Gradual change-point analysis based on Spearman matrices for multivariate time series

Abstract

It may happen that the behavior of a multivariate time series is such that the underlying joint distribution is gradually moving from one distribution to another between unknown times of change. Under this context of a possible gradual-change, tests of change-point detection in the dependence structure of multivariate series are developed around the associated sequence of Spearman matrices. It is formally established that the proposed test statistics for that purpose are asymptotically marginal-free under a general strong-mixing assumption, and written as functions of integrated Brownian bridges. Consistent estimators of the pair of times of change, as well as of the before-the-change and after-the-change Spearman matrices, are also proposed. A simulation study examines the sampling properties of the introduced tools, and the methodologies are illustrated on a synthetic dataset.

A Proofs of the main results

1.1 A.1: Proof of Proposition 1

From the definition of ${\mathscr {S}}_{\ell \ell '}^\textrm{Sp} = (\rho ^\textrm{Sp}_{\ell \ell ',1}, \ldots , \rho ^\textrm{Sp}_{\ell \ell ',T})$ and in view of (4), a direct computation yields

$$\begin{aligned} \varUpsilon _{\ell \ell '}^\textbf{k}= \langle \varvec{\textbf{w}}^\textbf{k}, \mathscr {S}_{\ell \ell '}^\textrm{Sp} \rangle= & {} {1\over T} \sum\nolimits_{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \rho ^\textrm{Sp}_{\ell \ell ',t} \\= & {} {1 \over T} \sum\nolimits_{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \left\{ \left( 1 - w_t^{\textbf{k}_0} \right) \rho ^\textrm{Sp}_{F_{\ell \ell '}} + w_t^{\textbf{k}_0} \rho ^\textrm{Sp}_{G_{\ell \ell '}} \right\} \\= & {} {1\over T} \sum\nolimits_{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \left\{ \rho ^\textrm{Sp}_{F_{\ell \ell '}} + w_t^{\textbf{k}_0} \left( \rho ^\textrm{Sp}_{G_{\ell \ell '}} - \rho ^\textrm{Sp}_{F_{\ell \ell '}} \right) \right\} \\= & {} \left( \rho ^\textrm{Sp}_{G_{\ell \ell '}} - \rho ^\textrm{Sp}_{F_{\ell \ell '}} \right) {1\over T} \sum\nolimits_{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) w_t^{\textbf{k}_0} \\= & {} \left( \rho ^\textrm{Sp}_{G_{\ell \ell '}} - \rho ^\textrm{Sp}_{F_{\ell \ell '}} \right) \left\langle \varvec{\textbf{w}}^\textbf{k}, \varvec{\textbf{w}}^{\textbf{k}_0} \right\rangle . \end{aligned}$$

It readily follows that $\varUpsilon ^\textbf{k}= (\varSigma _G^\textrm{Sp} - \varSigma _F^\textrm{Sp}) \, \langle \varvec{\textbf{w}}^\textbf{k}, \varvec{\textbf{w}}^{\textbf{k}_0} \rangle$. $\square$

1.2 A.2: Proof of Proposition 2

First recall that for any $\ell <\ell ' \in \{1,\ldots ,d\}$, one can associate $j \in \{ 1, \ldots , L \}$ via the rule $j = (\ell -1) d + \ell ' - {\ell +1 \atopwithdelims ()2}$. Letting ${{\widetilde{\textbf{u}}}}_j = (\textbf{1}_{\ell -1},u_\ell ,\textbf{1}_{\ell '-\ell -1},u_{\ell '},\textbf{1}_{d-\ell '})$, one can write

$$\begin{aligned} \sqrt{T} \, {\widehat{\varUpsilon }}^\textbf{k}_{\ell \ell '} = \sqrt{T} \, {\widehat{\varUpsilon }}^\textbf{k}_j= & {} {12\over \sqrt{T}} \sum\nolimits_{t=1}^T \left( \textbf{w}_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) (1-{\widehat{U}}_{t\ell }) (1-{\widehat{U}}_{t\ell '}) \nonumber \\= & {} {12\over \sqrt{T}} \sum\nolimits_{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \int _{[0,1]^2} \mathbb {I}\left( {\widehat{U}}_{t\ell } \le u_\ell , {\widehat{U}}_{t\ell '} \le u_{\ell '} \right) \textrm{d}u_\ell \textrm{d}u_{\ell '} \nonumber \\= & {} 12 \int _{[0,1]^2} \left\{ {1\over \sqrt{T}} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \mathbb {I}\left( {\widehat{\textbf{U}}}_t \le {{\widetilde{\textbf{u}}}}_j \right) \right\} \textrm{d}u_\ell \textrm{d}u_{\ell '} \nonumber \\= & {} 12 \int _{[0,1]^2} {\widehat{\mathbb {L}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_j) \, \textrm{d}u_\ell \textrm{d}u_{\ell '}, \end{aligned}$$

