MCMC Sample should be i.i.d I'm a bit not sure how to show that MCMC samples are i.i.d. In my opinion the trace plot should behave like white noise model because white noise model has a strong stationary properties i.e. the joint distribution from any lag is the same hence the identical distribution achieve. Is it correct ? And if one see the ACF plot one need to see whether the ACF are exponentially decreasing. But 0 correlation or 0 covariance doesn't imply independence, so how to proof it ?

MCMC chains do not generate independent samples. To convince yourself, look at the autocorrelation plots, the autocorrelations would be non-zero. That is why one of the metrics we use for judging the quality of the results we use metrics as effective sample size that tries to estimate how many independent samples does your chain contain.
