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On the distance to the stationary distribution

It is impossible to bound $D_{TV}(p,q)$ in terms of $x=D_{TV}(p,Mp)$ without further assumptions on the chain, like expansion. This is due to the phenomenon known as metastability. Rich examples are ...
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Convergence in probability of a supremum

As suggested by Anthony Quas, the supremum in question can be rewritten as $$s_n:=\sup_{b\ge\sqrt n/A}S_n(b),$$ where $$S_n(b):=\frac1{b^3}\sum_{i=1}^n Z_i^2\,1(Z_i<b)$$ and $Z_i:=1/X_i$, so that ...
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Does the average of correlated Gaussian random variables with mean zero and different variances converge in probability to their mean?

The following reference has several versions of the WLLN: Serfling, Robert J. 1980. Approximation Theorems of Mathematical Statistics. New York: Wiley. It refers to the following for a version with ...

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