Gaussian limits arise from many small centered contributions. Rare events retain their discreteness and lead instead to Poisson, compound-Poisson, or more general infinitely divisible experiments. Their likelihood ratios have exponential form, but their local geometry is not quadratic.
Poisson random measures
Definition. A Poisson random measure \(N\) on \((S,\mathcal S)\) with \(\sigma\)-finite mean measure \(\mu\), written \(N\sim\operatorname{PRM}(\mu)\), satisfies:
- \(N(A)\sim\operatorname{Poisson}\{\mu(A)\}\) when \(\mu(A)<\infty\);
- counts on disjoint sets are independent.
Proposition (Laplace functional). For measurable \(f\geq0\), \[ E_\mu\exp\left\{-\int f\,\mathrm{d}N\right\} = \exp\left\{ -\int(1-e^{-f})\,\mathrm{d}\mu \right\}. \]
The Laplace functional characterizes the law and plays the role of a moment-generating function for point processes.
Poisson likelihood ratios
Theorem. Let \(N\sim\operatorname{PRM}(\mu_0)\), and suppose \(\mu_\theta\ll\mu_0\) with density \(\lambda_\theta>0\) and \(\int|\lambda_\theta-1|\,\mathrm{d}\mu_0<\infty\). Then \[ \log\frac{\mathrm{d}P_{\mu_\theta}} {\mathrm{d}P_{\mu_0}}(N) = \int\log\lambda_\theta\,\mathrm{d}N -\int(\lambda_\theta-1)\,\mathrm{d}\mu_0. \]
Proof. For a simple function \(\lambda_\theta=\sum_jc_j1_{A_j}\) on disjoint sets, the counts are independent Poisson variables. The ratio of their mass functions is \[ \prod_j c_j^{N(A_j)} e^{-\mu_0(A_j)(c_j-1)}, \] whose logarithm is the displayed integral. Approximate a general density by simple functions and use the Poisson Laplace functional to pass to the limit. \(\square\)
Hellinger geometry of Poisson laws
Theorem. For Poisson random measures with mean measures \(\mu\) and \(\nu\), \[ \rho(P_\mu,P_\nu) = \exp\left\{ -\frac12\int \left( \sqrt{\frac{\mathrm{d}\mu}{\mathrm{d}\lambda}} - \sqrt{\frac{\mathrm{d}\nu}{\mathrm{d}\lambda}} \right)^2 \mathrm{d}\lambda \right\}, \] where \(\lambda\) dominates \(\mu,\nu\).
Thus the square roots of intensity measures determine the Hellinger geometry. Independent superposition adds intensities, while restrictions to disjoint regions factor the experiment.
Rare Bernoulli events
Theorem (law of small numbers). Let \(X_{n1},\ldots,X_{nn}\) be independent Bernoulli variables with probabilities \(p_{ni}\), and suppose \[ \max_i p_{ni}\to0, \qquad \sum_{i=1}^np_{ni}\to\lambda. \] Then \[ \sum_{i=1}^nX_{ni}\rightsquigarrow\operatorname{Poisson}(\lambda). \]
Proof. The probability-generating function is \[ \prod_i\{1+p_{ni}(z-1)\}. \] Its logarithm equals \[ (z-1)\sum_ip_{ni} +O\!\left(\sum_ip_{ni}^2\right). \] Since \(\sum_ip_{ni}^2\leq(\max_ip_{ni})\sum_ip_{ni}\to0\), the limit is \(\exp\{\lambda(z-1)\}\), the Poisson generating function. \(\square\)
Example (boundary experiment). If \(X_i\sim\operatorname{Bernoulli}(\theta/n)\), \(\theta\geq0\), then \[ N_n=\sum_iX_i\rightsquigarrow\operatorname{Poisson}(\theta). \] The statistic \(N_n\) is sufficient, so the full binary experiment converges to a Poisson count experiment. The parameter rate is \(n^{-1}\), not \(n^{-1/2}\), and the limit is discrete even though the sample size diverges.
Marked and compound-Poisson limits
If each rare event carries a mark \(Y\) with law \(F\), the point process converges to a marked Poisson process. Sums of marks have a compound-Poisson law: \[ S=\sum_{j=1}^NY_j, \qquad N\sim\operatorname{Poisson}(\lambda). \] Its characteristic function is \[ Ee^{itS} =\exp\left\{ \lambda\int(e^{ity}-1)F(\mathrm{d}y) \right\}. \] Change-point likelihoods and threshold estimators often have compound-Poisson local limits because crossing a local boundary changes only finitely many observations asymptotically.
Infinitely divisible experiments
Definition. A distribution is infinitely divisible if, for every \(m\), it is the law of a sum of \(m\) i.i.d. variables. Its characteristic exponent has Lévy–Khintchine form \[ \log Ee^{itX} = itb-\frac12\sigma^2t^2 +\int_{\mathbb R} \left(e^{itx}-1-itx1_{\{|x|\leq1\}}\right)\nu(\mathrm{d}x), \] where \(\nu\) is a Lévy measure.
The Gaussian component represents accumulated small fluctuations; the Lévy measure represents jumps. Limit experiments with independently scattered infinitely divisible likelihood processes extend both Gaussian and Poisson cases.
Decision transfer from non-Gaussian limits
Once likelihood-ratio vectors converge to a Poisson or infinitely divisible experiment, the same finite-prior argument used in local asymptotic minimax theory transfers its Bayes risks as lower bounds. The optimal procedure may be discrete, asymmetric, or threshold based; forcing a quadratic Gaussian approximation would give the wrong local risk benchmark.