Total variation distance ( $∥ μ - ν ∥_{TV}$ )

The total variation distance between probability distributions is

∥ μ - ν ∥_{TV} = sup_{A} ∣ μ (A) - ν (A) ∣.

Table of contents
- Coupling characterization of total variation distance Total variation distance
- Coupling inequality for total variation Total variation distance

Coupling characterization of total variation distance

The total variation distance is the least possible mismatch probability over all couplings:

∥ μ - ν ∥_{TV} = in f P (X \neq = Y) .

Coupling inequality for total variation

For any coupling of two Markov chains and its coalescence time

T

,

∥ L (X_{t}) - L (Y_{t}) ∥_{TV} \leq P (T > t) .

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 2 / b / Solution

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