{P}(X_1 >0, X_2<0)</math> can be computed either as <math>\nu_{1,2}( \mathbb{R}_+ \times \mathbb{R}_-)</math> or as <math>\nu_{2,1}( \mathbb{R}_- \times \mathbb{R}_+)</math>. Hence, for the finite-dimensional distributions to be consistent, it must hold that
<math>\nu_{1,2}( \mathbb{R}_+ \times \mathbb{R}_-) = \nu_{2,1}( \mathbb{R}_- \times \mathbb{R}_+)</math>.
The first condition
Continuing the example, the second condition implies that <math>\mathbb{P}(X_1>0) = \mathbb{P}(X_1>0, X_2 \in \mathbb{R})</math>. Also this is a trivial condition that will be satisfied by any consistent family of finite-dimensional distributions.
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