Kendall’s tau-based inference for gradually changing dependence structures

Abstract

Suppose that a sequence of random pairs $(X_1,Y_1)$, $\ldots$, $(X_n,Y_n)$ is subject to a gradual change in the sense that for $K_1 \le K_2 \in \{ 1, \ldots , n \}$, the joint distribution is F before $K_1$, G after $K_2$, and gradually moving from F to G between the two times of change $K_1$ and $K_2$. This setup elegantly generalizes the abrupt-change model that is usually assumed in the change-point analysis. Under this configuration, asymptotically unbiased estimates of Kendall’s tau up to and after the change are proposed, as well as tests and estimators of change points related to these measures. The asymptotic behaviour of the introduced estimators and test statistics is rigorously investigated, in particular by demonstrating a general result on weighted indexed U-statistics computed under a heterogeneous pattern. A simulation study is conducted to examine the sampling properties of the proposed methods under different scenarios of change in the dependence structure of bivariate series. An illustration is given on a time series of monthly atmospheric carbon dioxide concentrations and global temperature for the period 1959–2015.

Appendices

Proofs of the main results of the paper

1.1 Proof of Lemma 1

From the definitions of $A_n^{\varvec{\kappa }}$ and $B_n^{\varvec{\kappa }}$, one has

$$\begin{aligned} B_n^{\varvec{\kappa }}- A_n^{\varvec{\kappa }}= & {} 2 \left( {\bar{D}}_n - { {\bar{\omega }}^{\varvec{\kappa }}\over \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \, \langle {\varvec{\omega }}^{\varvec{\kappa }}, I_n \rangle \right) = {2 \over n} \sum _{j=1}^n \theta _j^{\varvec{\kappa }}\, \mathbb {I}\left( X_j \le \cdot , Y_j \le \cdot \right) \\ \text{ and }~~ A_n^{\varvec{\kappa }}+ B_n^{\varvec{\kappa }}= & {} 2 \left( {\bar{D}}_n + { (1-{\bar{\omega }}^{\varvec{\kappa }}) \over \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \, \langle {\varvec{\omega }}^{\varvec{\kappa }}, I_n \rangle \right) = {2 \over n} \sum _{j=1}^n \gamma _j^{\varvec{\kappa }}\, \mathbb {I}\left( X_j \le \cdot , Y_j \le \cdot \right) . \end{aligned}$$

Since $\textrm{d}( B_n^{\varvec{\kappa }}- A_n^{\varvec{\kappa }})$ gives the weights of $2\theta _1^{\varvec{\kappa }}/ n, \ldots , 2\theta _n^{\varvec{\kappa }}/ n$ respectively to the points $(X_1,Y_1)$, $\ldots$, $(X_n,Y_n)$, it follows that

$$\begin{aligned} {\widetilde{\tau }}_{F,n}^{\varvec{\kappa }}= & {} \int _{\mathbb {R}^2} \left( B_n^{\varvec{\kappa }}- A_n^{\varvec{\kappa }}\right) \textrm{d}\left( B_n^{\varvec{\kappa }}- A_n^{\varvec{\kappa }}\right) - 1 \\= & {} { 4 \over n^2 } \sum _{j,k=1}^n \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\, \mathbb {I}\left( X_j \le X_k, Y_j \le Y_k \right) - 1 \\= & {} { 1 \over n^2 } \sum _{j,k=1}^n \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\left\{ 4 \, \mathbb {I}\left( X_j \le X_k, Y_j \le Y_k \right) - 1 \right\} \\= & {} { 1 \over n^2 } \sum _{\begin{array}{c} j,k=1\\ j\ne k \end{array}}^n \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\left\{ 4 \, \mathbb {I}\left( X_j< X_k, Y_j < Y_k \right) - 1 \right\} - {3\over n^2} \sum _{j=1}^n (\theta _j^{\varvec{\kappa }})^2. \end{aligned}$$

Since it can be shown easily that $(1/n) \sum (\theta _j^{\varvec{\kappa }})^2 = 1 + ({\bar{\omega }}^{\varvec{\kappa }})^2 / \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle$,

$$\begin{aligned} {\widetilde{\tau }}_{F,n}^{\varvec{\kappa }}= & {} { 2 \over n^2 } \sum _{j<k} \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\left\{ 4 \, \mathbb {I}\left( X_j< X_k, Y_j< Y_k \right) - 1 \right\} + O \left( \frac{1}{n\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) \\= & {} { 2 \over n^2 } \sum _{j<k} \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\left\{ 2 \, \mathbb {I}\left( X_j< X_k, Y_j< Y_k \right) + 2 \, \mathbb {I}\left( X_k< X_j, Y_k< Y_j \right) - 1 \right\} \\{} & {} + \, O \left( \frac{1}{n\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) \\= & {} \left( 1 - {1\over n} \right) \left[ {n\atopwithdelims ()2}^{-1} \sum _{j<k} \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\, f \left\{ (X_j,Y_j), (X_k,Y_k) \right\} \right] + O \left( \frac{1}{n\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) \\= & {} \left( 1 - {1\over n} \right) \tau _{F,n}^{\varvec{\kappa }}+ O \left( \frac{1}{n\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) \\= & {} \tau _{F,n}^{\varvec{\kappa }}+ O \left( \frac{1}{n\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) . \end{aligned}$$

Arguments similar in all respects also yield ${\widetilde{\tau }}_{G,n}^{\varvec{\kappa }}= \tau _{G,n}^{\varvec{\kappa }}+ O({n^{-1}}\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle ^{-1})$.

1.2 Proof of Lemma 2

First observe that $\textrm{E}\{ f( (X_j,Y_j), (X_k,Y_k) ) \} = 2 \, \mathbb {P}(X_j<X_k, Y_j<Y_k) + 2 \, \mathbb {P}(X_k<X_j, Y_k<Y_j) - 1$. Now since $(X_j,Y_j) \sim (1-\omega _j^{{\varvec{\kappa }}_0}) F + \omega _j^{{\varvec{\kappa }}_0} \, G$ and $(X_k,Y_k) \sim (1-\omega _k^{{\varvec{\kappa }}_0}) F + \omega _k^{{\varvec{\kappa }}_0} \, G$,

$$\begin{aligned} \mathbb {P}(X_j<X_k, Y_j<Y_k)= & {} \int _{\mathbb {R}^2} \left\{ (1-\omega _j^{{\varvec{\kappa }}_0}) F(x,y) + \omega _j^{{\varvec{\kappa }}_0} \, G(x,y) \right\} \\ {}{} & {} \hspace{1cm} \times \, \left\{ (1-\omega _k^{{\varvec{\kappa }}_0}) \textrm{d}F(x,y) + \omega _k^{{\varvec{\kappa }}_0} \, \textrm{d}G(x,y) \right\} \\= & {} (1-\omega _j^{{\varvec{\kappa }}_0}) (1-\omega _k^{{\varvec{\kappa }}_0}) \int _{\mathbb {R}^2} F(x,y) \, \textrm{d}F(x,y) \\{} & {} + \, \omega _j^{{\varvec{\kappa }}_0} \, \omega _k^{{\varvec{\kappa }}_0} \int _{\mathbb {R}^2} G(x,y) \, \textrm{d}G(x,y) \\{} & {} + \, \omega _j^{{\varvec{\kappa }}_0} (1-\omega _k^{{\varvec{\kappa }}_0}) \int _{\mathbb {R}^2} G(x,y) \, \textrm{d}F(x,y) \\{} & {} + \, \omega _k^{{\varvec{\kappa }}_0} (1-\omega _j^{{\varvec{\kappa }}_0}) \int _{\mathbb {R}^2} F(x,y) \, \textrm{d}G(x,y) \\= & {} (1-\omega _j^{{\varvec{\kappa }}_0}) (1-\omega _k^{{\varvec{\kappa }}_0}) \left( \tau _F + 1 \over 4 \right) + \omega _j^{{\varvec{\kappa }}_0} \, \omega _k^{{\varvec{\kappa }}_0} \left( \tau _G + 1 \over 4 \right) \\{} & {} + \, \omega _j^{{\varvec{\kappa }}_0} (1-\omega _k^{{\varvec{\kappa }}_0}) \left( \tau _{FG} + 1 \over 4 \right) + \omega _k^{{\varvec{\kappa }}_0} (1-\omega _j^{{\varvec{\kappa }}_0}) \left( \tau _{FG} + 1 \over 4 \right) . \end{aligned}$$

