We obtain the asymptotic distribution of . A similar argument can be made for the off‐diagonals of
. Under the conditions of Theorem 1,
where is
Under IID sampling, the infinite sum above reduces to
Applying the delta method for function , we obtain
3.4 Confidence Regions for Means
Suppose that is an estimate of the limiting Monte Carlo variance–covariance matrix,
for IID sampling, and
for MCMC sampling. Let
be the
‐quantile of a
distribution. The CLT yields a large‐sample confidence region around
as
Let denote the determinant. The volume of this ellipsoidal confidence region, which depends on
,
, and
, is given by
Sometimes a joint sampling distribution may be difficult to obtain, or the limiting variance–covariance matrix may be too complicated to estimate. In such cases, one can consider hyperrectangular confidence regions. Let