  The recent paper "Simple confidence intervals for MCMC without CLTs" by J.S.
Rosenthal, showed the derivation of a simple MCMC confidence interval using
only Chebyshev's inequality, not CLT. That result required certain assumptions
about how the estimator bias and variance grow with the number of iterations
$n$. In particular, the bias is $o(1/\sqrt{n})$. This assumption seemed mild.
It is generally believed that the estimator bias will be $O(1/n)$ and hence
$o(1/\sqrt{n})$. However, questions were raised by researchers about how to
verify this assumption. Indeed, we show that this assumption might not always
hold. In this paper, we seek to simplify and weaken the assumptions in the
previously mentioned paper, to make MCMC confidence intervals without CLTs more
widely applicable.
