Appendix
Proof of Theorem 2
From Theorem 1.2.18 of Muirhead (1982) and Theorem 3.15 of Gupta et al. (2013), we get
$$\begin{aligned} \sqrt{n}\left( \begin{array}{c} \hat{\varvec{\mu }}-\varvec{\mu }\\ \text{ vech }(\hat{\varvec{\Sigma }})-\text{ vech }(\varvec{\Sigma })\\ \end{array} \right) \mathop {\longrightarrow }\limits ^{d} \mathcal {N}\left( \mathbf {0},\mathbf {\Omega }\right) \end{aligned}$$
as \(n \longrightarrow \infty \) where the symbol \(\text{ vech }\) stands for the vech operator, i.e., \(\text{ vech }(\mathbf {A})=(a_{11}, \ldots ,a_{k1}, \ldots ,\)
\(a_{ii}, \ldots ,a_{ki}, \ldots a_{kk})^\prime \) for an arbitrary symmetric matrix \(\mathbf {A}=(a_{ij})\), and
$$\begin{aligned} \mathbf {\Omega }=\left( \begin{array}{ll} \varvec{\Sigma }&{}\quad \mathbf {0} \\ \mathbf {0} &{}\quad \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \\ \end{array} \right) . \end{aligned}$$
In the following we make use of the operator vec defined by \(\text{ vec }(\mathbf {A})=(a_{11}, \ldots ,a_{k1}, \ldots ,a_{1i}, \ldots ,a_{ki},\)
\(a_{1k}, \ldots ,a_{kk})^\prime \). The symbol \(\mathbf {I}_{k^2}\) denotes the identity matrix of order \(k^2\); \(\mathbf {K}_k\) is a commutation matrix; \(\mathbf {D}_k\) is a \(k^2\times k(k+1)/2\) duplication matrix such that \(\mathbf {D}_k \text{ vech }(\mathbf {A})= \text{ vec }(\mathbf {A})\) and \(\mathbf {D}_k^+=(\mathbf {D}_k^\prime \mathbf {D}_k)^{-1}\mathbf {D}_k^\prime \) with the property \(\mathbf {D}_k^+ \text{ vec }(\mathbf {A})= \text{ vech }(\mathbf {A})\) (cf., Harville 1997). Finally, the symbol \(\otimes \) denotes the Kronecker product.
Next, we derive the asymptotic distribution of \(\hat{R}_{\mathrm{GMV}}\), \(\hat{V}_{\mathrm{GMV}}\), and \(\hat{s}\) which is further used in the proof of the theorem. From the proof of Theorem 1 in Bodnar et al. (2009), we get
$$\begin{aligned} \frac{\partial R_{\mathrm{GMV}}}{\partial \varvec{\mu }}= & {} \frac{\varvec{\Sigma }^{-1}\mathbf {1}}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}},\\ \frac{\partial V_{\mathrm{GMV}}}{\partial \varvec{\mu }}= & {} \mathbf {0},\\ \frac{\partial s}{\partial \varvec{\mu }}= & {} 2\mathbf {Q}\varvec{\mu },\\ \frac{\partial R_{\mathrm{GMV}}}{\partial \text{ vech }(\varvec{\Sigma })}= & {} \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }} \left( V_{\mathrm{GMV}}(\mathbf {1}\otimes \varvec{\mu }) -R_{\mathrm{GMV}}V_{\mathrm{GMV}}(\mathbf {1}\otimes \mathbf {1})\right) ,\\ \frac{\partial V_{\mathrm{GMV}}}{\partial \text{ vech }(\varvec{\Sigma })}= & {} -V_{\mathrm{GMV}}^{-2}\frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}(\mathbf {1}\otimes \mathbf {1}),\\ \frac{\partial s}{\partial \text{ vech }(\varvec{\Sigma })}= & {} -\frac{\partial (\mathrm{vec}\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}(R_{\mathrm{GMV}}^2 (\mathbf {1}\otimes \mathbf {1})-2R_{\mathrm{GMV}}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })), \end{aligned}$$
where \(\mathbf {Q}=\varvec{\Sigma }^{-1}-\frac{\varvec{\Sigma }^{-1}\mathbf {1}\mathbf {1}'\varvec{\Sigma }^{-1}}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}\) and (cf., Harville 1997, p. 368)
$$\begin{aligned} \frac{\partial (\text{ vec }(\varvec{\Sigma }^{-1}))^\prime }{\partial \text{ vech }\varvec{\Sigma }}=-\mathbf {D}^\prime _k(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1}){\mathbf {D}^{+}_k}^\prime \mathbf {D}^\prime _k. \end{aligned}$$
(30)
The application of the delta-method (cf., Theorem 3.7 in DasGupta 2008) leads to