Since $\tau _{GF} = \tau _{FG}$, it follows that $4 \,\mathbb {P}(X_j<X_k, Y_j<Y_k) - 1 = (1-\omega _j^{{\varvec{\kappa }}_0}) (1-\omega _k^{{\varvec{\kappa }}_0}) \tau _F + \omega _j^{{\varvec{\kappa }}_0} \, \omega _k^{{\varvec{\kappa }}_0} \, \tau _G + \{ \omega _j^{{\varvec{\kappa }}_0} (1-\omega _k^{{\varvec{\kappa }}_0}) + \omega _k^{{\varvec{\kappa }}_0} (1-\omega _j^{{\varvec{\kappa }}_0}) \} \tau _{FG}$. Because $\mathbb {P}(X_j<X_k, Y_j<Y_k) = \mathbb {P}(X_k<X_j, Y_k<Y_j)$, one computes

$$\begin{aligned} 2 {n\atopwithdelims ()2} \, \textrm{E}(\tau _{F,n}^{\varvec{\kappa }})= & {} \sum _{j\ne k} \theta _j^{\varvec{\kappa }}\theta _k^{\varvec{\kappa }}\left\{ 4 \, \mathbb {P}(X_j<X_k, Y_j<Y_k) - 1 \right\} \\= & {} \sum _{j,k=1}^n \theta _j^{\varvec{\kappa }}\theta _k^{\varvec{\kappa }}\left\{ 4 \, \mathbb {P}(X_j<X_k, Y_j<Y_k) - 1 \right\} \\= & {} \left\{ \sum _{j=1}^n (1-\omega _j^{{\varvec{\kappa }}_0}) \theta _j^{\varvec{\kappa }}\right\} ^2 \tau _F + \left( \sum _{j=1}^n \omega _j^{{\varvec{\kappa }}_0} \, \theta _j^{\varvec{\kappa }}\right) ^2 \tau _G \\{} & {} + \, 2 \left\{ \sum _{j=1}^n (1-\omega _j^{{\varvec{\kappa }}_0}) \theta _j^{\varvec{\kappa }}\right\} \left( \sum _{j=1}^n \omega _j^{{\varvec{\kappa }}_0} \, \theta _j^{\varvec{\kappa }}\right) \tau _{FG} \\= & {} n^2 \left\{ \left( 1 - \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) ^2 \tau _F + n^2 (\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 \, \tau _G \right. \\ {}{} & {} \left. + \, 2 \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \left( 1 - \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) \tau _{FG} \right\} , \end{aligned}$$

where the last equality is based on Identities ${\mathcal {I}}_1$–${\mathcal {I}}_2$. Hence,

$$\begin{aligned} \textrm{E}(\tau _{F,n}^{\varvec{\kappa }}) = \left( 1 - \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) ^2 \tau _F + (\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 \, \tau _G + 2 \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \left( 1 - \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) \tau _{FG} + O(n^{-1}). \end{aligned}$$

Similar computations using Identities ${\mathcal {I}}_3$–${\mathcal {I}}_4$ yield

$$\begin{aligned} \textrm{E}(\tau _{G,n}^{\varvec{\kappa }}) = \left( 1 - \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) ^2 \tau _F + (\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 \, \tau _G + 2 \, \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \left( 1 - \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) \tau _{FG} + O(n^{-1}). \end{aligned}$$

When ${\varvec{\kappa }}= {\varvec{\kappa }}_0$, one has $\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} = 0$ and $\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} = 1$. One may then conclude that $\textrm{E}(\tau _{F,n}^{\varvec{\kappa }}) = \tau _F + O(n^{-1})$ and $\textrm{E}(\tau _{G,n}^{\varvec{\kappa }}) = \tau _G + O(n^{-1})$.

1.3 Proof of Theorem 1

The proof is centered around the following Lemma which can be seen as a version of a result of Chakrabortty and Kuchibhotla (2018) (see also Theorem 3.5.3 in Peña and Giné (1999) and Remark 3.5.4 therein) that has been adapted to the current context. The result is presented next without proof, but can be established using a decoupling inequality by de la Peña (1992) and a symmetrisation inequality (see its Lemma A.1).

Lemma S1

Let $\mathcal {Z} = \{ \textbf{Z}_1, \ldots , \textbf{Z}_n \}$ be a set of independent d-variate random vectors and $\mathcal {Z}'$ be an independent copy of $\mathcal {Z}$. Let also $\mathcal {E} = \{ e_1, \ldots , e_n \}$ be a set of independent Rademacher random variables and $\mathcal {E}'$ an independent copy of $\mathcal {E}$ such that $\mathcal {E} \cup \mathcal {E}'$ is independent of $\mathcal {Z} \cup \mathcal {Z}'$. For any $p>1$,

$$\begin{aligned} \textrm{E}\left\{ \left| \sum _{j<k} {\widetilde{\beta }}^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k) \right| ^p \right\} \le (192)^p \, \textrm{E}\left\{ \left| \sum _{j<k} e_j e_k' \, \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \right| ^p \right\} , \end{aligned}$$

where ${\widetilde{\beta }}^\theta _{jk}(\textbf{z}_1,\textbf{z}_2) = \beta ^\theta _{jk}(\textbf{z}_1,\textbf{z}_2) - \textrm{E}_{\textbf{Z}_j}\{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{z}_2)\} - \textrm{E}_{\textbf{Z}_k} \{ \beta ^\theta _{jk}(\textbf{z}_1,\textbf{Z}_k) \} + \textrm{E}_{\textbf{Z}_j,\textbf{Z}_k} \{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k) \}$.

In the sequel, it is understood that $T_n^\theta = T({\mathcal {F}}_n^\theta )$, where ${\mathcal {F}}^\theta _n = \{\beta ^\theta _{jk}: \mathbb {R}^d \times \mathbb {R}^d \rightarrow \mathbb {R}, j<k \in \{ 1, \ldots , d \} \}$. Define a version ${\widetilde{T}}_n^\theta$ of $T_n^\theta$ in such a way that

$$\begin{aligned} T_n^\theta - {\widetilde{T}}_n^\theta = \sum _{j<k} {\widetilde{\beta }}^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k). \end{aligned}$$

An application of Lemma S1. then ensures that it holds uniformly in $\theta \in \Theta _n$ that $\textrm{E}( |T_n^\theta - {\widetilde{T}}_n^\theta |^p) \le (192)^p \, \textrm{E}(|\mathbb {U}^\theta _n|^p)$, where

$$\begin{aligned} \mathbb {U}^\theta _n = \sum _{j<k} e_j e_k' \, \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') = \sum _{j=1}^n e_j \left\{ {1\over 2} \sum _{k\ne j} e_k' \, \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \right\} . \end{aligned}$$

By letting $\chi _{j,\theta } = \sum _{k \ne j} e_k' \, \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') / 2$, one observes that

$$\begin{aligned} \mathbb {U}^\theta _n = \sum _{j=1}^n e_j \, \chi _{j,\theta }, \end{aligned}$$

so that when conditioned on $\mathcal {E}'\cup \mathcal {Z} \cup \mathcal {Z}'$, it is a sum of independent zero-mean random variables. Proposition A.1.6 of van der Vaart and Wellner (1996) then ensures that there exists a universal constant K (independent of p, n and $\theta$) such that it holds, for any $\theta \in \Theta _n$, that

$$\begin{aligned} \left\{ \textrm{E}_{\mathcal {E}} \left( \left| \mathbb {U}^\theta _n \right| ^p \right) \right\} ^{1/p}\le & {} { K p \over \log p } \left[ \textrm{E}_{\mathcal {E}} \left( \left| \mathbb {U}^\theta _n \right| \right) + \left\{ \textrm{E}_{\mathcal {E}} \left( \max _{1\le j\le n} \left| e_j \, \chi _{j,\theta } \right| ^p \right) \right\} ^{1/p} \right] \\= & {} { K p \over \log p } \left[ \textrm{E}_{\mathcal {E}} \left( \left| \mathbb {U}^\theta _n \right| \right) + \left\{ \textrm{E}_{\mathcal {E}} \left( \max _{1\le j\le n} \left| \chi _{j,\theta } \right| ^p \right) \right\} ^{1/p} \right] . \end{aligned}$$

Invoking Jensen’s inequality,

$$\begin{aligned} \left\{ \textrm{E}\left( \left| \mathbb {U}^\theta _n \right| ^p \right) \right\} ^{1/p}\le & {} \textrm{E}_{\mathcal {E}',\mathcal {Z},\mathcal {Z}'} \left[ \left\{ \textrm{E}_{\mathcal {E}} \left( \left| \mathbb {U}^\theta _n \right| ^p \right) \right\} ^{1/p} \right] \\\le & {} { K p \over \log p } \left\{ \textrm{E}\left( \left| \mathbb {U}^\theta _n \right| \right) + \textrm{E}\left( \max _{1\le j\le n} \left| \chi _{j,\theta } \right| \right) \right\} \\\le & {} { K p \over \log p } \left[ \left\{ \textrm{E}\left( \left| \mathbb {U}^\theta _n \right| ^2 \right) \right\} ^{1/2} + \left\{ \sum _{j=1}^n \textrm{E}\left( \chi _{j,\theta }^2 \right) \right\} ^{1/2} \right] . \end{aligned}$$

Since $\textrm{E}(|\mathbb {U}^\theta _n|^2) = \textrm{E}_{\mathcal {E}',\mathcal {Z},\mathcal {Z}'} \{ \textrm{E}_{\mathcal {E}}(|\mathbb {U}^\theta _n|^2)\} = \sum _{j=1}^n \textrm{E}( \chi _{j,\theta }^2)$, one then has

$$\begin{aligned} \left\{ \textrm{E}\left( \left| \mathbb {U}^\theta _n \right| ^p \right) \right\} ^{1/p} \le { 2 K p \over \log p } \left\{ \sum _{j=1}^n \textrm{E}( \chi _{j,\theta }^2) \right\} ^{1/2}. \end{aligned}$$

