Multinomial central limit theorem
Proposition taught in Categorical data and the Dirichlet distribution, part of Bayesian statistics
Multinomial central limit theorem
| Type | Proposition |
| Taught in | Categorical data and the Dirichlet distribution |
| Part | Bayesian statistics |
Let {\boldsymbol{x}}\sim\operatorname{Multinomial}(n,{\boldsymbol{p}}) and \widehat{\boldsymbol{p}}={\boldsymbol{x}}/n. For fixed interior {\boldsymbol{p}}, define the covariance matrix
{\color{blue}\boldsymbol{\Sigma}}({\boldsymbol{p}}) =\operatorname{diag}({\boldsymbol{p}})-{\boldsymbol{p}}{\boldsymbol{p}}^{\mathsf T}.
Then
\sqrt n\,(\widehat{\boldsymbol{p}}-{\boldsymbol{p}}) \xrightarrow{\mathcal D} \operatorname{N}_K\!\left( {\boldsymbol{0}}, {\color{blue}\boldsymbol{\Sigma}}({\boldsymbol{p}}) \right).
The covariance matrix is singular because the components of \widehat{\boldsymbol{p}}-{\boldsymbol{p}} sum to zero. On the (K-1)-dimensional simplex, the count cloud therefore tightens at the usual n^{-1/2} rate and becomes locally normal.
Used in
- Taught in Categorical data and the Dirichlet distribution, where the derivation and examples are.
- All 7 questions on Categorical data and the Dirichlet distribution
See also
Categorical and multinomial models · Dirichlet distribution · The Beta distribution is the two-category Dirichlet · Dirichlet moments · Interior Dirichlet mode · Dirichlet–Multinomial conjugacy