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 Incoming links: C-relevant pair of random variables

Relevance of independent self-sums Created 2026-09-24 Updated 2026-11-05

Let

U, V, X, Y

take values in the vector space

F_{2}^{n}

over the two-element finite field. In this additive group, subtraction equals addition. Suppose

(U, V)

is a C-relevant pair of random variables relative to

(X, Y)

, and take mutually independent copies

U_{1}, U_{2}, V_{1}, V_{2}

. Then

(U_{1} + U_{2}, V_{1} + V_{2})

is

2 C

-relevant to

(X, Y)

.

For a proof, let

S = U_{1} + U_{2}

and take an independent representative of

X

. The entropy submodularity for three independent sums gives

H (S + X) + H (U) \leq H (S) + H (U + X),

so the definition of Entropic Ruzsa distance yields

d_{R} (S; X) \leq d_{R} (U; X) + \frac{1}{2} d_{R} (U; U) \leq 2 d_{R} (U; X) .

The last step is the Entropic Ruzsa triangle inequality. Apply the same argument to

V_{1} + V_{2}, Y

and add the inequalities.^[ref]

The binary finite field hypothesis makes the displayed self-sum argument agree with the difference convention in Entropic Ruzsa distance. On a general abelian group, signs must be retained explicitly.

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