Because $e_1, \ldots , e_n$ are i.i.d. Rademacher random variables,

$$\begin{aligned} \textrm{E}_{\mathcal {E}'}(\chi _j^2)= & {} {1\over 4} \sum _{k\ne j} \sum _{\ell \ne j} \textrm{E}_{\mathcal {E}'} \left\{ e_k' \, e_\ell ' \, \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \, \beta ^\theta _{j\ell }(\textbf{Z}_j,\textbf{Z}_\ell ') \right\} \\= & {} {1\over 4} \sum _{k\ne j} \textrm{E}_{\mathcal {E}'} \left\{ (e_k')^2 \, \right\} \left\{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \right\} ^2 \\= & {} {1\over 4} \sum _{k\ne j} \left\{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \right\} ^2. \end{aligned}$$

Now since $\textrm{E}(\chi _{j,\theta }^2) = \textrm{E}_{\mathcal {E},\mathcal {Z},\mathcal {Z}'} \{ \textrm{E}_{\mathcal {E}'}(\chi _{j,\theta }^2) \}$, it follows that

$$\begin{aligned} \sum _{j=1}^n \textrm{E}\left( \chi _{j,\theta }^2 \right) = {1\over 4} \sum _{j=1}^n \sum _{k\ne j} \textrm{E}\left[\left\{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \right\} ^2\right]= {1\over 2} \sum _{j<k} \textrm{E}\left[\left\{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \right\} ^2\right]. \end{aligned}$$

Since

$$\begin{aligned} \sum _{j<k} \textrm{E}[\lbrace \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \rbrace ^2]\le \max _{\theta \in \Theta _n} \sum _{j<k} \textrm{E}[\lbrace \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k') \rbrace ^2]=\sigma ^2_n, \end{aligned}$$

one has for any $p>3$ and $\theta \in \Theta _n$ that

$$\begin{aligned} \textrm{E}\left( |\mathbb {U}^\theta _n|^p \right) \le \left( 2 K p \, \sigma _n \over \log p \right) ^p \le \left( 2 K p \, \sigma _n \right) ^p. \end{aligned}$$

As a consequence, it holds for any $\theta \in \Theta _n$ that

$$\begin{aligned} \textrm{E}\left( \left| T^\theta _n - {\widetilde{T}}^\theta _n \right| ^p \right) \le (192)^p \, \textrm{E}\left( \left| \mathbb {U}^\theta _n \right| ^p \right) \le \left( 384 K p \, \sigma _n \right) ^p. \end{aligned}$$

In view of the above equation, an application of Markov’s inequality then ensures that for any $p>3$,

$$\begin{aligned} \max _{\theta \in \Theta _n} \mathbb {P}\left( \left| T_n^\theta - {\widetilde{T}}_n^\theta \right| > 392 K p \, \sigma _n \log n \right) \le \left( \log n \right) ^{-p}. \end{aligned}$$

Since by assumption, $\textrm{card}(\Theta _n) \le C \, n^a$, one may conclude that for all $p>3$,

$$\begin{aligned}{} & {} \mathbb {P}\left( \max _{\theta \in \Theta _n} \left| T_n^\theta - {\widetilde{T}}_n^\theta \right|> 392 K p \, \sigma _n \log n \right) \\{} & {} \quad \le C \, n^a \max _{\theta \in \Theta _n} \mathbb {P}\left( \left| T_n^\theta - {\widetilde{T}}_n^\theta \right| > 392 K p \, \sigma _n \log n \right) \le C \, n^a \left( \log n \right) ^{-p}. \end{aligned}$$

By setting $p= (a+2) \log n / \log \log n$, one then obtains

$$\begin{aligned} \mathbb {P}\left\{ \max _{\theta \in \Theta _n} \left| T_n^\theta - {\widetilde{T}}_n^\theta \right| > { 392 K (a+2) \sigma _n \log ^2 n \over \log \log n} \right\} \le Cn^{-2}. \end{aligned}$$

Applying the Borel–Cantelli Theorem, one can conclude that it holds uniformly in $\theta \in \Theta$ that $T_n^\theta = {\widetilde{T}}_n^\theta + O_\mathrm{a.s.} \{ \sigma _n \log ^2 n (\log \log n)^{-1} \}$, where

$$\begin{aligned} {\widetilde{T}}_n^\theta= & {} T_n^\theta - \sum _{j<k} {\widetilde{\beta }}_{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \\= & {} \sum _{j<k} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) - {\widetilde{\beta }}_{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} \\= & {} \sum _{j<k} \left\{ \textrm{E}_{\textbf{Z}_j} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} + \textrm{E}_{\textbf{Z}_k} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} - \textrm{E}_{\textbf{Z}_j,\textbf{Z}_k} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} \right\} \\= & {} \sum _{j=1}^n \left[ {1\over 2} \sum _{\begin{array}{c} k=1\\ k\ne j \end{array}}^n \textrm{E}_{\textbf{Z}_k} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} \right] + \sum _{k=1}^n \left[ {1\over 2} \sum _{\begin{array}{c} j=1\\ j\ne k \end{array}}^n \textrm{E}_{\textbf{Z}_j} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} \right] \\{} & {} - \, \sum _{j<k} \textrm{E}_{\textbf{Z}_j,\textbf{Z}_k} \left\{ \beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) \right\} \\= & {} \sum _{j=1}^n B_j^\theta (\textbf{Z}_j) - \textrm{E}(T_n^\theta ). \end{aligned}$$

Upon noting that

$$\begin{aligned} \sum _{j=1}^n \textrm{E}\left( B^\theta _j(\textbf{Z}_j) \right) = \sum _{j \ne k=1}^n \textrm{E}_{\textbf{Z}_j,\textbf{Z}_k} \left\{ \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k) \right\} = 2 \, \textrm{E}(T_n^\theta ), \end{aligned}$$

one can finally write

$$\begin{aligned} T_n^\theta - \textrm{E}(T_n) = \sum _{j=1}^n \left\{ B_j^\theta (\textbf{Z}_j) - \textrm{E}\left( B^\theta _j(\textbf{Z}_j) \right) \right\} + O_\mathrm{a.s.} \left( \sigma _n \log ^2 n \over \log \log n \right) . \end{aligned}$$

1.4 Proof of Proposition 1

1.4.1 Proof of Proposition 1 (a)

The asymptotic tightness of ${\varvec{\kappa }}\rightarrow \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle ( S_{F,\varvec{\theta },n}^{\varvec{\kappa }}, S_{G,\varvec{\theta },n}^{\varvec{\kappa }}, S_{F,\varvec{\gamma },n}^{\varvec{\kappa }}, S^{\varvec{\kappa }}_{G,\varvec{\gamma },n})$ in the space of bounded functions of the form $z: \varDelta \rightarrow \mathbb {R}^4$ can be deduced from the joint asymptotic tightness in $\ell ^\infty (\varDelta ) \times \ell ^\infty (\varDelta ) \times \ell ^\infty (\varDelta ) \times \ell ^\infty (\varDelta )$ of

$$\begin{aligned} ({\varvec{\kappa }}_1,{\varvec{\kappa }}_2,{\varvec{\kappa }}_3,{\varvec{\kappa }}_4) \mapsto \begin{pmatrix} \langle {\varvec{\omega }}^{{\varvec{\kappa }}_1},{\varvec{\omega }}^{{\varvec{\kappa }}_1} \rangle S_{F,\varvec{\theta },n}^{{\varvec{\kappa }}_1} \\ \langle {\varvec{\omega }}^{{\varvec{\kappa }}_2},{\varvec{\omega }}^{{\varvec{\kappa }}_2} \rangle S_{G,\varvec{\theta },n}^{{\varvec{\kappa }}_2} \\ \langle {\varvec{\omega }}^{{\varvec{\kappa }}_3},{\varvec{\omega }}^{{\varvec{\kappa }}_3} \rangle S_{F,\varvec{\gamma },n}^{{\varvec{\kappa }}_3}\\ \langle {\varvec{\omega }}^{{\varvec{\kappa }}_4}, {\varvec{\omega }}^{{\varvec{\kappa }}_4} \rangle S^{{\varvec{\kappa }}_4}_{G,\varvec{\gamma },n}\end{pmatrix}^\top , \end{aligned}$$

which in view of Lemma 1.4.3 of van der Vaart and Wellner (1996) will follow upon proving the asymptotic tightness in $\ell ^\infty (\varDelta )$ of each element in the above vector. To this end, first note that according to Theorem 1.5.7 of van der Vaart and Wellner (1996), the asymptotic tightness of the stochastic processes ${\varvec{\kappa }}\mapsto {\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}:= \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, S_{F,\varvec{\theta },n}^{\varvec{\kappa }}$ will be achieved if:

(${\mathcal {A}}_1$) ${\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}$ is asymptotically tight in $\mathbb {R}$ for each ${\varvec{\kappa }}\in \varDelta$;

(${\mathcal {A}}_2$) For any $\eta ,\epsilon >0$, there exists $\delta >0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {P}\left( \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \\ \Vert {\varvec{\kappa }}- {\varvec{\kappa }}'\Vert \le \delta \end{array}} \left| {\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}- {\widetilde{T}}_{F,\varvec{\theta },n}^{{\varvec{\kappa }}'} \right| > \epsilon \right) \le \eta . \end{aligned}$$

(6)