$$\begin{aligned}\sqrt{n}\left( \left( \begin{array}{c} \hat{R}_{\mathrm{GMV}} \\ \hat{V}_{\mathrm{GMV}}\\ \hat{s}\end{array}\right) -\left( \begin{array}{c} R_{\mathrm{GMV}} \\ V_{\mathrm{GMV}}\\ s\end{array}\right) \right) \mathop {\rightarrow }\limits ^{d} \mathcal {N}\left( \left( \begin{array}{c} 0 \\ 0\\ 0\end{array}\right) ,\left( \begin{array}{ccc} \sigma ^2_1 &{} \sigma _{12} &{} \sigma _{13}\\ \sigma _{12} &{} \sigma ^2_2 &{} \sigma _{23} \\ \sigma _{13} &{} \sigma _{23} &{} \sigma ^2_3 \end{array}\right) \right) ,\end{aligned}$$
where
$$\begin{aligned} \sigma ^2_1= & {} ((\partial R_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial R_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))') \mathbf {\Omega }((\partial R_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial R_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))')^\prime ,\\ \sigma _{12}= & {} ((\partial R_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial R_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))') \mathbf {\Omega }((\partial V_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial V_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))')^\prime ,\\ \sigma _{13}= & {} ((\partial R_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial R_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))') \mathbf {\Omega }((\partial s/\partial \varvec{\mu })' \; (\partial s/\partial (\text{ vech }\varvec{\Sigma }))')^\prime ,\\ \sigma ^2_2= & {} ((\partial V_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial V_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))') \mathbf {\Omega }((\partial V_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial V_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))')^\prime ,\\ \sigma _{23}= & {} ((\partial V_{\mathrm{GMV}}/\partial \varvec{\mu })' \; (\partial V_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))') \mathbf {\Omega }((\partial s/\partial \varvec{\mu })' \; (\partial s/\partial (\text{ vech }\varvec{\Sigma }))')^\prime ,\\ \sigma ^2_3= & {} ((\partial s/\partial \varvec{\mu })' \; (\partial s/\partial (\text{ vech }\varvec{\Sigma }))') \mathbf {\Omega }((\partial s/\partial \varvec{\mu })' \; (\partial s/\partial (\text{ vech }\varvec{\Sigma }))')^\prime . \end{aligned}$$
We get that
$$\begin{aligned} \sigma ^2_1= & {} (\partial R_{\mathrm{GMV}}/\partial \varvec{\mu })'\varvec{\Sigma }(\partial R_{\mathrm{GMV}}/\partial \varvec{\mu })\\&+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} (\partial R_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))'\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime }(\partial R_{\mathrm{GMV}}/\partial (\text{ vech }\varvec{\Sigma }))\\= & {} \frac{1}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_{\mathrm{GMV}}(\mathbf {1}\otimes \varvec{\mu }) -R_{\mathrm{GMV}}V_{\mathrm{GMV}}(\mathbf {1}\otimes \mathbf {1})\right) ' \left( \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\right) '\\&\times \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }} \left( V_{\mathrm{GMV}}(\mathbf {1}\otimes \varvec{\mu }) -R_{\mathrm{GMV}}V_{\mathrm{GMV}}(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} V_{\mathrm{GMV}}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_{\mathrm{GMV}}(\mathbf {1}\otimes \varvec{\mu }) -R_{\mathrm{GMV}}V_{\mathrm{GMV}}(\mathbf {1}\otimes \mathbf {1})\right) '\\&\times \mathbf {D}_k{\mathbf {D}^{+}_k}(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\mathbf {D}_k\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \mathbf {D}^\prime _k(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1}){\mathbf {D}^{+}_k}^\prime \mathbf {D}^\prime _k\\&\times \left( V_{\mathrm{GMV}}(\mathbf {1}\otimes \varvec{\mu }) -R_{\mathrm{GMV}}V_{\mathrm{GMV}}(\mathbf {1}\otimes \mathbf {1})\right) . \end{aligned}$$
