Sampling from the Transition Kernel

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Let $(Omega, Sigma, P)$ be a probability space. Assume that $K(x,A) : mathbbR^n times mathbbB(mathbbR^n) to [0,1]$ is a Transition Kernel, i.e., for every fixed $x$, $K$ is a probability measure and for every fixed $A$, $K$ is measurable.



Suppose, $Y$ is a given random variable and $Sigma_Y$ be the sigma algebra generated by $Y$. Then, is it possible to obtain some random variable $Z$ such that $E[f(Z)|Sigma_Y] (omega) = int f K(Y(omega),bullet)$, $P$-a.s. for every $f$ in some nice class of functions defined on $mathbbR^n$?



A textbook I am reading says $Z sim K(Y,bullet)$ and then uses the above equality for all $f in C_b(mathbbR^n)$ (bounded continuous functions).



Thanks,



P.S. Most probably this follows from existence of Markov Processes for a given transition kernel.




probability-theory




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    Let $(Omega, Sigma, P)$ be a probability space. Assume that $K(x,A) : mathbbR^n times mathbbB(mathbbR^n) to [0,1]$ is a Transition Kernel, i.e., for every fixed $x$, $K$ is a probability measure and for every fixed $A$, $K$ is measurable.



    Suppose, $Y$ is a given random variable and $Sigma_Y$ be the sigma algebra generated by $Y$. Then, is it possible to obtain some random variable $Z$ such that $E[f(Z)|Sigma_Y] (omega) = int f K(Y(omega),bullet)$, $P$-a.s. for every $f$ in some nice class of functions defined on $mathbbR^n$?



    A textbook I am reading says $Z sim K(Y,bullet)$ and then uses the above equality for all $f in C_b(mathbbR^n)$ (bounded continuous functions).



    Thanks,



    P.S. Most probably this follows from existence of Markov Processes for a given transition kernel.




    probability-theory




    share|cite|improve this question























      up vote
      1
      down vote

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      up vote
      1
      down vote

      favorite











      Let $(Omega, Sigma, P)$ be a probability space. Assume that $K(x,A) : mathbbR^n times mathbbB(mathbbR^n) to [0,1]$ is a Transition Kernel, i.e., for every fixed $x$, $K$ is a probability measure and for every fixed $A$, $K$ is measurable.



      Suppose, $Y$ is a given random variable and $Sigma_Y$ be the sigma algebra generated by $Y$. Then, is it possible to obtain some random variable $Z$ such that $E[f(Z)|Sigma_Y] (omega) = int f K(Y(omega),bullet)$, $P$-a.s. for every $f$ in some nice class of functions defined on $mathbbR^n$?



      A textbook I am reading says $Z sim K(Y,bullet)$ and then uses the above equality for all $f in C_b(mathbbR^n)$ (bounded continuous functions).



      Thanks,



      P.S. Most probably this follows from existence of Markov Processes for a given transition kernel.




      probability-theory




      share|cite|improve this question













      Let $(Omega, Sigma, P)$ be a probability space. Assume that $K(x,A) : mathbbR^n times mathbbB(mathbbR^n) to [0,1]$ is a Transition Kernel, i.e., for every fixed $x$, $K$ is a probability measure and for every fixed $A$, $K$ is measurable.



      Suppose, $Y$ is a given random variable and $Sigma_Y$ be the sigma algebra generated by $Y$. Then, is it possible to obtain some random variable $Z$ such that $E[f(Z)|Sigma_Y] (omega) = int f K(Y(omega),bullet)$, $P$-a.s. for every $f$ in some nice class of functions defined on $mathbbR^n$?



      A textbook I am reading says $Z sim K(Y,bullet)$ and then uses the above equality for all $f in C_b(mathbbR^n)$ (bounded continuous functions).



      Thanks,



      P.S. Most probably this follows from existence of Markov Processes for a given transition kernel.





      probability-theory





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      share|cite|improve this question




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      edited Jul 16 at 21:18
























      asked Jul 16 at 19:34









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