}^{(h)}\left( \boldsymbol{\beta },\boldsymbol{\Sigma }_{hi}\right) , \\
\mathbf{\Psi}_{\lambda }\left( \boldsymbol{\beta }\right) & =\sum_{h=1}^{H}\eta
_{h}\mathbf{\Psi}_{\lambda }^{(h)}\left( \boldsymbol{\beta }\right)
=\sum_{h=1}^{H}\eta _{h}\lim_{n_{h}\rightarrow \infty }\frac{1}{n_{h}}\sum_{i=1}^{n_{h}}\mathbf{\Psi}_{i,\lambda }^{(h)}\left( \boldsymbol{\beta }\right) ,\end{aligned}$$where $\mathbf{\Omega}_{i,\lambda }^{(h)}\left( \boldsymbol{\beta },\boldsymbol{\Sigma }_{hi}\right) $ and $\mathbf{\Psi}_{i,\lambda }^{(h)}\left( \boldsymbol{\beta }\right) $ have the same expressions (not meaning) as $\mathbf{\Omega}_{hi,\lambda }\left( \boldsymbol{\beta },\boldsymbol{\Sigma }_{hi}\right) $and $\mathbf{\Psi}_{hi,\lambda }\left( \boldsymbol{\beta }\right) $respectively, given in (\[hhi\]), (\[ghi\]).

The idea of the previous theorem matches the philosophy of the asymptotic result developed in Castilla et al. (2018), where other family of divergences was considered and the derivations of the results were quite different. It is also appropriate to consider the following results in the place of Corollary \[cor1\], b and c.

\[cor1b\]The following ones are (weak) consistent estimators as $n_{h}$ goes to infinity, for every $h=1,...,H$:

a)

:   $\mathbf{\Psi}_{\lambda }^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})=\frac{1}{n_{h}}\sum_{i=1}^{n_{h}}\mathbf{\Psi}_{i,\lambda
    }^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$ is a consistent estimator of $\mathbf{\Psi}_{\lambda }^{(h)}\left( \boldsymbol{\beta }_{0}\right) $, for every $h=1,...,H$.

b)

:   $\mathbf{\Omega}_{\lambda }^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\{\widehat{\boldsymbol{\Sigma }}_{hi}\}_{h=1,...,H:i=1,...,n_{h}})=\frac{1}{n_{h}}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{i,\lambda }^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\widehat{\boldsymbol{\Sigma }}_{hi})$ is a consistent estimator of $\mathbf{\Omega}_{\lambda }^{(h)}\left( \boldsymbol{\beta }_{0}\right) $, for every $h=1,...,H$, whenever $\Sigma _{hi}=\mathrm{Var}[\widehat{\boldsymbol{Y}}_{hi}]$ is consistently estimated through $\widehat{\boldsymbol{\Sigma }}_{hi}$ for all $(h,i)\in \{1,...,H\}\times \{1,...,m_{hi}\}$.

The following result is useful for any sample of polytomous logistic regression with complex 