(13)

where ${\widehat{\mathbb {L}}}^\textbf{k}$ is the empirical process defined for $\textbf{u}\in [0,1]^d$ by

$$\begin{aligned} {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) = {1\over \sqrt{T}} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \, \mathbb {I}\left( {\widehat{\textbf{U}}}_t \le \textbf{u}\right) . \end{aligned}$$

It will now be shown that ${\widehat{\mathbb {L}}}^\textbf{k}$ is asymptotically equivalent to

$$\begin{aligned} {{\widetilde{\mathbb {L}}}}^\textbf{k}(\textbf{u}) = {1\over \sqrt{T}} \sum\nolimits_{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \, \mathbb {I}\left( \textbf{U}_t \le \textbf{u}\right) . \end{aligned}$$

Letting $\widehat{\textbf{F}}^{-1}(\textbf{u}) = ({\widehat{F}}_1^{-1}(u_1), \ldots , {\widehat{F}}_d^{-1}(u_d))$, one can write

$$\begin{aligned} \sup _{\textbf{u}\in [0,1]^d} \left| {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\textbf{u}) \right| \le \sup _{\textbf{u}\in [0,1]^d} \left| {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\widehat{\textbf{F}}^{-1}(\textbf{u})) \right| + \sup _{\textbf{u}\in [0,1]^d} \left| {{\widetilde{\mathbb {L}}}}^\textbf{k}(\widehat{\textbf{F}}^{-1}(\textbf{u})) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\textbf{u}) \right| . \end{aligned}$$

(14)

Under the assumption that the $\alpha$-mixing coefficients of $(\textbf{U}_t)_{t\in \mathbb {Z}}$ are such that $\alpha (r) = O(r^{-4-d(1+\epsilon )})$ for some $\epsilon \in (0,1/4]$, one can invoke Proposition 1 of Quessy (2019) and conclude that ${{\widetilde{\mathbb {L}}}}^\textbf{k}$ converges weakly in the space $\varDelta \times \ell ^\infty ([0,1]^d)$ to a centered Gaussian process $\mathbb {L}^\textbf{k}$ such that for $(\textbf{k},\textbf{u}),(\textbf{k}',\textbf{u}') \in \varDelta \times [0,1]^d$,

$$\begin{aligned} \textrm{E}\left\{ \mathbb {L}^\textbf{k}(\textbf{u}) \, \mathbb {L}^{\textbf{k}'}(\textbf{u}') \right\} = \varLambda (\textbf{k},\textbf{k}') \, \sum\nolimits_{t\in \mathbb {Z}} \textrm{cov}\left\{ \mathbb {I}(\textbf{U}_0 \le \textbf{u}), \mathbb {I}(\textbf{U}_t \le \textbf{u}') \right\} , \end{aligned}$$

where for $\textbf{k}= (k_1,k_2)$ and $\textbf{k}' = (k_1',k_2')$,

$$\begin{aligned} \varLambda (\textbf{k},\textbf{k}') = \int _{k_1}^{k_2} \int _{k_1'}^{k_2'} \left\{ \min (s,s') - s s' \over (k_2-k_1) (k_2'-k_1') \right\} \textrm{d}s \, \textrm{d}s'. \end{aligned}$$

Since $\sqrt{T}(\widehat{\textbf{F}}^{-1}(\textbf{u})-\textbf{u})$ converges weakly and ${{\widetilde{\mathbb {L}}}}^\textbf{k}$ is asymptotically continuous, the second expression on the right of inequality (14) converges in probability to zero. To deal with the first expression on the right of inequality (14), note that

$$\begin{aligned} {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\widehat{\textbf{F}}^{-1}(\textbf{u})) = {1\over \sqrt{T}} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \left\{ \mathbb {I}\left( \widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}\right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}) \right) \right\} . \end{aligned}$$

