C-relevant pair of random variables

(U, V)

is

C

-relevant to

(X, Y)

when

d_{R} (U; X) + d_{R} (V; Y) \leq C d_{R} (X; Y) .

Table of contents
- Relevance of independent self-sums C-relevant pair of random variables

Relevance of independent self-sums

If

(U, V)

is

C

-relevant to

(X, Y)

and

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

are mutually independent copies, then

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

is

2 C

-relevant to

(X, Y)

. The entropy submodularity for three independent sums implies

d_{R} (U_{1} + U_{2}; X) \leq \frac{1}{2} (2 d_{R} (U; X) + d_{R} (U; U)),

and similarly for

V

. The Entropic Ruzsa triangle inequality bounds

d_{R} (U; U) \leq 2 d_{R} (U; X), d_{R} (V; V) \leq 2 d_{R} (V; Y),

which proves the claim after addition.

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