Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 4

Write $W=Z-\mathbb{E}Z$ and $\psi (\lambda )=\log {\mathbb{E}}_P{e}^{\lambda W}$. For any $Q\ll P$ and real $\lambda$, define the exponential tiltPositivity of relative entropy givesThus every $Q$ with ${\mathbb{E}}_{Q}W\ge t$ has $D(Q\Vert P)\ge {\sup }_{\lambda \ge 0}\{\lambda t-\psi (\lambda )\}={\psi }^{\ast }(t)$.

For $0<t<b$, continuity of ${\psi }^{\prime }$ supplies ${\lambda }_t>0$ with ${\psi }^{\prime }({\lambda }_t)=t$. Under $P_{{\lambda }_t}$, $\mathbb{E}W=t$, and direct substitution givesThis tilt attains the constrained infimum and proves the identity.

For any coupling $\pi$ of $X\sim P$ and $Y\sim Q$ and every $A\subseteq \Omega$,Its absolute value is at most ${\mathbb{P}}_{\pi }(X\ne Y)$. Taking the supremum over $A$ and then the infimum over couplings proves

For each $i$, the total variation distance between $\mathrm{Bernoulli}(p_i)$ and $\mathrm{Poisson}(p_i)$ isIndeed, compare their masses at zero, one, and the Poisson tail; the positive excess of Bernoulli mass at one is $p_i(1-e^{-p_i})$. A maximal coupling therefore gives a pair $(X_i,N_i)$ with mismatch probability at most $p_i^2$. Couple these pairs independently. Thenby the union bound. Since ${\sum }_i{N}_i\sim \mathrm{Poisson}({\sum }_i{p}_i)=\mathrm{Poisson}(\nu )$, part (b) yields