Kolmogorov extension theorem
Topic of Random variables and laws, part of Stochastic processes
Kolmogorov extension theorem
| Type | Topic |
| Taught in | Random variables and laws |
| Part | Stochastic processes |
A consistent family of finite-dimensional distributions determines a stochastic process on the product space. It is an existence result: it does not give continuous sample paths or any other regularity a model may need.
In Random variables and laws
A consistent family of finite-dimensional distributions defines a stochastic process on a product space, under the hypotheses of the Kolmogorov extension theorem. The theorem is an existence result. It does not by itself produce continuous sample paths, or any other regularity the model may need. A Poisson process and a queue are built from the explicit clocks on Poisson processes and Birth–death chains and queues, which is a stronger description than consistency of the finite-dimensional laws alone.
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See also
Stochastic process · Measurable random variable · Simple variables and expectation · Lebesgue decomposition · Atom of a distribution