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# Transition probabilities

• Kai Lai Chung
Chapter
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 104)

## Abstract

On the probability triple (Ω, , 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 {x n , n≧0} or simply 0. This Borel field 0 is in general a subfield of , but in a discussion which is concerned solely with the sequence {x n , n≧0} only sets in 0 will occur. In the case where all the x n are discrete with the state space I, the probabilities of all sets in 0 are completely determined by the finite-dimensional joint probabilities
$$P\{ x_0 (\omega ) = i_0 ,x_1 (\omega ) = i_1 ,...,x_n (\omega ) = i_n \}$$
for all n≧0 and all inI. The above probability can be expressed as a product of conditional probabilities as follows:
$$P\{ x_0 (\omega ) = i_0 \} \prod\limits_{t = 1}^n {P\{ x_t (\omega ) = i_t |x_s (\omega ) = i_s ,0 \leqq s < t\} } .$$

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## Copyright information

© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1960

## Authors and Affiliations

• Kai Lai Chung
• 1
1. 1.Syracuse UniversityUSA