Burn-in total variation comparison
= Burn-in total variation comparison
Applying a <Markov kernel> to two initial laws cannot increase their <total variation distance>, even for the distribution of an entire subsequent finite trajectory. If a bounded summand has absolute value at most $B$, the <variance> of a length-$N$ path sum differs between these laws by at most $6B^2N^2$ times their <total variation distance>. Thus a geometrically long enough burn-in makes this difference negligible relative to $N$.