Independent product experiments are not necessarily identically distributed. Different instruments, covariates, exposure times, or noise levels contribute unequal information. Hellinger tensorization and triangular-array central limit theorems replace the single-distribution i.i.d. argument.
Triangular product experiments
Definition. A triangular product experiment has \[ P_{n,\theta} = \bigotimes_{i=1}^{m_n}P_{ni,\theta}. \] Relative to \(\theta_0\), \[ \log\frac{\mathrm{d}P_{n,\theta}} {\mathrm{d}P_{n,\theta_0}} = \sum_{i=1}^{m_n} \log\frac{\mathrm{d}P_{ni,\theta}} {\mathrm{d}P_{ni,\theta_0}}. \]
Independence gives additive log likelihoods, additive KL divergences, and multiplicative Hellinger affinities. It does not guarantee a Gaussian limit: one must ensure no component remains macroscopically influential.
Infinitesimal arrays
Definition. For a fixed parameter pair \((\theta,\theta')\), the product array is Hellinger infinitesimal if \[ \max_{1\leq i\leq m_n} H^2(P_{ni,\theta},P_{ni,\theta'}) \longrightarrow0. \]
This prevents one observation from deciding the hypothesis by itself. Combined with convergence of the sum of component Hellinger contributions, it leads to an infinitely divisible likelihood-ratio limit; a Lindeberg condition selects the Gaussian case.
Componentwise quadratic mean expansions
Suppose each component has a local square-root expansion \[ \sqrt{\frac{\mathrm{d}P_{ni,\theta_0+t}} {\mathrm{d}P_{ni,\theta_0}}} = 1+\frac12t^\top\dot\ell_{ni}+r_{ni}(t) \quad\text{in }L_2(P_{ni,\theta_0}), \] with \[ E_{\theta_0}\dot\ell_{ni}=0, \qquad I_{ni}=E_{\theta_0} (\dot\ell_{ni}\dot\ell_{ni}^\top). \] Let \[ I_n=\sum_{i=1}^{m_n}I_{ni}. \]
Theorem (triangular LAN). Assume \(I_n\) is nonsingular and choose local shifts \[ t_n=I_n^{-1/2}h. \] Suppose:
- the accumulated quadratic-mean remainders are negligible;
- for every \(\varepsilon>0\), \[ \sum_{i=1}^{m_n} E_{\theta_0}\!\left[ \|I_n^{-1/2}\dot\ell_{ni}\|^2 1\{\|I_n^{-1/2}\dot\ell_{ni}\|>\varepsilon\} \right]\to0; \]
- \[ I_n^{-1/2} \left(\sum_iI_{ni}\right) I_n^{-1/2}=I_p. \]
Then \[ \log\frac{\mathrm{d}P_{n,\theta_0+I_n^{-1/2}h}} {\mathrm{d}P_{n,\theta_0}} = h^\top\Delta_n-\frac12\|h\|^2+o_p(1), \] where \[ \Delta_n = \sum_i I_n^{-1/2}\dot\ell_{ni} \rightsquigarrow N_p(0,I_p). \]
Proof outline. Expand each square-root likelihood ratio and then \(2\log(1+s_{ni})\) as in the i.i.d. DQM proof. Summing linear terms gives \(\Delta_n\); the normalization of component densities supplies the quadratic drift. The displayed Lindeberg condition gives the triangular-array multivariate CLT and makes the sum of large score contributions negligible. The accumulated DQM remainder controls all remaining terms. \(\square\)
Heteroscedastic normal regression
Example. Let \[ Y_{ni}=x_{ni}^\top\beta+\varepsilon_{ni}, \qquad \varepsilon_{ni}\sim N(0,\sigma_{ni}^2) \] independently, with known variances. The score and information are \[ \dot\ell_{ni} =\frac{x_{ni}\varepsilon_{ni}}{\sigma_{ni}^2}, \qquad I_n=\sum_i\frac{x_{ni}x_{ni}^\top}{\sigma_{ni}^2}. \] If \[ \max_i \left\| I_n^{-1/2}\frac{x_{ni}}{\sigma_{ni}} \right\|\to0, \] no design point dominates, and LAN holds at rate \(I_n^{1/2}\). The efficient estimator is weighted least squares.
Information accumulation at unequal rates
Different parameter directions can accumulate information at different rates. A matrix normalization \(r_n\), not a scalar \(\sqrt n\), is then natural: \[ r_n^{-\top}I_nr_n^{-1}\to I. \] Weak identification appears when one eigenvalue of \(I_n\) grows slowly or remains bounded; the corresponding local scale is wider and may have a nonstandard limit.
Gaussian versus jump limits
For infinitesimal independent likelihood increments, the possible limits are governed by a Lévy–Khintchine decomposition. A Lindeberg condition removes jumps and yields LAN. If rare increments retain finite total intensity, a Poisson or compound-Poisson component survives. Thus “independent and numerous” is not enough for Gaussianity; the tail allocation of likelihood information determines the limit experiment.
Decision consequences
Once the triangular experiment converges to a Gaussian shift with information \(I\), the same local asymptotic minimax and convolution theorems apply. The proof depends only on likelihood-ratio convergence, not on identical distributions. What changes is the construction and estimation of the accumulated information matrix.