To show ${\mathcal {A}}_1$, let $W_{F,j}^{\varvec{\kappa }}= \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle ~ \theta _j^{\varvec{\kappa }}\{ V_{F,j} - \textrm{E}(V_{F,j}) \} = \{ \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle + ({\bar{\omega }}^{\varvec{\kappa }})^2 - {\bar{\omega }}^{\varvec{\kappa }}\omega ^{\varvec{\kappa }}_j \} \{ V_{F,j} - \textrm{E}(V_{F,j}) \}$ and note that since $\omega _j^{\varvec{\kappa }}\le 1$ for all $j \in \{ 1, \ldots , n \}$,

$$\begin{aligned} \sup _{{\varvec{\kappa }}\in \varDelta } \max \left( \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle , {\bar{\omega }}^{\varvec{\kappa }}\right) \le 1. \end{aligned}$$

Now the fact that $|V_{F,j}| \le 2$ entails $|W_{F,j}^{\varvec{\kappa }}| \le 16$ for all $j \in \{ 1, \ldots , n \}$, so that Bernstein’s inequality ensures that for any $x>0$,

$$\begin{aligned} \mathbb {P}\left( \left| {\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}\right|> x \right) = \mathbb {P}\left( \left| {1\over \sqrt{n}} \sum _{j=1}^n W_{F,j}^{\varvec{\kappa }}\right| > x \right) \le \exp \left\{ - x^2 \over 2 (64) \max (x/\sqrt{n},64) \right\} . \end{aligned}$$

Hence, for an arbitrary $\epsilon \in (0,1)$, choosing n such that $n \ge -2 \ln \epsilon$ entails $\sqrt{- 2 (64)^2\ln \epsilon } / \sqrt{n} \le 64$. The choices ${\widetilde{C}} = \sqrt{2} (64)$ and $x = {\widetilde{C}} \sqrt{- \ln \epsilon }$ then allow to conclude that for any $\epsilon > 0$,

$$\begin{aligned} \mathbb {P}\left( \left| {\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}\right| > {\widetilde{C}} \sqrt{- \ln \epsilon } \right) \le \epsilon . \end{aligned}$$

In other words, the sequence ${\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}$ is asymptotically tight in $\mathbb {R}$ and ${\mathcal {A}}_1$ is satisfied. Next, in order to establish ${\mathcal {A}}_2$, define

$$\begin{aligned} \mathbb {G}^{\varvec{\kappa }}_{F,n} = {1\over \sqrt{n}} \sum _{j=1}^n \omega _j^{\varvec{\kappa }}\left\{ V_{F,j} - \textrm{E}(V_{F,j}) \right\} ~\text{ and }~~ \mathbb {H}_{F,n} = {1\over \sqrt{n}} \sum _{j=1}^n \left\{ V_{F,j} - \textrm{E}(V_{F,j}) \right\} . \end{aligned}$$

With this notation, one can write ${\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}= \{ \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle + ({\bar{\omega }}^{\varvec{\kappa }})^2 \} \, \mathbb {H}_{F,n} - {\bar{\omega }}^{\varvec{\kappa }}\, \mathbb {G}_{F,n}^{\varvec{\kappa }}$. Before going any further, consider the following lemma.

Lemma S2

Let $\rho _n(\kappa ,\kappa ') = |\lfloor n\kappa \rfloor -\lfloor n\kappa ' \rfloor | / n$ for $\kappa ,\kappa ' \in [0,1]$. For any ${\varvec{\kappa }}, {\varvec{\kappa }}' \in \varDelta$ and any integer $p \ge 1$,

$$\begin{aligned} \sum _{i=1}^n \left| \omega _i^{\varvec{\kappa }}- \omega _i^{{\varvec{\kappa }}'} \right| ^p \le n \left\{ \rho _n(\kappa _1,\kappa _1') + \rho _n(\kappa _2,\kappa _2') \right\} . \end{aligned}$$

Some computations making use of Lemma S2. allow to establish that

$$\begin{aligned} \max \left\{ \left| \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle - \langle {\varvec{\omega }}^{{\varvec{\kappa }}'}, {\varvec{\omega }}^{{\varvec{\kappa }}'} \rangle \right| , \left| {\bar{\omega }}^{\varvec{\kappa }}- {\bar{\omega }}^{{\varvec{\kappa }}'} \right| \right\}\le & {} 4 \left\| {\lfloor n{\varvec{\kappa }} \rfloor \over n} - {\lfloor n{\varvec{\kappa }}' \rfloor \over n} \right\| _1 \\\le & {} \left\| {\varvec{\kappa }}-{\varvec{\kappa }}' \right\| _1 + {1\over n} \,. \end{aligned}$$

Using the basic fact that $(a^2 - b^2) = (a-b)(a+b)$,

$$\begin{aligned} \left| {\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}- {\widetilde{T}}_{F,\varvec{\theta },n}^{{\varvec{\kappa }}'} \right|\le & {} 128 \left( \Vert {\varvec{\kappa }}- {\varvec{\kappa }}' \Vert _1 + {1\over n} \right) \\{} & {} \times \left( |\mathbb {H}_{F,n}| + |\mathbb {H}_{G,n}| + 2 \max \left\{ |\mathbb {G}^{\varvec{\kappa }}_{F,n}|, |\mathbb {G}^{{\varvec{\kappa }}'}_{F,n}| \right\} \right) \\{} & {} + \, 2 \left| \mathbb {G}^{\varvec{\kappa }}_{F,n} - \mathbb {G}^{{\varvec{\kappa }}'}_{F,n} \right| . \end{aligned}$$

It follows that for any $\epsilon , \delta > 0$ and $n >2 / \delta$,

$\displaystyle \mathbb {P}\left( \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert \le \delta \end{array}} \left| {\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}- {\widetilde{T}}_{F,\varvec{\theta },n}^{{\varvec{\kappa }}'} \right| > \epsilon \right)$

$$\begin{aligned} \le \sum _{D \in \{F,G\}} \left\{ \mathcal {Q}_{F,n}^{(1)} \left( \delta ,{\epsilon \over 6} \right) + \mathcal {Q}_{F,n}^{(2)} \left( \delta ,{\epsilon \over 6} \right) + \mathcal {Q}_{F,n}^{(3)} \left( \delta ,{\epsilon \over 6} \right) \right\} , \end{aligned}$$

where

$$\begin{aligned} \mathcal {Q}_{F,n}^{(1)}(\delta ,\epsilon )= & {} \mathbb {P}\left( \sup _{{\varvec{\kappa }}\in \varDelta } \left| \mathbb {G}^{\varvec{\kappa }}_{F,n} \right|> \frac{\epsilon }{512 \, \delta } \right) , \\ \mathcal {Q}_{F,n}^{(2)}(\delta ,\epsilon )= & {} \mathbb {P}\left( \left| \mathbb {H}_{F,n} \right|> \frac{\epsilon }{256 \, \delta } \right) \\ \text{ and }~ \mathcal {Q}_{F,n}^{(3)}(\delta ,\epsilon )= & {} \mathbb {P}\left( \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert \le \delta \end{array}} \left| \mathbb {G}^{\varvec{\kappa }}_{F,n} -\mathbb {G}^{{\varvec{\kappa }}'}_{F,n} \right| > \epsilon \right) . \end{aligned}$$

The conclusion that ${\widetilde{T}}_{F,\varvec{\theta },n}^{\varvec{\kappa }}$ satisfies ${\mathcal {A}}_2$ is based on the following lemma.

Lemma S3

Let $V_{1,n},\ldots , V_{n,n}$ be a triangular array of mean-zero random variables such that $| V_{j,n}| \le 2$ and for ${\varvec{\kappa }}\in \varDelta$, let

$$\begin{aligned} \mathbb {G}_n^{{\varvec{\kappa }}} = {1\over \sqrt{n}} \sum _{j=1}^n \omega _j^{\varvec{\kappa }}\, V_{j,n}. \end{aligned}$$

Then there exists ${\widetilde{K}}>0$ independent of n such that for any $\epsilon , \eta >0$, one has for any $\delta \le {\widetilde{K}}\eta ^{5/3} \epsilon ^{5/2}$ and $n > 2 / \delta$ that

$$\begin{aligned} \mathbb {P}\left( \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert \le \delta \end{array}} \left| \mathbb {G}_n^{{\varvec{\kappa }}} - \mathbb {G}_n^{{\varvec{\kappa }}'} \right| > \epsilon \right) \le \eta . \end{aligned}$$

Moreover, there exists ${\bar{K}}>0$ such that for any $\epsilon >0$,

$$\begin{aligned} \mathbb {P}\left( \sup _{{\varvec{\kappa }}\in \varDelta } |\mathbb {G}_n^{\varvec{\kappa }}| > M \right) \le \epsilon ~~\text{ for } \text{ all } ~~ M \ge { {\bar{K}} \over \epsilon ^{1/6} } . \end{aligned}$$

Lemma S3. ensures that one can find ${\widetilde{K}}>0$ such that for any $\epsilon , \eta >0$,

$$\begin{aligned} \mathcal {Q}_{F,n}^{(1)} \left( \delta , \frac{\epsilon }{6} \right) \le {\eta \over 6} ~~\text{ for } \text{ any }~~ \delta \le {\widetilde{K}} \left( \eta \over 6 \right) ^{5/3} \left( \epsilon \over 6 \right) ^{5/2} ~\text{ and }~~ n > {2\over \delta } . \end{aligned}$$

Since ${\widetilde{K}}$ is independent of n, it guarantees the existence of ${\widetilde{C}}>0$ such that for $n \ge - 2\ln (\eta /6)$ and as long as $\delta \le {\widetilde{C}} (\epsilon /6) \{ -\log (\eta /6) \}^{-1/2}$,

$$\begin{aligned} \mathcal {Q}_{F,n}^{(3)} \left( \delta , \frac{\epsilon }{6} \right) \le {\eta \over 6} . \end{aligned}$$

Equation (6) then holds for any $\epsilon ,\eta >0$, which establishes ${\mathcal {A}}_2$. The proof of the asymptotic tightness of ${\varvec{\kappa }}\rightarrow \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle S_{F,\varvec{\theta },n}^{\varvec{\kappa }}$ is therefore complete. Identical arguments can be invoked to show the asymptotic tightness of the maps ${\varvec{\kappa }}\mapsto \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle S_{G,\varvec{\theta },n}^{\varvec{\kappa }}$, ${\varvec{\kappa }}\mapsto \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle S_{F,\varvec{\gamma },n}^{\varvec{\kappa }}$ and ${\varvec{\kappa }}\mapsto \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle S_{G,\varvec{\gamma },n}^{\varvec{\kappa }}$.

