Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 164 / 4 / ii / Solution

Write $\delta =d_R(X,Y)$, $A=U_1+U_2$, $B=V_1+V_2$, and $S=A+B$. Distances depend only on distributions, so all variables used in any one application may be realized as independent random variables.

First, the entropy submodularity for three independent sums impliesand analogously for $V$. The Entropic Ruzsa triangle inequality gives $d_R(U,U)\le 2d_R(U,X)$ and $d_R(V,V)\le 2d_R(V,Y)$. Consequently the relevance of independent self-sums gives

Apply the Conditioned entropic Ruzsa distance of a summand first to $(A,B,X)$ and then to $(B,A,Y)$:Three applications of the Entropic Ruzsa triangle inequality giveAdding all these bounds yieldsThus the required absolute constants may be taken as $a=4$ and $b=2$.

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