Kolmogorov extension theorem: Difference between revisions

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==Explanation of the conditions==

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 generalises this obvious statement to hold for any number of time points <math>t_i</math>, and any control sets <math>F_i</math>.