Sampler

• Given :
  • $P_X$ is a distribution on a measure space (E,B)
  • A seq of $X_1,X_2,\dots$ of random variables is a sampler if for all $A\in B$
  • $P_X(A)=li{m}_{N\to \infty }\frac{1}{N}{\Sigma }_{i=1}^N{1}_A\circ X_i$
  • $1_A$ is an indicator function for A
• $X_1,X_2,\dots$ need not have the same distribution or need to be independant
• Dream : All $X_i$ are uniformly distributed on [0,1]. (Impossible)