Let \(\mathbf {N}_{k}=\mathbf {D}_k{\mathbf {D}^{+}_k}\). Taking into account that for arbitrary \(k\times k\) matrix \(\mathbf {A}\) the following equalities hold:
$$\begin{aligned} \mathbf {N}_{k}= & {} \mathbf {D}_k{\mathbf {D}^{+}_k}=\frac{1}{2}(\mathbf {I}_{k^2}+\mathbf {K}_k),\\ \mathbf {N}_{k}(\mathbf {A}\otimes \mathbf {A})= & {} (\mathbf {A}\otimes \mathbf {A})\mathbf {N}_{k},\\ \mathbf {D}_k{\mathbf {D}^{+}_k}(\mathbf {A}\otimes \mathbf {A})\mathbf {D}_k= & {} (\mathbf {A}\otimes \mathbf {A})\mathbf {D}_k,\\ \mathbf {N}_{k}= & {} \mathbf {N}_{k}^2=\mathbf {N}_{k}^\prime \end{aligned}$$
we get
$$\begin{aligned}&\mathbf {D}_k{\mathbf {D}^{+}_k}(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\mathbf {D}_k\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k)(\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \mathbf {D}^\prime _k(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1}){\mathbf {D}^{+}_k}^\prime \mathbf {D}^\prime _k\\&\quad =2(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\mathbf {N}_{k}(\varvec{\Sigma }\otimes \varvec{\Sigma })(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})=2\mathbf {N}_{k}(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\\&\quad =(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k). \end{aligned}$$
It implies that
$$\begin{aligned} \sigma ^2_1= & {} V_{\mathrm{GMV}}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_\mathrm{GMV}(\mathbf {1}'\otimes \varvec{\mu }')-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}'\otimes \mathbf {1}')\right) \\&\times (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k)\left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) -R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} V_\mathrm{GMV}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_\mathrm{GMV}(\mathbf {1}'\otimes \varvec{\mu }')-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}'\otimes \mathbf {1}')\right) \\&\times (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu })-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \\&+\,\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_\mathrm{GMV}(\mathbf {1}'\otimes \varvec{\mu }')-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}'\otimes \mathbf {1}')\right) \\&\times (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\left( V_\mathrm{GMV}(\varvec{\mu }\otimes \mathbf {1}) -R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} V_\mathrm{GMV}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\big (V_\mathrm{GMV}^2\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }+R_\mathrm{GMV}^2V_\mathrm{GMV}^2(\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1})^2\\&-\,2R_\mathrm{GMV}V_\mathrm{GMV}^2\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\varvec{\mu }'\varvec{\Sigma }^{-1}\mathbf {1}\big )\\&+\,\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\big (V_\mathrm{GMV}^2(\varvec{\mu }'\varvec{\Sigma }^{-1}\mathbf {1})^2+R_\mathrm{GMV}^2V_\mathrm{GMV}^2(\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1})^2\\&-\,2R_\mathrm{GMV}V_\mathrm{GMV}^2\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\varvec{\mu }'\varvec{\Sigma }^{-1}\mathbf {1}\big )\\= & {} V_\mathrm{GMV}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( \frac{\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}-\left( \frac{\varvec{\mu }'\varvec{\Sigma }^{-1}\mathbf {1}}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}\right) ^2\right) \\= & {} V_\mathrm{GMV}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}sV_\mathrm{GMV}, \end{aligned}$$
where we use that the equality \(\mathbf {K}_k(\mathbf {A}\otimes \mathbf {a})=(\mathbf {a}\otimes \mathbf {A})\) holds for an arbitrary \(k\times n\) matrix \(\mathbf {A}\) and an arbitrary \(k\times 1\) vector \(\mathbf {a}\).
