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  "body": "\n## Definition\n\nLet $\\Theta \\subseteq \\R^k$ be open, and for each $n$ let $\\set{P_{n,\\theta} : \\theta \\in \\Theta}$ be a family of probability measures on a measurable space $(\\Omega_n, \\mathcal{F}_n)$.\nLet $\\theta \\in \\Theta$.\nThe sequence of models is *locally asymptotically normal* (LAN) at $\\theta$ if the following holds.\nThere are random vectors $\\Delta_{n,\\theta}$ on $\\Omega_n$ and a $k \\times k$ [[positive definite matrix|0QQCG3JWJ4]] $I_\\theta$ such that:\n\n1. $\\Delta_{n,\\theta} \\xrightarrow{d} \\mathcal{N}_k(0, I_\\theta)$ under $P_{n,\\theta}$, in the sense of [[convergence in distribution|0JDAQHWA9N]] to a [[multivariate normal|0G558GJQMW]] law;\n2. for every sequence $h_n \\to h$ in $\\R^k$,\n$$\\log \\frac{dP_{n, \\theta + h_n/\\sqrt{n}}}{dP_{n,\\theta}} = h^T \\Delta_{n,\\theta} - \\frac{1}{2} h^T I_\\theta h + o_{P_{n,\\theta}}(1).$$\nHere the log-likelihood ratio uses the [[Radon–Nikodym derivative|1NTKG64F7N]] of the absolutely continuous part.\nThe term $o_{P_{n,\\theta}}(1)$ is a sequence of random variables that tends to $0$ in [[probability|0EFDDXTVD6]] under $P_{n,\\theta}$.\n\n## Remarks\n\nExample: for i.i.d. $\\mathcal{N}(\\theta, 1)$ observations, $P_{n,\\theta} = \\mathcal{N}(\\theta, 1)^{\\otimes n}$, the log-likelihood ratio is exactly $h\\, \\Delta_n - h^2/2$ with $\\Delta_n = n^{-1/2}\\sum_i (X_i - \\theta) \\sim \\mathcal{N}(0, 1)$, so the model is LAN with $I_\\theta = 1$ and no remainder.\n\nMore generally, if $X_1, \\dots, X_n$ are i.i.d. from a model that is differentiable in quadratic mean at $\\theta$ with nonsingular [[Fisher information|G1Z3WSTE8Q]] $I_\\theta$, the model is LAN at $\\theta$ with $\\Delta_{n,\\theta} = n^{-1/2}\\sum_i s(\\theta; X_i)$, where $s$ is the [[score|CP97GRVTQ0]].\n\nNon-example: for i.i.d. Uniform$(0, \\theta)$ observations and $h \\ne 0$, $P_{n, \\theta + h/\\sqrt{n}}$ and $P_{n,\\theta}$ become asymptotically singular.\nThe reason is that the scale at which they differ is $1/n$; clause 2 fails, and the model is not LAN.\n\nUnder LAN, $P_{n,\\theta + h/\\sqrt{n}}$ and $P_{n,\\theta}$ are mutually [[contiguous|DQGJMP9WHC]], since the limit law $\\mathcal{N}(-\\frac{1}{2} h^T I_\\theta h, \\, h^T I_\\theta h)$ of the log-likelihood ratio has $E e^{L} = 1$.\n",
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