1.4.2 Proof of Proposition 1 (b)

First define $\varDelta _n = \lbrace {\varvec{\kappa }}\in \varDelta : n{\varvec{\kappa }}\in \mathbb {N}^2\rbrace$ and note that $\lbrace \tau _{F,n}^{\varvec{\kappa }}, {\varvec{\kappa }}\in \varDelta \rbrace = \lbrace \tau _{F,n}^{\varvec{\kappa }}, {\varvec{\kappa }}\in \varDelta _n \rbrace$. In order to apply Theorem 1, let $\theta := {\varvec{\kappa }}$, $\Theta _n:= \varDelta _n \subset \mathbb {R}^2$ and $\beta _{jk}^\theta (\textbf{z}_1,\textbf{z}_2):= \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, \theta _j^{\varvec{\kappa }}\, \theta _k^{\varvec{\kappa }}\, f\{ (X_j,Y_j),(X_k,Y_k) \}$, and define

$$\begin{aligned} T^\theta _n = \sum _{j<k}\beta _{jk}^\theta (\textbf{Z}_j,\textbf{Z}_k) = \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle {n\atopwithdelims ()2} \, \tau _{F,n}^{\varvec{\kappa }}. \end{aligned}$$

Upon noting that $( f\{ (X_j,Y_j),(X_k,Y_k) \} )^2 \le 1$ and since $\omega ^{\varvec{\kappa }}_j \le 1$ implies that $\langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \le 1$ and ${\bar{\omega }}^{\varvec{\kappa }}\le 1$, it follows that

$$\begin{aligned} \sum _{j<k} \textrm{E}\left\{ \left( \beta ^\theta _{jk}(\textbf{Z}_j,\textbf{Z}_k) \right) ^2 \right\}\le & {} {\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle ^2 } \, n^2 \left\{ \frac{1}{n}\sum _{j=1}^n \left( \theta _j^{\varvec{\kappa }}\right) ^2\right\} \\= & {} {\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle ^2 } \, n^2 \left( 1+\frac{{\bar{\omega }}^{\varvec{\kappa }}}{\langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) ^2 \le 4 n^2. \end{aligned}$$

Since $\textrm{card}(\varDelta _n) = O(n^2)$, Theorem 1 with $\sigma _n = O(n)$ ensures that

$$\begin{aligned} \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle {n\atopwithdelims ()2} { T_{F,n}^{\varvec{\kappa }}\over \sqrt{n} }= & {} \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \sum _{j=1}^n \left\{ B_j^{{\varvec{\kappa }},{\varvec{\kappa }}_0}(X_j,Y_j) - \textrm{E}\left( B_j^{{\varvec{\kappa }},{\varvec{\kappa }}_0}(X_j,Y_j) \right) \right\} \\{} & {} + \, O_{a.s.}\left( n \log ^2 n \over \log \log n \right) \end{aligned}$$

uniformly in ${\varvec{\kappa }}\in \varDelta$, where

$$\begin{aligned} B_j^{{\varvec{\kappa }},{\varvec{\kappa }}_0}(x,y) = \theta _j^{\varvec{\kappa }}\sum _{\begin{array}{c} k=1\\ k\ne j \end{array}}^n \theta _k^{\varvec{\kappa }}\, \textrm{E}\left[ f \left\{ (x,y), (X_k,Y_k) \right\} \right] . \end{aligned}$$

At this point, note that since $(X_k,Y_k) \sim (1-\omega _k^{{\varvec{\kappa }}_0}) F + \omega _k^{{\varvec{\kappa }}_0} \, G$,

$$\begin{aligned} \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right]= & {} 2 \, \mathbb {P}\left( X_k> \cdot , Y_k > \cdot \right) + 2 \, \mathbb {P}\left( X_k< \cdot , Y_k < \cdot \right) - 1 \\= & {} 2 (1-\omega _k^{{\varvec{\kappa }}_0}) \left( F + {\bar{F}} \right) + 2 \, \omega _k^{{\varvec{\kappa }}_0} \left( G + {\bar{G}} \right) - 1. \end{aligned}$$

It follows that

$$\begin{aligned} B_j^{{\varvec{\kappa }},{\varvec{\kappa }}_0}= & {} \theta _j^{\varvec{\kappa }}\sum _{\begin{array}{c} k=1\\ k\ne j \end{array}}^n \theta _k^{\varvec{\kappa }}\, \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right] \\= & {} \theta _j^{\varvec{\kappa }}\sum _{k=1}^n \theta _k^{\varvec{\kappa }}\, \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right] - (\theta _j^{\varvec{\kappa }})^2 \, \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right] \\= & {} \theta _j^{\varvec{\kappa }}\left\{ 2(F+{\bar{F}}) \sum _{k=1}^n (1-\omega _k^{{\varvec{\kappa }}_0}) \theta _k^{\varvec{\kappa }}+ 2(G+{\bar{G}}) \sum _{k=1}^n \omega _k^{{\varvec{\kappa }}_0} \, \theta _k^{\varvec{\kappa }}- \sum _{k=1}^n \theta _k^{\varvec{\kappa }}\right\} \\{} & {} - \, (\theta _j^{\varvec{\kappa }})^2 \, \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right] \\= & {} n \, \theta _j^{\varvec{\kappa }}\left\{ 2 \left( 1-\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) (F+{\bar{F}}) + 2 \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}(G+{\bar{G}}) - 1 \right\} \\{} & {} - \, (\theta _j^{\varvec{\kappa }})^2 \, \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right] , \end{aligned}$$

where the last line used Identities ${\mathcal {I}}_1$–${\mathcal {I}}_2$ in Sect. E. One then has

$$\begin{aligned} \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \sum _{j=1}^n (\theta _j^{\varvec{\kappa }})^2 \, \textrm{E}\left[ f \left\{ (\cdot ,\cdot ), (X_k,Y_k) \right\} \right] \le n \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \left( 1+\frac{{\bar{\omega }}^{\varvec{\kappa }}}{\langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) = O(n), \end{aligned}$$

so that one can conclude that uniformly in ${\varvec{\kappa }}\in \varDelta$,

$$\begin{aligned} {n\atopwithdelims ()2} { T_{F,n}^{\varvec{\kappa }}\over \sqrt{n} }= & {} 2 \, n^{3/2} \left( 1-\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) S_{F,\varvec{\theta },n}^{\varvec{\kappa }}+ 2 \, n^{3/2} \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \, S_{G,\varvec{\theta },n}^{\varvec{\kappa }}+ O_{a.s.}\left( \frac{n \log ^2 n }{\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \log \log n } \right) . \end{aligned}$$

One finally has

$$\begin{aligned} T_{F,n}^{\varvec{\kappa }}= 4 \left( 1-\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) S_{F,\varvec{\theta },n}^{\varvec{\kappa }}+ 4 \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \, S_{G,\varvec{\theta },n}^{\varvec{\kappa }}+ O_{a.s.} \left( \frac{\log ^2 n }{\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \sqrt{n}\log \log n } \right) . \end{aligned}$$

Arguments in all points similar using Identities ${\mathcal {I}}_3$–${\mathcal {I}}_4$ in Sect. E enable to show that

$$\begin{aligned} T_{G,n}^{\varvec{\kappa }}= 4 \left( 1-\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) S_{F,\varvec{\gamma },n}^{\varvec{\kappa }}+ 4 \, \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \, S_{G,\varvec{\gamma },n}^{\varvec{\kappa }}+ O_{a.s.} \left( \frac{\log ^2 n }{\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \sqrt{n}\log \log n } \right) . \end{aligned}$$

The formula for $T_{FG,n} ^{\varvec{\kappa }}$ obtains similarly.

1.4.3 Proof of Proposition 1 (c)

Upon noting that $\omega _j^{\varvec{\kappa }}\le 1$ for each $j \in \{1,\ldots ,n\}$, one has $|\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}| \le 1 + \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle ^{-1}$ and $|\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}| \le 1 + \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle ^{-1}$ for any ${\varvec{\kappa }}\in \varDelta$. Combining parts (a) and (b) of Proposition 1 then yields the result.