Analogically, for the quantities \(\sigma _{12}\), \(\sigma _{13}\), \(\sigma _{23}\), \(\sigma ^2_2\), and \(\sigma ^2_3\), we get
$$\begin{aligned} \sigma ^2_2= & {} (\partial V_\mathrm{GMV}/\partial \varvec{\mu })'\varvec{\Sigma }(\partial V_\mathrm{GMV}/\partial \varvec{\mu })\\&+\,\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} (\partial V_\mathrm{GMV}/\partial (\text{ vech }\varvec{\Sigma }))'\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime }(\partial V_\mathrm{GMV}/\partial (\text{ vech }\varvec{\Sigma }))\\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) ' \left( \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\right) '\\&\times \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }} \left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( -V_\mathrm{GMV}^2(\mathbf {1}'\otimes \mathbf {1}')\right) (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k)\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}2V_\mathrm{GMV}^4\left( \mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\right) ^2=\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}2V_\mathrm{GMV}^2, \end{aligned}$$
$$\begin{aligned} \sigma ^2_3= & {} (\partial s/\partial \varvec{\mu })'\varvec{\Sigma }(\partial s/\partial \varvec{\mu })\\&+\,\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} (\partial s/\partial (\text{ vech }\varvec{\Sigma }))'\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime }(\partial s/\partial (\text{ vech }\varvec{\Sigma }))\\= & {} 4\varvec{\mu }'\mathbf {Q}'\varvec{\Sigma }\mathbf {Q}\varvec{\mu }+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) '\\&\times \,\left( \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\right) '\\&\times \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\nonumber \\&\times \left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} 4s+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) '\\&\times (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k)\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} 4s+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) '\\&\times (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\varvec{\mu }\otimes \mathbf {1}) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\&+\, \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) '\\&\times (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} 4s+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (2R_\mathrm{GMV}^4(\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1})^2-8R_\mathrm{GMV}^3\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }\\&+\,8R_\mathrm{GMV}^2(\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu })^2\\&+\,4R_\mathrm{GMV}^2\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }-8R_\mathrm{GMV}\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }+2(\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu })^2\Big )\\= & {} 4s+2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }-R_\mathrm{GMV}^2/V_\mathrm{GMV}\Big )^2 =4s+2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}s, \end{aligned}$$
$$\begin{aligned} \sigma _{12}= & {} (\partial V_\mathrm{GMV}/\partial \varvec{\mu })'\varvec{\Sigma }(\partial R_\mathrm{GMV}/\partial \varvec{\mu })\\&+\,\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} (\partial V_\mathrm{GMV}/\partial (\text{ vech }\varvec{\Sigma }))'\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime }(\partial R_\mathrm{GMV}/\partial (\text{ vech }\varvec{\Sigma }))\\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) ' \left( \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\right) '\\&\times \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }} \left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu })-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k)\\&\times \left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu })-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu })\right. \nonumber \\&\left. -\,R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \\&+\,\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\left( V_\mathrm{GMV}(\varvec{\mu }\otimes \mathbf {1})-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) \Big )\\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (-2V_\mathrm{GMV}^3\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }+2R_\mathrm{GMV}V_\mathrm{GMV}^3(\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1})^2\Big )=0, \end{aligned}$$