Invoking Lemma 1 in Quessy (2019) that establishes that

$$\begin{aligned} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) a_t = {1 \over K_2-K_1} \sum\nolimits_{j=K_1}^{K_2-1} \left( \sum\nolimits_{t=1}^j a_t - {j\over T} \sum\nolimits _{t=1}^T a_t \right) , \end{aligned}$$

where $K_1 = \lfloor T k_1 \rfloor$ and $K_2 = \lfloor T k_2 \rfloor$, one has

$$\begin{aligned} \left| {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\widehat{\textbf{F}}^{-1}(\textbf{u})) \right|= & {} {1 \over \sqrt{T} (K_2-K_1)} \left| \sum\nolimits_{j=K_1}^{K_2-1} \left[ \sum\nolimits_{t=1}^j \left\{ \mathbb {I}\left( \widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}\right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}) \right) \right\} \right. \right. \\{} & {} \left. \left. - \, {j\over T} \sum\nolimits_{t=1}^T \left\{ \mathbb {I}\left( \widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}\right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}) \right) \right\} \right] \right| \\\le & {} {1 \over \sqrt{T} (K_2-K_1)} \sum\nolimits_{j=K_1}^{K_2-1} \left| \sum\nolimits_{t=1}^j \left\{ \mathbb {I}\left( \widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}\right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}) \right) \right\} \right| \\{} & {} + \, {1 \over \sqrt{T} (K_2-K_1)} \sum\nolimits_{j=K_1}^{K_2-1} {j\over T} \left| \sum\nolimits _{t=1}^T \left\{ \mathbb {I}\left( \widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}\right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}) \right) \right\} \right| . \end{aligned}$$

Now proceeding as in Fermanian et al. (2004), define the grid $G_T = \{(i_1,...,i_d)/T: i_1, \ldots , i_d \in \{1,\ldots ,T\} \}$ and observe that $\mathbb {I}(\widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}) = \mathbb {I}(\textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}))$ when $\textbf{u}\in G_T$. From the fact that

$$\begin{aligned} \left| \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{s}) \right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{s}-\textbf{1}/T) \right) \right|= & {} \left| \sum\nolimits_{\ell =1}^d \prod _{\ell '=1}^{\ell -1} \mathbb {I}\left( U_{t\ell '} \le {\widehat{F}}_{\ell '}^{-1}(s_{\ell '}) \right) \prod _{\ell '=\ell +1}^d \mathbb {I}\left( U_{t\ell '} \le {\widehat{F}}_{\ell '}^{-1}(s_{\ell '}-1/T) \right) \right. \\{} & {} \left. \times \, \left\{ \mathbb {I}\left( U_{t\ell } \le {\widehat{F}}_{\ell }^{-1}(s_\ell ) \right) - \mathbb {I}\left( U_{t\ell } \le {\widehat{F}}_{\ell }^{-1}(s_\ell -1/T) \right) \right\} \right| \\\le & {} \sum\nolimits_{\ell =1}^d \left| \mathbb {I}\left( U_{t\ell } \le \widehat{F}_{\ell }^{-1}(s_\ell ) \right) - \mathbb {I}\left( U_{t\ell } \le \widehat{F}_{\ell }^{-1}(s_\ell -1/T) \right) \right| , \end{aligned}$$

one can write

$$\begin{aligned} \sup _{\textbf{u}\in [0,1]^d} \left| \sum\nolimits_{t=1}^j \left\{ \mathbb {I}\left( \widehat{\textbf{F}}(\textbf{U}_t) \le \textbf{u}\right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{u}) \right) \right\} \right|\le & {} \max _{\textbf{s} \in G_T} \left| \sum\nolimits_{t=1}^j \left\{ \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{s}) \right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{s}-\textbf{1}/T) \right) \right\} \right| \\\le & {} \max _{\textbf{s} \in G_T} \sum\nolimits_{t=1}^j \left| \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{s}) \right) - \mathbb {I}\left( \textbf{U}_t \le \widehat{\textbf{F}}^{-1}(\textbf{s}-\textbf{1}/T) \right) \right| \\\le & {} d. \end{aligned}$$

