# Lec19 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 19: Stationary Markov Chains Lecturer: Jim Pitman Symmetry Ideas: General idea of symmetry: make a transformation, and some- thing stays the same. In probability theory, the transformation may be con- ditioning, or some rearrangement of variables. What stays the same is the distribution of something. For a sequence of random variables X 0 ,X 1 2 ,... , various notions of symmetry: (1) Independence. X 0 and X 1 are independent. Distribution of X 1 given X 0 does not involve X 0 ; that is, ( X 1 | X 0 A ) d = X 1 . (2) Identical distribution. The X n are identically distributed: X 0 d = X n for every n . Of course IID = LLN / CLT. Stationary: ( X 1 2 ) d = ( X 0 1 ) (shifting time by 1) By measure theory, this is the same as ( X 1 2 ,...,X n ) d = ( X 0 1 n - 1 ) , for all n. Obviously IID = Stationary. But the converse is not necessarily true. E.g., X n = X 0 for all n , for any non-constant random variable X 0 . Another example : Stationary MC, i.e., start a MC with transition matrix P with initial distribution π . Then easily, the following are equivalent:

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## This note was uploaded on 05/14/2011 for the course STAT 150 taught by Professor Evans during the Spring '08 term at Berkeley.

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Lec19 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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