# Tag Info

### 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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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 ...