It follows that

$$\begin{aligned} \sup _{\textbf{u}\in [0,1]^d} \left| {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\widehat{\textbf{F}}^{-1}(\textbf{u})) \right|\le & {} {d\over \sqrt{T}(K_2-K_1)} \sum\nolimits_{j=K_1}^{K_2-1} \left( 1 + {j\over T} \right) \\= & {} {d\over \sqrt{T}} \left( 1 + { K_1+K_2-1 \over 2 T } \right) \approx {d\over \sqrt{T}} \left( 1 + { k_1 + k_2 \over 2 } \right) . \end{aligned}$$

This shows that as $T\rightarrow \infty$

$$\begin{aligned} \sup _{\textbf{u}\in [0,1]^d} \left| {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) - {{\widetilde{\mathbb {L}}}}^\textbf{k}(\textbf{u}) \right| {\mathop {\longrightarrow }\limits ^{\mathbb {P}}}0. \end{aligned}$$

As a consequence, referring back to (13),

$$\begin{aligned} \sqrt{T} \, {\widehat{\varUpsilon }}^\textbf{k}_{\ell \ell '} = \sqrt{T} \, {\widehat{\varUpsilon }}^\textbf{k}_j = 12 \int _{[0,1]^2} {{\widetilde{\mathbb {L}}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_j) \, \textrm{d}u_\ell \textrm{d}u_{\ell '} + o_\mathbb {P}(1). \end{aligned}$$

One can then write

$$\begin{aligned} { \sqrt{T} \, \mathscr {V}\left( {\widehat{\varUpsilon }}^\textbf{k}\right) \over 12 } = \left( \int _{[0,1]^2} {{\widetilde{\mathbb {L}}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_1) \, \textrm{d}u_1 \textrm{d}u_2, \ldots , \int _{[0,1]^2} {{\widetilde{\mathbb {L}}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_L) \, \textrm{d}u_{d-1} \textrm{d}u_d \right) + o_\mathbb {P}(1). \end{aligned}$$

An application of the Continuous mapping Theorem then ensures that $\sqrt{T} \, \mathscr {V}({\widehat{\varUpsilon }}^\textbf{k}) / 12$ converges weakly to a vector of processes of the form $\mathbb {V}(\textbf{k}) = \left( \mathbb {V}_1(\textbf{k}), \ldots , \mathbb {V}_L(\textbf{k}) \right)$, where

$$\begin{aligned} \mathbb {V}_j(\textbf{k}) = \int\nolimits_{[0,1]^2} \mathbb {L}^\textbf{k}({{\widetilde{\textbf{u}}}}_j) \, \textrm{d}u_\ell \, \textrm{d}u_{\ell '}, \quad j \in \{1, \ldots , L\}. \end{aligned}$$

Therefore, $\mathbb {V}$ is a vector of centered Gaussian processes such that for $j' = (m-1) d + m' - {m+1 \atopwithdelims ()2}$ associated to $m<m' \in \{1,\ldots ,d\}$, its covariance structure is managed for $\textbf{k}, \textbf{k}' \in \varDelta$ by

$$\begin{aligned} \textrm{E}\left\{ \mathbb {V}_j(\textbf{k}) \, \mathbb {V}_{j'}(\textbf{k}') \right\}= & {} \int\nolimits_{[0,1]^4} \textrm{E}\left\{ \mathbb {L}^\textbf{k}({{\widetilde{\textbf{u}}}}_j) \, \mathbb {L}^{\textbf{k}'}({{\widetilde{\textbf{u}}}}_{j'}) \right\} \textrm{d}u_\ell \, \textrm{d}u_{\ell '} \, \textrm{d}u_m \, \textrm{d}u_{m'} \\= & {} \varLambda (\textbf{k},\textbf{k}') \int\nolimits_{[0,1]^4} \sum\nolimits _{t\in \mathbb {Z}} \textrm{cov}\left\{ \mathbb {I}(\textbf{U}_0 \le {{\widetilde{\textbf{u}}}}_j), \mathbb {I}(\textbf{U}_t \le {{\widetilde{\textbf{u}}}}_{j'}) \right\} \, \textrm{d}u_\ell \, \textrm{d}u_{\ell '} \, \textrm{d}u_m \, \textrm{d}u_{m'} \\= & {} \varLambda (\textbf{k},\textbf{k}') \sum\nolimits _{t\in \mathbb {Z}} \textrm{cov}\left\{ \int _{[0,1]^2} \mathbb {I}(\textbf{U}_0 \le {{\widetilde{\textbf{u}}}}_j) \,\right. \\{} & {} \left. \textrm{d}u_\ell \, \textrm{d}u_{\ell '}, \int _{[0,1]^2} \mathbb {I}(\textbf{U}_t \le {{\widetilde{\textbf{u}}}}_{j'}) \, \textrm{d}u_m \, \textrm{d}u_{m'} \right\} \\= & {} \varLambda (\textbf{k},\textbf{k}') \sum\nolimits_{t\in \mathbb {Z}} \textrm{cov}\left\{ (1-U_{0\ell })(1-U_{0\ell '}), (1-U_{tm})(1-U_{tm'}) \right\} \\= & {} \varLambda (\textbf{k},\textbf{k}') \, \varOmega _{jj'}. \end{aligned}$$

