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which we assume is positive‐definite. A CLT for a Monte Carlo average,
1 IID. Let . If , then, as ,
2 MCMC. Let be polynomially ergodic of order where such that , then if is positive‐definite, as ,
Typically, MCMC algorithms exhibit positive correlation implying that
3.2 Quantiles
Let
An asymptotic distribution for sample quantiles is available under both IID Monte Carlo and MCMC.
Theorem 2.
Let
1 IID. Let , then
2 MCMC. [11] If the Markov chain is polynomially ergodic of order and , then
The density value,
3.3 Other Estimators
For many estimators, a delta method argument can yield a limiting normal distribution. For example, a CLT for
We obtain the asymptotic distribution of