1.5 Proof of Proposition 2

Since $\alpha _n^{{\varvec{\kappa }}_0,{\varvec{\kappa }}_0} = 0$ and $\beta _n^{{\varvec{\kappa }}_0,{\varvec{\kappa }}_0} = 1$, a consequence of Proposition 1 is that $T_{F,n}^{\varvec{\kappa }}= 4 \, S_{F,\varvec{\theta },n}^{\varvec{\kappa }}+ o_\mathbb {P}(1)$ and $T_{G,n}^{\varvec{\kappa }}= 4 \, S_{G,\varvec{\gamma },n}^{\varvec{\kappa }}+ o_\mathbb {P}(1)$. The asymptotic joint normality of $T_{F,n}^{\varvec{\kappa }}$ and $T_{G,n}^{\varvec{\kappa }}$ will then follow from that of $S_{F,\varvec{\theta },n}^{\varvec{\kappa }}$ and $S_{G,\varvec{\gamma },n}^{\varvec{\kappa }}$. To this end, define for $\textbf{a} = (a_1,a_2)^\top \in \mathbb {R}^2$ the random variable $Z_{n,\textbf{a}} = a_1 \, S_{F,\varvec{\theta },n}^{\varvec{\kappa }}+ a_2 \, S_{G,\varvec{\theta },n}^{\varvec{\kappa }}$, which because $\langle {\varvec{\omega }}^{{\varvec{\kappa }}_0},{\varvec{\omega }}^{{\varvec{\kappa }}_0}\rangle$ is bounded away from 0, can be viewed as a sum of bounded and centred independent random variables for n sufficiently large. One can then conclude that $Z_{n,\textbf{a}}$ is asymptotically normal with mean 0 and variance

$$\begin{aligned} \sigma _{Z_{\textbf{a}}}^2= & {} \lim _{n\rightarrow \infty } \textrm{var}(Z_{n,\textbf{a}}) \\= & {} \lim _{n\rightarrow \infty } \frac{1}{n} \sum _{i=1}^n \textrm{var}\left( a_1 \, \delta _{1,j}^{{\varvec{\kappa }}_0} \, V_{F,j} + a_2 \, \delta _{2,j}^{{\varvec{\kappa }}_0} \, V_{G,j}\right) \\= & {} \lim _{n\rightarrow \infty } \left\{ \frac{1}{n} \sum _{i=1}^n a_1^2 \left( \delta _{1,j}^{{\varvec{\kappa }}_0} \right) ^2 \textrm{var}(V_{F,j}) + \frac{1}{n} \sum _{i=1}^n a_2^2 \left( \delta _{2,j}^{{\varvec{\kappa }}_0} \right) ^2 \textrm{var}( V_{G,j}) \right. \\{} & {} \left. + \, \frac{2}{n} \sum _{i=1}^n a_1 a_2 \, \delta _{1,j}^{{\varvec{\kappa }}_0} \, \delta _{2,j}^{{\varvec{\kappa }}_0} \, \textrm{Cov}( V_{F,j}, V_{G,j}) \right\} \\= & {} \textbf{a}^\top \varSigma ({\varvec{\kappa }}_0,{\varvec{\kappa }}_0) \, \textbf{a}. \end{aligned}$$

The asymptotic joint normality of $(S_{F,\varvec{\theta },n}^{\varvec{\kappa }},S_{G,\varvec{\gamma },n}^{\varvec{\kappa }})$, and thus of $(T_{F,n}^{\varvec{\kappa }},T_{G,n}^{\varvec{\kappa }})$, follows from an application of the Cramér–Wold device.

1.6 Proof of Proposition 3

Invoking the formulas for $\textrm{E}(\tau _{F,n}^{\varvec{\kappa }})$ and $\textrm{E}(\tau _{F,n}^{\varvec{\kappa }})$ stated in Lemma 2, as well as that for $\textrm{E}(\tau _{FG,n}^{\varvec{\kappa }})$ at the beginning of Section 3, one can write

$$\begin{aligned}{} & {} \textrm{E}\left( \tau _{G,n}^{\varvec{\kappa }}+ \tau _{F,n}^{\varvec{\kappa }}- 2 \, \tau _{FG,n}^{\varvec{\kappa }}\right) \\{} & {} \quad = \left\{ (1-\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 + (1-\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 - 2 (1-\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}) (1-\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}) \right\} \tau _F \\{} & {} \qquad + \, \left\{ (\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 + (\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0})^2 - 2 \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right\} \tau _G \\{} & {} \qquad + \, 2 \left\{ \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} (1-\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}) + \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} (1-\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}) \right. \\{} & {} \qquad \left. - \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} - \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} + 2 \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right\} \tau _{FG} + O(n^{-1}) \\{} & {} \quad = \left( \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} - \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) ^2 \left( \tau _F + \tau _G - 2 \tau _FG \right) + O(n^{-1}). \end{aligned}$$

For the definition of $\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}$ and $\beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}$, it comes easily that $\alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} - \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} = \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}_0) / \varLambda ^2({\varvec{\kappa }},{\varvec{\kappa }})$. Thus,

$$\begin{aligned} \textrm{E}\left( \tau _{G,n}^{\varvec{\kappa }}+ \tau _{F,n}^{\varvec{\kappa }}- 2 \, \tau _{FG,n}^{\varvec{\kappa }}\right) = \left( \langle {\varvec{\omega }}^{{\varvec{\kappa }}_0}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \over \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \right) ^2 \left( \tau _F + \tau _G - 2 \, \tau _{FG} \right) + O(n^{-1}). \end{aligned}$$

One can then deduce from Proposition 1 (c) that as long as $\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle$ converges to $\varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) > 0$ as $n\rightarrow \infty$, $\tau _{G,n}^{\varvec{\kappa }}+ \tau _{F,n}^{\varvec{\kappa }}- 2 \, \tau _{FG,n}^{\varvec{\kappa }}$ is $\sqrt{n}$-consistent for

$$\begin{aligned} \left( \langle {\varvec{\omega }}^{{\varvec{\kappa }}_0}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \over \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \right) ^2 \left( \tau _F + \tau _G - 2 \, \tau _{FG} \right) . \end{aligned}$$

Defining $M_n^{\varvec{\kappa }}= \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) \, |\tau _{G,n}^{\varvec{\kappa }}+ \tau _{F,n}^{\varvec{\kappa }}- 2 \, \tau _{FG,n}|$, one can then write

$$\begin{aligned} M_n^{\varvec{\kappa }}= \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) \left( \langle {\varvec{\omega }}^{{\varvec{\kappa }}_0}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \over \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \right) ^2 \left| \tau _F + \tau _G - 2 \, \tau _{FG} + O_\mathbb {P}(n^{-1/2}) \right| . \end{aligned}$$

From the assumption that ${\varvec{\kappa }}\in \varDelta _\epsilon$, a consequence of eq. (4) is that $\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \ge \epsilon /24$ for n sufficiently large. Hence, since $\langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{{\varvec{\kappa }}'} \rangle = \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}') + O(n^{-1})$, one has $\varLambda ({\varvec{\kappa }},{\varvec{\kappa }})\ge \epsilon /48$ uniformly in ${\varvec{\kappa }}\varDelta _\epsilon$ for n taken sufficiently large. One then deduces that as $n\rightarrow \infty$ and uniformly in ${\varvec{\kappa }}\in \varDelta _\epsilon$,

$$\begin{aligned} \sup _{{\varvec{\kappa }}\in \varDelta _\epsilon } \left| M_n^{\varvec{\kappa }}- M^{\varvec{\kappa }}\right| = O_{\mathbb {P}}(n^{-1/2}), \end{aligned}$$

where $M^{\varvec{\kappa }}= \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}_0) \, |\tau _G^{\varvec{\kappa }}+ \tau _F^{\varvec{\kappa }}- 2 \, \tau _{FG}| / \varLambda ^2({\varvec{\kappa }},{\varvec{\kappa }})$. Since ${\varvec{\kappa }}_0$ is the unique maximiser of $M^{\varvec{\kappa }}$,

$$\begin{aligned} M^{{\varvec{\kappa }}_0} \ge M^{{\widehat{{\varvec{\kappa }}}}_{0,\epsilon }} \ge M_n^{{\widehat{{\varvec{\kappa }}}}_{0,\epsilon }} - O_{\mathbb {P}}(n^{-1/2}) \ge M_n^{{\varvec{\kappa }}_0} - O_{\mathbb {P}}(n^{-1/2}) \ge M^{{\varvec{\kappa }}_0} - O_{\mathbb {P}}(n^{-1/2}). \end{aligned}$$

It follows that $n\rightarrow \infty$, $M^{{\widehat{{\varvec{\kappa }}}}_{0,\epsilon }} \rightarrow M^{{\varvec{\kappa }}_0}$ in probability. This entails that ${\widehat{{\varvec{\kappa }}}}_{0,\epsilon }$ converges in probability to ${\varvec{\kappa }}_0$, and the proof is complete.