Hence, $\textrm{E}\{ \mathbb {V}(\textbf{k})^\top \mathbb {V}(\textbf{k}') \} = \varLambda (\textbf{k},\textbf{k}') \, \varOmega$ and

$$\begin{aligned} \textrm{E}\left\{ \left( \mathbb {V}(\textbf{k}) \, \varOmega ^{-1/2} \right) ^\top \left( \mathbb {V}(\textbf{k}') \, \varOmega ^{-1/2} \right) \right\} = \varOmega ^{-1/2} \, \textrm{E}\left\{ \mathbb {V}(\textbf{k})^\top \mathbb {V}(\textbf{k}') \right\} \varOmega ^{-1/2} = \varLambda (\textbf{k},\textbf{k}') \, I_L. \end{aligned}$$

Since the covariance function of the integrated Brownian bridge is $\textrm{E}\{ {{\widetilde{\mathbb {B}}}}(\textbf{k}) \, {{\widetilde{\mathbb {B}}}}(\textbf{k}') \} = \varLambda (\textbf{k},\textbf{k}')$, the covariance structure $\varLambda (\textbf{k},\textbf{k}') \, I_L$ is the same as that of a vector $({{\widetilde{\mathbb {B}}}}_1(\textbf{k}), \ldots , {{\widetilde{\mathbb {B}}}}_L(\textbf{k}))$ of independent integrated Brownian bridges. It follows that $\sqrt{T} \, \mathscr {V}({\widehat{\varUpsilon }}^\textbf{k}) \, \varOmega ^{-1/2} / 12$ converges weakly to $({{\widetilde{\mathbb {B}}}}_1(\textbf{k}), \ldots , {{\widetilde{\mathbb {B}}}}_L(\textbf{k}))$. $\square$

1.3 A.3: Proof of Proposition 3

From the conclusion of Proposition 2 and the Continuous mapping Theorem,

$$\begin{aligned} \sqrt{T} \, S_{T,p} = \sup _{\textbf{k}\in \varDelta } \eta (\textbf{k}) \left\| \sqrt{T} \, \mathscr {V}({\widehat{\varUpsilon }}^\textbf{k}) \, \varOmega ^{-1/2} \over 12 \right\| _p \rightsquigarrow \sup _{\textbf{k}\in \varDelta } \eta (\textbf{k}) \left\| \left( {{\widetilde{\mathbb {B}}}}_1(\textbf{k}), \ldots , {{\widetilde{\mathbb {B}}}}_L(\textbf{k}) \right) \right\| _p, \end{aligned}$$

which completes the proof. $\square$

1.4 A.4: Proof of Proposition 4

The first step of the proof consists to write

$$\begin{aligned} {\widehat{\varUpsilon }}^\textbf{k}_{\ell \ell '}= & {} 12 \int _{[0,1]^2} \left[ {1\over T} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \left\{ \mathbb {I}({\widehat{\textbf{U}}}_t \le {{\widetilde{\textbf{u}}}}_j) - \textrm{E}\left( \mathbb {I}(\textbf{U}_t \le \textbf{u}) \right) + \textrm{E}\left( \mathbb {I}(\textbf{U}_t \le \textbf{u}) \right) \right\} \right] \textrm{d}u_\ell \textrm{d}u_{\ell '} \\= & {} {12 \over \sqrt{T}} \int _{[0,1]^2} {\widehat{\mathbb {L}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_j) \, \textrm{d}u_\ell \textrm{d}u_{\ell '} + 12 \, \langle \textbf{w}^\textbf{k}, \textbf{w}^{\textbf{k}_0} \rangle \left\{ (\varSigma _G)_{\ell \ell '} - (\varSigma _F)_{\ell \ell '} \right\} , \end{aligned}$$

