Let (mathscr{T}=left{boldsymbol{X}_{1}, ldots, boldsymbol{X}_{n} ight}) be iid data from a pdf (g(boldsymbol{x} mid boldsymbol{theta})) with Fisher matrix

Let \(\mathscr{T}=\left\{\boldsymbol{X}_{1}, \ldots, \boldsymbol{X}_{n}\right\}\) be iid data from a pdf \(g(\boldsymbol{x} \mid \boldsymbol{\theta})\) with Fisher matrix \(\mathbf{F}(\boldsymbol{\theta})\). Explain why, under the conditions where (4.7) holds,

\[ \boldsymbol{S}_{\mathscr{T}}(\boldsymbol{\theta}):=\frac{1}{n} \sum_{i=1}^{n} \boldsymbol{S}\left(\boldsymbol{X}_{i} \mid \boldsymbol{\theta}\right) \]

for large \(n\) has approximately a multivariate normal distribution with expectation vector \(\mathbf{0}\) and covariance matrix \(\mathbf{F}(\theta)\).