1.7 Proof of Proposition 4

First note that $S_{F,\varvec{\theta },n}^{\varvec{\kappa }}= S_{G,\varvec{\theta },n}^{\varvec{\kappa }}$ and $S_{F,\varvec{\gamma },n}^{\varvec{\kappa }}= S_{G,\varvec{\gamma },n}^{\varvec{\kappa }}$ when $F=G$, so that Proposition 1 entails that uniformly in ${\varvec{\kappa }}\in \varDelta$,

$$\begin{aligned} \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, T_{F,n}^{\varvec{\kappa }}= & {} 4 \, \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, S_{F,\varvec{\theta },n}^{\varvec{\kappa }}+ o_{a.s.}(1) \\ \text{ and } ~ \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, T_{G,n}^{\varvec{\kappa }}= & {} 4 \, \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, S_{F,\varvec{\gamma },n}^{\varvec{\kappa }}+ o_{a.s.}(1). \end{aligned}$$

Since $\textrm{E}[ f \{ (X_j,Y_j), (X_k,Y_k) \} ] = \textrm{E}\{ (F+{\bar{F}})(X_1,Y_1) \}$ when $F=G$,

$$\begin{aligned} {n\atopwithdelims ()2} \left\{ \textrm{E}(\tau _{F,n}^{\varvec{\kappa }}) - \textrm{E}(\tau _{G,n}^{\varvec{\kappa }}) \right\}= & {} \frac{1}{2} \, \textrm{E}\lbrace (F+{\bar{F}})(X_1,Y_1)\rbrace \sum _{j\ne k}(\theta _j \theta _k - \gamma _j\gamma _k) \,\\= & {} \frac{1}{2} \, \textrm{E}\lbrace (F+{\bar{F}})(X_1,Y_1)\rbrace \left[- \sum _{j=1}^n \left\{ (\theta ^{\varvec{\kappa }}_j)^2 - (\gamma ^{\varvec{\kappa }}_j)^2 \right\} \right]\\= & {} \frac{n}{2} \, \textrm{E}\lbrace (F+{\bar{F}})(X_1,Y_1)\rbrace \left( \frac{2{\bar{\omega }}^{\varvec{\kappa }}-1}{\langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle } \right) , \end{aligned}$$

where the fact that $\sum _{j, k}(\theta _j \theta _k - \gamma _j\gamma _k) =0$ was used to derive the second line above. Consequently, $\sqrt{n} \, \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \left| \textrm{E}(\tau _{F,n}^{\varvec{\kappa }}) - \textrm{E}(\tau _{G,n}^{\varvec{\kappa }}) \right| = O(n^{-1})$. Thus,

$$\begin{aligned} \sqrt{n} \, L_n^{\varvec{\kappa }}= & {} \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \, \sqrt{n} \left( \tau _{G,n}^{\varvec{\kappa }}- \tau _{F,n}^{\varvec{\kappa }}\right) \\= & {} 4 \, \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \left( S_{F,n}^{\varvec{\gamma }} - S_{F,n}^{\varvec{\theta }} \right) + o_{a.s.}(1) \\= & {} {4 \, \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \over \sqrt{n}} \sum _{j=1}^n \left( \gamma _j^{\varvec{\kappa }}- \theta _j^{\varvec{\kappa }}\right) \left( V_{F,j} - \mu _F^{\varvec{\kappa }}\right) +o_{a.s.}(1) \\= & {} {4\over \sqrt{n}} \sum _{j=1}^n (\omega _j^{\varvec{\kappa }}- {\bar{\omega }}^{\varvec{\kappa }}) \left( V_{F,j} - \mu _F^{\varvec{\kappa }}\right) + o_{a.s.}(1) \end{aligned}$$

uniformly in ${\varvec{\kappa }}\in \varDelta$. As a consequence, $\sqrt{n} \, L_n^{\varvec{\kappa }}$ is asymptotically normal with mean zero for each ${\varvec{\kappa }}\in \varDelta$, whereas for arbitrary ${\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta$,

$$\begin{aligned} n \, \textrm{Cov}\left( L_n^{\varvec{\kappa }}, L_n^{{\varvec{\kappa }}'} \right)= & {} {16 \over n} \sum _{j=1}^n (\omega _j^{\varvec{\kappa }}- {\bar{\omega }}^{\varvec{\kappa }}) (\omega _j^{{\varvec{\kappa }}'} - {\bar{\omega }}^{{\varvec{\kappa }}'}) \, \textrm{var}(V_{F,j}) + o(1) \\= & {} {16 \, \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{{\varvec{\kappa }}'} \rangle } \, \sigma ^2_F + o(1), \end{aligned}$$

where $\sigma ^2_F = \textrm{var}\{ (F+{\bar{F}})(X,Y) \}$ for $(X,Y) \sim F$. Since $\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{{\varvec{\kappa }}'} \rangle \rightarrow \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}')$ as $n\rightarrow \infty$, one concludes that

$$\begin{aligned} \lim _{n\rightarrow \infty } n \, \textrm{Cov}\left( L_n^{\varvec{\kappa }}, L_n^{{\varvec{\kappa }}'} \right) = { 16 \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}') }\, \sigma ^2_F. \end{aligned}$$

Finally, upon noting that $\varLambda ({\varvec{\kappa }},{\varvec{\kappa }}') = \textrm{Cov}\{ \widetilde{\mathbb {B}}({\varvec{\kappa }}), \widetilde{\mathbb {B}}({\varvec{\kappa }}') \}$, where $\widetilde{\mathbb {B}}$ is the integrated Brownian bridge defined in (5), one deduces the asymptotic representation $\sqrt{n} \, L_n^{\varvec{\kappa }}= 4 \, \sigma _F \, \widetilde{\mathbb {B}}({\varvec{\kappa }}) + o_\mathbb {P}(1)$. The result follows upon noting that Lemma S3. ensures the asymptotic tightness of $\sqrt{n} \, L_n^{\varvec{\kappa }}$ in $\ell ^\infty (\varDelta )$.

Proof of two technical lemmas

1.1 Proof of Lemma S2.

Assume without loss of generality that $\kappa _1 < \kappa _1'$. On one side, if $\kappa _1 <\kappa _2 \le \kappa _2'$, one has for $K_1 = \lfloor n\kappa _1 \rfloor$ and $K_2 = \lfloor n\kappa _2 \rfloor$ that

$$\begin{aligned} \left| \omega _j^{(\kappa _1,\kappa _2)} - \omega _j^{(\kappa _1,\kappa _2')} \right|= & {} \left\{ \begin{array}{ll} \left( j - K_1 \over K_2-K_1 \right) \left( K_2'-K_2 \over K_2'-K_1 \right) , &{} \quad j \in (K_1,K_2); \\ 1 - \left( j-K_1 \over K_2'-K_1 \right) , &{}\quad j \in [K_2,K_2'); \\ 0, &{} \text{ otherwise } \end{array} \right. \\\le & {} \left\{ \begin{array}{ll} \left( K_2'-K_2 \over K_2'-K_1 \right) , &{}\quad j \in (K_1,K_2); \\ \left( K_2'-K_2 \over K_2'-K_1 \right) , &{} \quad j \in [K_1,K_2'); \\ 0, &{} \quad \text{ otherwise } \end{array} \right. \\= & {} { \rho _n(\kappa _2,\kappa _2') \over \rho _n(\kappa _1,\kappa _2') } \, \mathbb {I}\left\{ j \in (\lfloor n\kappa _1 \rfloor ,\lfloor n\kappa _2' \rfloor \right\} . \end{aligned}$$

On the other side, if $\kappa _1 \le \kappa _2' \le \kappa _2$,

$$\begin{aligned} \left| \omega _j^{(\kappa _1,\kappa _2)} - \omega _j^{(\kappa _1,\kappa _2')} \right|= & {} \left\{ \begin{array}{ll} \left( j - K_1 \over K_2-K_1 \right) \left( K_2-K_2' \over K_2'-K_1 \right) , &{} \quad j \in (K_1,K_2'); \\ 1 - \left( j-K_1 \over K_2-K_1 \right) , &{}\quad j \in [K_2',K_2); \\ 0, &{} \quad \text{ otherwise } \end{array} \right. \\\le & {} { \rho _n(\kappa _2,\kappa _2') \over \rho _n(\kappa _1,\kappa _2) } \, \mathbb {I}\left\{ j \in (\lfloor n\kappa _1 \rfloor ,\lfloor n\kappa _2 \rfloor \right\} . \end{aligned}$$

Also note that

$$\begin{aligned} \left| \omega _j^{(\kappa _1,\kappa _2')} - \omega _j^{(\kappa _1',\kappa _2')} \right|= & {} \left\{ \begin{array}{ll} {j - K_1 \over K_2'-K_1}, &{}\quad j \in (K_1,K_1']; \\ \left( K_2'-j \over K_2'-K_1 \right) \left( K_1'-K_1 \over K_2'-K_1' \right) , &{} \quad j \in (K_1',K_2'); \\ 0, &{}\quad \text{ otherwise } \end{array} \right. \\\le & {} { \rho _n(\kappa _1,\kappa _1') \over \rho _n(\kappa _1,\kappa _2') } \, \mathbb {I}\left\{ j \in (\lfloor n\kappa _1 \rfloor ,\lfloor n\kappa _2' \rfloor \right\} . \end{aligned}$$

The result follows from the fact that

$$\begin{aligned} \sum _{j=1}^n \left| \omega _j^{\varvec{\kappa }}- \omega _j^{{\varvec{\kappa }}'} \right| ^p\le & {} \sum _{j=1}^n \left| \omega _j^{\varvec{\kappa }}- \omega _j^{{\varvec{\kappa }}'} \right| \\\le & {} \sum _{j=1}^n \left| \omega _j^{(\kappa _1,\kappa _2)} - \omega _j^{(\kappa _1,\kappa _2')} \right| + \sum _{j=1}^n \left| \omega _j^{(\kappa _1',\kappa _2)} - \omega _j^{(\kappa _1',\kappa _2')} \right| , \end{aligned}$$

combined with the inequality

$$\begin{aligned} \sum _{j=1}^n \mathbb {I}\left\{ j \in (\lfloor n\kappa \rfloor ,\lfloor n\kappa ' \rfloor ) \right\} \le n \, \rho _n(\kappa ,\kappa '), \quad \text{ for } \text{ any } \kappa <\kappa ' \in (0,1). \end{aligned}$$

