= 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 $2C$-relevant to $(X,Y)$. The <entropy submodularity for three independent sums> implies
$$
d_R(U_1+U_2;X)
\leq\frac12\bigl(2d_R(U;X)+d_R(U;U)\bigr),
$$
and similarly for $V$. The <Entropic Ruzsa triangle inequality> bounds
$$
d_R(U;U)\leq2d_R(U;X),
\qquad
d_R(V;V)\leq2d_R(V;Y),
$$
which proves the claim after addition.
Back to article page
