An important example when the local asymptotic normality holds is in the case of i.i.d sampling from a regular parametric model.
The notion of local asymptotic normality was introduced by Le Cam (1960) and is fundamental in the treatment of estimator and test efficiency.
Suppose { X1, X2, …, Xn } is an iid sample, where each Xi has density function f(x, θ).
The likelihood function of the model is equal to If f is twice continuously differentiable in θ, then Plugging in
, gives By the central limit theorem, the first term (in parentheses) converges in distribution to a normal random variable Δθ ~ N(0, Iθ), whereas by the law of large numbers the expression in second parentheses converges in probability to Iθ, which is the Fisher information matrix: Thus, the definition of the local asymptotic normality is satisfied, and we have confirmed that the parametric model with iid observations and twice continuously differentiable likelihood has the LAN property.