}(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) \}\{\boldsymbol{y}_{hij}-\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\right] \otimes \boldsymbol{x}_{hi}.\end{aligned}$$Hence, it holds$$\sqrt{m}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}-\boldsymbol{\beta }_{0})\overset{\mathcal{L}}{\underset{m\mathcal{\rightarrow }\infty }{\longrightarrow }}\mathcal{N}\left( \boldsymbol{0}_{d(k+1)},\mathbf{\Psi}_{\lambda }^{-1}\left( \boldsymbol{\beta }_{0}\right) \mathbf{\Omega}_{\lambda
}\left( \boldsymbol{\beta }_{0}\right) \mathbf{\Psi}_{\lambda }^{-1}\left( 
\boldsymbol{\beta }_{0}\right) \right) ,$$with$$\begin{aligned}
\mathbf{\Omega}_{\lambda }\left( \boldsymbol{\beta }\right) & =\lim_{m\rightarrow
\infty }\left\{ 
\begin{array}{lc}
\begin{array}{l}
\frac{1}{m}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}{m}_{hi}w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) ) \\ 
\times \mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T},\end{array}
& \lambda >0 \\ 
\frac{1}{m}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}w_{hi}^{2}m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}^{\ast }\left( \boldsymbol{\beta }\right)
)\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & \lambda =0\end{array}\right.  \\
\mathbf{\Psi}_{\lambda }\left( \boldsymbol{\beta }\right) & =\lim_{m\rightarrow
\infty }\left\{ 
\begin{array}{lc}
\frac{1}{m}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}w_{hi}m_{hi}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right)
\}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & 
\lambda >0 \\ 
\frac{1}{m}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}w_{hi}m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}^{\ast }\left( \boldsymbol{\beta }\right) )\otimes 
\boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & \lambda =0\end{array}\right. .\end{aligned}$$The formal proof is omitted, but the derivation of the expressions is almost the same considering $\boldsymbol{U}_{\lambda }(\boldsymbol{\beta },\boldsymbol{x}_{hi})={\sum\limits_{j=1}^{m_{hi}}}\boldsymbol{V}_{j,\lambda }(\boldsymbol{\beta },\boldsymbol{x}_{hi})$, with $\boldsymbol{V}_{j,\lambda }(\boldsymbol{\beta },\boldsymbol{x}_{hi})$ i.i.d. random variables $j=1,...,m_{hi}$. This idea matches the philosophy of the asymptotic result developed in Castilla et al. (2019), where for $H=1$ and $w_{1i}=1$, $i=1,...,m_{1i}$.

The following result is useful for any sample of polytomous logistic 