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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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:
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online