The Average Estimation Of Functions On Markov Chain

Suppose that ? finit set,? is a probability measure on ? and f is a real-valued function on ?.In many practical problems,we often need to calculate E?(f)=?x??f(x)?(x).But when th number of elements in ? is too large,it is impossible to calculate E?(f).So we can use the Markov chain Monte Carlo method to estimate E?(f).Namely,we pick Random variables?1,?2,...,?N which are i.i.d.And 1/N?Ni=1 f(?i)is a approxi-mate estimation of E?(f).But in many situations,it is hard to get ?i(i = 1,2,...,N)by computer.Then we can get a ergodic Markov chain {?0,?1,?2,...}.And the stationary distribution of the chain is ?.Then 1/N ?Ni=1 f(?i)is a estimation of E?(f).Because of the stationary distributions of the first samples of the Markov chain is different from?,it is better to use 1/N-s?N0=s+1f(?i)to estimate E?(f).This article introduce how to determine the s and N when given the precision requirement(Proposition5.10)and related preliminary knowledge in detail.