APPLD STOCHASTC PROCESSE
APPLD STOCHASTC PROCESSE STA 624
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Date Created: 10/23/15
Notes7 April 187 Sta624 Spring 2006 Example of MC for continuous state space cont random variables But still discrete time 1 Transition probability matrix needs to be replaced by a transition probability kernel pzly which is a density for any given y value 2 in the following example is taken to be Uniform 17y7 1 State space is the interval 07 1 Example of a MC X0 N any number between 0 and 1 or could be from any distribution on 01 interval X1 Unif 17 X071 X2 Unif 17 X171 Xn Unif17 Xn17 1 This is a MC 1f the distribution of Xn is convergent at all here it does7 the stationary distribution must satisfy the following integral equation pltziygtfltygtdy 7 M We may check that distribution fz 2x solves the above equation Therefore Xn7 as n large7 will have a distribution approximately equal tofz2zfor0ltmlt1 Remark this is the cont version of the equation 7r0P 7r0 for discrete MC Since 17yltzlt1 39 MW We compute7 using fy 2y 1 1 0 pzlyfydy 7 I1yltmlt12dy 7 2 1 dy 7 2x 7w 1 Notice here we only used the random variables from uniform distributions and end up with a random variable with density x 2x The idea of MCMC is to use an easy in the iteration but end up with a random variable with complicated x that we desire Suppose X0 N any number between 0 and 1 X1 Unif 017 X0 X2 Unif 017 X1 Xn Unif017 Xn17 Find the stationary distribution x of this MC This will also be the approx distribution of Xn for large n