$$\begin{aligned} \sigma _{13}= & {} (\partial R_\mathrm{GMV}/\partial \varvec{\mu })'\varvec{\Sigma }(\partial s/\partial \varvec{\mu })\\&+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} (\partial R_\mathrm{GMV}/\partial (\text{ vech }\varvec{\Sigma }))'\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime }(\partial s/\partial (\text{ vech }\varvec{\Sigma }))\\= & {} \left( 2\mathbf {Q}\varvec{\mu }\right) '\varvec{\Sigma }\frac{\varvec{\Sigma }^{-1}\mathbf {1}}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu })-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) '\nonumber \\&\times \left( \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\right) '\\&\times \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\nonumber \\&\left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} 2\left( \varvec{\Sigma }^{-1}\varvec{\mu }-\varvec{\Sigma }^{-1}\mathbf {1}\left( \frac{\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}\right) \right) '\varvec{\Sigma }\frac{\varvec{\Sigma }^{-1}\mathbf {1}}{\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}}\\&+\,\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( V_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu })-R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k)\\&\times \left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} 0+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (2R_\mathrm{GMV}^2V_\mathrm{GMV}\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }-2R_\mathrm{GMV}V_\mathrm{GMV}\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }\\&-\,2R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu })^2+2V_\mathrm{GMV}\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }\varvec{\mu }'\varvec{\Sigma }^{-1}\varvec{\mu }-2R_\mathrm{GMV}^3V_\mathrm{GMV}(\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1})^2\\&+\,4R_\mathrm{GMV}^2V_\mathrm{GMV}\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }-2R_\mathrm{GMV}V_\mathrm{GMV}(\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu })^2=0, \end{aligned}$$
and
$$\begin{aligned} \sigma _{23}= & {} (\partial V_\mathrm{GMV}/\partial \varvec{\mu })'\varvec{\Sigma }(\partial s/\partial \varvec{\mu })\\&+\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} (\partial V_\mathrm{GMV}/\partial (\text{ vech }\varvec{\Sigma }))'\mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime }(\partial s/\partial (\text{ vech }\varvec{\Sigma }))\\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) ' \left( \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\right) '\\&\times \mathbf {D}_k^+(\mathbf {I}_{k^2}+\mathbf {K}_k) (\varvec{\Sigma }\otimes \varvec{\Sigma })\mathbf {D}_k^{+ \, \prime } \frac{\partial (\text{ vec }\varvec{\Sigma }^{-1})'}{\partial \text{ vech }\varvec{\Sigma }}\\&\times \left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})(\mathbf {I}_{k^2}+\mathbf {K}_k)\\&\times \left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (\left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\\&\times \left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\mathbf {1}\otimes \varvec{\mu }) +(\varvec{\mu }\otimes \varvec{\mu })\right) \\&+\, \left( -V_\mathrm{GMV}^2(\mathbf {1}\otimes \mathbf {1})\right) '(\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\nonumber \\&\times \left( R_\mathrm{GMV}^2 (\mathbf {1}\otimes \mathbf {1})-2R_\mathrm{GMV}(\varvec{\mu }\otimes \mathbf {1}) +(\varvec{\mu }\otimes \varvec{\mu })\right) \Big )\\= & {} \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\Big (-2V_\mathrm{GMV}^2R_\mathrm{GMV}^2(\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1})^2\\&+\,4R_\mathrm{GMV}V_\mathrm{GMV}^2\mathbf {1}'\varvec{\Sigma }^{-1}\mathbf {1}\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu }-2V_\mathrm{GMV}^2(\mathbf {1}'\varvec{\Sigma }^{-1}\varvec{\mu })^2\Big )=0. \end{aligned}$$
Hence,
$$\begin{aligned} \sqrt{n}\left( \left( \begin{array}{c} \hat{R}_\mathrm{GMV} \\ \hat{V}_\mathrm{GMV} \\ \hat{s} \end{array}\right) - \left( \begin{array}{c} R_\mathrm{GMV} \\ V_\mathrm{GMV} \\ s \end{array}\right) \right) \mathop {\longrightarrow }\limits ^{d} \mathcal {N}\left( \mathbf {0},\varvec{\Xi }\right) \,. \end{aligned}$$
