 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 $6B^2{N}^2$ times their total variation distance. Thus a geometrically long enough burn-in makes this difference negligible relative to $N$.