Abstract
On the probability triple (Ω, ℱ, P) let an arbitrary sequence of random variables {x n , n≧0} be given. The Borel field generated by this sequence of random variables will be denoted by ℱ{x n , n≧0} or simply ℱ0. This Borel field ℱ0 is in general a subfield of ℱ, but in a discussion which is concerned solely with the sequence {x n , n≧0} only sets in ℱ0 will occur. In the case where all the x n are discrete with the state space I, the probabilities of all sets in ℱ0 are completely determined by the finite-dimensional joint probabilities
for all n≧0 and all in∈I. The above probability can be expressed as a product of conditional probabilities as follows:
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© 1960 Springer-Verlag OHG. Berlin · Göttingen · Heidelberg
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Chung, K.L. (1960). Transition probabilities. In: Markov Chains with Stationary Transition Probabilities. Die Grundlehren der Mathematischen Wissenschaften, vol 104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-49686-8_2
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DOI: https://doi.org/10.1007/978-3-642-49686-8_2
Publisher Name: Springer, Berlin, Heidelberg
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