(31)
$$\begin{aligned} \varvec{\Xi }=\left( \begin{array}{ccc} V_\mathrm{GMV}(1+s\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}) &{} 0 &{} 0 \\ 0 &{} 2V_\mathrm{GMV}^2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} &{} 0 \\ 0 &{} 0 &{} 4s+2s^2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2} \\ \end{array} \right) \end{aligned}$$
-
(a)
Rewriting (17), we get
$$\begin{aligned} \hat{\alpha }_\mathrm{TP}= F \left( \gamma \sqrt{(\hat{\varvec{\mu }}-r_0\mathbf {1})^\prime \hat{\varvec{\Sigma }}^{-1} (\hat{\varvec{\mu }}-r_0\mathbf {1})}\right) =F \left( \gamma \sqrt{\hat{s}+\frac{(\hat{R}_\mathrm{GMV}-r_0)^2}{\hat{V}_\mathrm{GMV}}}\right) . \end{aligned}$$
The application of (31) and the delta method lead to
$$\begin{aligned} \sqrt{n}(\hat{\alpha }_{\mathrm{TP}}-\alpha _{\mathrm{TP}})\mathop {\longrightarrow }\limits ^{d} \mathcal {N}(0,\sigma ^2_{\alpha }) \end{aligned}$$
where
$$\begin{aligned} \sigma _{\alpha }^2= & {} \left( \frac{f\left( \gamma \sqrt{s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}}\right) }{2\sqrt{s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}}}\right) ^2 \gamma ^2\\&\qquad \times \left( 2\frac{(R_\mathrm{GMV}-r_0)}{V_\mathrm{GMV}},\,-\frac{(R_\mathrm{GMV}-r_0)^2}{V_\mathrm{GMV}^2},\,1\right) \varvec{\Xi } \left( \begin{array}{c} 2(R_\mathrm{GMV}-r_0)/V_\mathrm{GMV} \\ -(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}^2 \\ 1 \end{array}\right) \\= & {} \left( \frac{f\left( \gamma \sqrt{s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}}\right) }{2\sqrt{s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}}}\right) ^2\gamma ^2\\&\times \left( 4\frac{(R_\mathrm{GMV}-r_0)^2}{V_\mathrm{GMV}}\left( 1+s\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\right) +2\frac{(R_\mathrm{GMV}-r_0)^4}{V_\mathrm{GMV}^2}\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}+ 4s+2s^2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\right) \\= & {} \frac{2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\left( s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}\right) ^2+4\left( s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}\right) }{4\left( s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}\right) }\\&\times f^2\left( \gamma \sqrt{s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}}\right) \gamma ^2\\= & {} \left( 1+\frac{1}{2}\left( s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}\right) \frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\right) f^2\left( \gamma \sqrt{s+(R_\mathrm{GMV}-r_0)^2/V_\mathrm{GMV}}\right) \gamma ^2\\= & {} \left( 1+\frac{\psi ^{\prime \prime }(0)}{2\left( \psi ^{\prime }(0)\right) ^2}(\varvec{\mu }-r_0\mathbf {1})^\prime \varvec{\Sigma }^{-1}(\varvec{\mu }-r_0\mathbf {1})\right) f^2\left( \gamma \sqrt{(\varvec{\mu }-r_0\mathbf {1})^\prime \varvec{\Sigma }^{-1}(\varvec{\mu }-r_0\mathbf {1})}\right) \gamma ^2\,. \end{aligned}$$
-
(b)
Similarly, we get
$$\begin{aligned} \sqrt{n}(\hat{r}_0(\alpha _{\mathrm{TP}})-r_0(\alpha _{\mathrm{TP}}))\mathop {\longrightarrow }\limits ^{d} \mathcal {N}(0,\sigma ^2_{r_0}) \end{aligned}$$
where
$$\begin{aligned} \sigma ^2_{r_0}= & {} \left( 1,\,-\frac{\sqrt{\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2-s}}{2\sqrt{V_\mathrm{GMV}}},\,\frac{\sqrt{V_\mathrm{GMV}}}{2\sqrt{\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2-s}}\right) \varvec{\Xi } \left( \begin{array}{c} 1 \\ -\frac{\sqrt{\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2-s}}{2\sqrt{V_\mathrm{GMV}}} \\ \frac{\sqrt{V_\mathrm{GMV}}}{2\sqrt{\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2-s}} \end{array}\right) \\= & {} V_\mathrm{GMV}\left( 1+s\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\right) +\frac{\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2-s}{2}V_\mathrm{GMV}\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\\&+\left( 4s+2s^2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}\right) \frac{V_\mathrm{GMV}}{4(\gamma ^{-2}d_{1-\alpha _\mathrm{TP}}^2-s)}\\= & {} V_\mathrm{GMV}\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2\frac{2+\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2\frac{\psi ^{\prime \prime }(0)}{\left( \psi ^{\prime }(0)\right) ^2}}{2(\gamma ^{-2}d_{1-\alpha _{\mathrm{TP}}}^2-s)} \end{aligned}$$
The theorem is proved. \(\square \)
