Local asymptotic normality for local experiments

Let $P_{n,h}$ be probability laws on a common measurable space for each integer $n\ge 1$ and local parameter $h\in {\mathbb{R}}^d$, where $d<\mathrm{\infty }$ and $P_{n,0}$ is the base law. Suppose that for every fixed $h$ we are supplied a nonnegative measurable function $L_{n,h}$ with $\int L_{n,h}d{P}_{n,0}=1.$ It defines an absolutely continuous proxy ${\bar{P}}_{n,h}(A):={\int }_A{L}_{n,h}d{P}_{n,0}\mathrm{.}$ The actual local law is allowed to have a singular part; the approximation condition below is what connects it to this proxy.

The experiments are locally asymptotically normal (LAN) at $h=0$ with information matrix $K$ and central sequence ${\mathrm{\Delta }}_n$ if $K$ is a symmetric positive semidefinite $d\times d$ matrix, ${\mathrm{\Delta }}_n$ is an ${\mathbb{R}}^d$-valued statistic, ${\mathrm{\Delta }}_n\Rightarrow N_d(0,K)\text{under }{P}_{n,0},$ and for every fixed $h\in {\mathbb{R}}^d$, $\mathrm{\parallel }{P}_{n,h}-{\bar{P}}_{n,h}{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\to 0,\log L_{n,h}=h^T{\mathrm{\Delta }}_n-\frac{1}{2}{h}^{T}Kh+o_{P_{n,0}}(1),$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$ is the event-supremum total variation distance (Total variation distance for probability laws) and $o_{P_{n,0}}(1)$ means convergence to zero in $P_{n,0}$-probability. We set $\log 0=-\mathrm{\infty }$; the finite-valued expansion entails $P_{n,0}(L_{n,h}=0)\to 0$. Convergence in distribution is as in Convergence in distribution of random elements, and $N_d(0,K)$ allows singular $K$ (Multivariate normal law, including singular covariance).

If $P_{n,h}={\bar{P}}_{n,h}$, this is the usual dominated LAN comparison. The proxy formulation also covers QMD paths whose local alternatives are not absolutely continuous with respect to the base law, provided the singular product mass tends to zero in total variation. This definition is pointwise in each fixed $h$; it does not assert uniformity over bounded sets of $h$.