where ${\widehat{\mathbb {L}}}^\textbf{k}$ is the empirical process defined for $\textbf{u}\in [0,1]^d$ by

$$\begin{aligned} {\widehat{\mathbb {L}}}^\textbf{k}(\textbf{u}) = {1\over \sqrt{T}} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \left\{ \mathbb {I}({\widehat{\textbf{U}}}_t \le \textbf{u}) - \textrm{E}\left( \mathbb {I}(\textbf{U}_t \le \textbf{u}) \right) \right\} . \end{aligned}$$

By arguments similar as those in the proof of Proposition 2, ${\widehat{\mathbb {L}}}^\textbf{k}$ is asymptotically equivalent to

$$\begin{aligned} {{\widetilde{\mathbb {L}}}}^\textbf{k}(\textbf{u}) = {1\over \sqrt{T}} \sum\nolimits _{t=1}^T \left( w_t^\textbf{k}- {\bar{w}}^\textbf{k}\right) \left\{ \mathbb {I}(\textbf{U}_t \le \textbf{u}) - \textrm{E}\left( \mathbb {I}(\textbf{U}_t \le \textbf{u}) \right) \right\} . \end{aligned}$$

It is easy to see that ${{\widetilde{\mathbb {L}}}}^\textbf{k}$ converges weakly to a non-degenerate centered Gaussian process whose covariance structure is characterized by the finite-dimensional structure of ${{\widetilde{\mathbb {L}}}}^\textbf{k}$. As a consequence,

$$\begin{aligned} \mathscr {V}({\widehat{\varUpsilon }}^\textbf{k})= & {} {12 \over \sqrt{T}} \left( \int _{[0,1]^2} {{\widetilde{\mathbb {L}}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_1) \, \textrm{d}u_1 \textrm{d}u_2, \ldots , \int _{[0,1]^2} {{\widetilde{\mathbb {L}}}}^\textbf{k}({{\widetilde{\textbf{u}}}}_L) \, \textrm{d}u_{d-1} \textrm{d}u_d \right) \\{} & {} + \, 12 \, \left\langle \textbf{w}^\textbf{k}, \textbf{w}^{\textbf{k}_0} \right\rangle \, \left\{ \mathscr {V}\left( \varSigma _G^\textrm{Sp} - \varSigma _F^\textrm{Sp} \right) \right\} + o_\mathbb {P}(1/\sqrt{T}). \end{aligned}$$

Since $\varSigma _F^\textrm{Sp} \ne \varSigma _G^\textrm{Sp}$, one has in probability that

$$\begin{aligned} \left\| \mathscr {V}({\widehat{\varUpsilon }}^\textbf{k}) \right\| \longrightarrow 12 \, \varLambda (\textbf{k},\textbf{k}_0) \, \left\| \mathscr {V}\left( \varSigma _G^\textrm{Sp} - \varSigma _F^\textrm{Sp} \right) \right\| > 0, \end{aligned}$$

where $\varLambda (\textbf{k},\textbf{k}_0) = \lim _{T\rightarrow \infty } \langle \textbf{w}^\textbf{k}, \textbf{w}^{\textbf{k}_0} \rangle$. Invoking the argmax continuous mapping Theorem (see Theorem 3.2.2 of van der Vaart and Wellner 1996), one can conclude that in probability,

$$\begin{aligned} {\widehat{\textbf{k}}}_0 = \underset{\varvec{\textbf{k}}\in \varDelta ^\star }{\textrm{argsup}} \, { \Vert \mathscr {V}({\widehat{\varUpsilon }}^\textbf{k}) \Vert \over \langle \textbf{w}^\textbf{k}, \textbf{w}^\textbf{k}\rangle ^{1/2} } \longrightarrow \left\{ 12 \left\| \mathscr {V}\left( \varSigma _G^\textrm{Sp} - \varSigma _F^\textrm{Sp} \right) \right\| \right\} \underset{\varvec{\kappa }\in \varDelta ^\star }{\textrm{argsup}} \, { \left| \varLambda (\textbf{k},\textbf{k}_0) \right| \over \{ \varLambda (\textbf{k},\textbf{k}) \}^{1/2} } = \textbf{k}_0. \end{aligned}$$

$\square$