1.2 Proof of Lemma S3.

The proof uses arguments that are similar to the ones used through Example 2.2.12 of van der Vaart and Wellner (1996). For ${\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta$, let

$$\begin{aligned} d_n({\varvec{\kappa }},{\varvec{\kappa }}') = \left\| {\lfloor n{\varvec{\kappa }} \rfloor \over n} - {\lfloor n{\varvec{\kappa }}' \rfloor \over n} \right\| _1 \end{aligned}$$

and set $\widetilde{\varDelta }_n = \lbrace {\varvec{\kappa }}\in \varDelta : {\varvec{\kappa }}= (i/n,j/n), i,j \in [0,n]\cap \mathbb {N}\rbrace$. Observe that since $d_n({\varvec{\kappa }},{\varvec{\kappa }}') \le \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1 + n^{-1}$ and since $\Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1 = d_n({\varvec{\kappa }},{\varvec{\kappa }}')$ when ${\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n$, one deduces from the definition of the $\omega _1^{\varvec{\kappa }}, \ldots , \omega _n^{\varvec{\kappa }}$ that

$$\begin{aligned} \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1 \le \delta - n^{-1} \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right|\le & {} \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \\ d_n({\varvec{\kappa }},{\varvec{\kappa }}') \le \delta \end{array}} \left| \mathbb {G}_n^{{\varvec{\kappa }}} - \mathbb {G}_n^{{\varvec{\kappa }}'} \right| = \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \varDelta : \\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1 \le \delta \end{array}} \left| \mathbb {G}_n^{{\varvec{\kappa }}} - \mathbb {G}_n^{{\varvec{\kappa }}'} \right| . \end{aligned}$$

(7)

Now since $V_{1,n}, \ldots , V_{n,n}$ are centred and bounded by 1,

$$\begin{aligned} { n^3 \over 2^6 \, 6! } \, \textrm{E}\left\{ \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| ^6 \right\}\le & {} \sum _{i=1}^n \left| \omega _i^{\varvec{\kappa }}- \omega _i^{{\varvec{\kappa }}'} \right| ^6 \\{} & {} + \, \left\{ \sum _{i=1}^n \left| \omega _i^{\varvec{\kappa }}- \omega _i^{{\varvec{\kappa }}'} \right| ^2 \right\} \left\{ \sum _{i=1}^n \left| \omega _i^{\varvec{\kappa }}- \omega _i^{{\varvec{\kappa }}'} \right| ^4 \right\} \\{} & {} + \, \left\{ \sum _{i=1}^n \left| \omega _i^{\varvec{\kappa }}- \omega _i^{{\varvec{\kappa }}'} \right| ^3 \right\} ^2 + \left\{ \sum _{i=1}^n \left| \omega _i^{\varvec{\kappa }}- \omega _i^{{\varvec{\kappa }}'} \right| ^2 \right\} ^3 \\\le & {} n \, d_n({\varvec{\kappa }},{\varvec{\kappa }}') +2 n^2 d_n^2 ({\varvec{\kappa }},{\varvec{\kappa }}') + n^3 d_n^3 ({\varvec{\kappa }},{\varvec{\kappa }}') \\\le & {} 4n^3 d_n^3({\varvec{\kappa }},{\varvec{\kappa }}') , \end{aligned}$$

where the one-to-last inequality is a consequence of Lemma 4, whereas the last one derives from the fact that $d_n({\varvec{\kappa }},{\varvec{\kappa }}') = 0$ if ${\varvec{\kappa }}={\varvec{\kappa }}'$ and $d_n({\varvec{\kappa }},{\varvec{\kappa }}') \ge 1/n$ otherwise. Since $d_n({\varvec{\kappa }},{\varvec{\kappa }}') = \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1$ for any ${\varvec{\kappa }},{\varvec{\kappa }}'\in \widetilde{\varDelta }_n$,

$$\begin{aligned} \left[ \textrm{E}\left\{ \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| ^6 \right\} \right] ^{1/6} \le 2 \left( 4 \times 6! \right) ^{1/6} \left\| {\varvec{\kappa }}-{\varvec{\kappa }}' \right\| _1^{1/2}. \end{aligned}$$

Because the process $\{ \mathbb {G}_n^{{\varvec{\kappa }}}: {\varvec{\kappa }}\in {\widetilde{\varDelta }}_n \}$ is separable, Theorem 2.2.4 of van der Vaart and Wellner (1996) with $\psi (x) = x^6$ allows to deduce that there exists $K>0$ (independent of n) such that for all ${\widetilde{\eta }},\delta >0$,

$$\begin{aligned} \left[ \textrm{E}\left\{ \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n:\\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| ^6 \right\} \right] ^{1/6}\le & {} K \left[ \int _0^{{\widetilde{\eta }}} \left\{ D \left( \epsilon ,\sqrt{\Vert \cdot \Vert _1} \right) \right\} ^{1/6} \textrm{d}\epsilon \right. \\{} & {} \left. + \, \delta \left\{ D \left( {\widetilde{\eta }},\sqrt{\Vert \cdot \Vert _1} \right) \right\} ^{1/3} \right] , \end{aligned}$$

where $y\mapsto D(y,\sqrt{\Vert \cdot \Vert _1})\rbrace ^{1/2})$ is the so-called packing number of the set ${\widetilde{\varDelta }}_n$, i.e., the maximum number of y-separated points in ${\widetilde{\varDelta }}_n$ (see e.g., Definition 2.2.3 of van der Vaart and Wellner (1996)). Using the relationship between $D(y,\sqrt{\Vert \cdot \Vert _1})$ and the covering numbers $N(y,\sqrt{\Vert \cdot \Vert _1})$ (i.e., the minimal number of balls of radius y needed to cover ${\widetilde{\varDelta }}_n$, see again Definition 2.2.3 of van der Vaart and Wellner (1996)), one can further write

$$\begin{aligned} \left[ \textrm{E}\left\{ \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n:\\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| ^6 \right\} \right] ^{1/6}\le & {} K \left[ 2 \int _0^{{\widetilde{\eta }}/2} \left\{ N \left( \epsilon ,\sqrt{\Vert \cdot \Vert _1} \right) \right\} ^{1/6} \textrm{d}\epsilon \right. \\{} & {} \left. + \, \delta \left\{ N \left( {\widetilde{\eta }}/2,\sqrt{\Vert \cdot \Vert _1} \right) \right\} ^{1/3} \right] . \end{aligned}$$

Since a $\sqrt{\Vert \cdot \Vert _1}$-ball of radius $\epsilon$ centred at $(z_1,z_2) \in {\widetilde{\varDelta }}_n$ is of the form $\{ (t_1,t_2): |t_1-z_1|+ |t_2-z_2| \le \epsilon ^2 \}$, which covers an $\Vert \cdot \Vert _\infty$-ball of radius $\epsilon ^2/2$ centred at $(z_1,z_2)$, one has

$$\begin{aligned} N \left( y,\sqrt{\Vert \cdot \Vert _1} \right) \le N \left( {y^2\over 2}, \Vert \cdot \Vert _\infty \right) \le \left( y^2 \over 2 \right) ^{-2}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left[ \textrm{E}\left\{ \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n:\\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| ^6 \right\} \right] ^{1/6}\le & {} K \left[ 2 \int _0^{{\widetilde{\eta }}/2} \left( 4\epsilon ^{-4} \right) ^{1/6} \textrm{d}\epsilon + 4 \, \delta \, {\widetilde{\eta }}^{-4/3} \right] \\\le & {} K \left( 2 \, {\widetilde{\eta }}^{1/3} + 4 \, \delta \, {\widetilde{\eta }}^{-4/3} \right) . \end{aligned}$$

Taking ${\widetilde{\eta }} = \delta ^{3/5}$ ensures that

$$\begin{aligned} \left[ \textrm{E}\left\{ \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n:\\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| ^6 \right\} \right] ^{1/6} \le 6 \, K \, \delta ^{1/5}. \end{aligned}$$

Hence, by the Markov inequality, one has for all $\epsilon ,\delta >0$ that

$$\begin{aligned} \mathbb {P}\left( \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n:\\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| >\epsilon \right) \le \epsilon ^{-6} \, (6K)^6 \delta ^{6/5}. \end{aligned}$$

Hence, for every $\epsilon ,\eta >0$, there exists $\delta \equiv (\eta (\epsilon /6K)^6)^{5/6}$ such that

$$\begin{aligned} \mathbb {P}\left( \sup _{\begin{array}{c} {\varvec{\kappa }},{\varvec{\kappa }}' \in \widetilde{\varDelta }_n:\\ \Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta \end{array}} \left| \mathbb {G}_n^{\varvec{\kappa }}- \mathbb {G}_n^{{\varvec{\kappa }}'} \right| >\epsilon \right) \le \eta . \end{aligned}$$

In view of Eq. (7), the first part of the Lemma follows from the above inequality together with the fact that $\Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1^{1/2} \le \delta$ if and only if $\Vert {\varvec{\kappa }}-{\varvec{\kappa }}'\Vert _1 \le \delta ^2$ and the fact that $n>2\delta ^{-1}$ implies $\delta - n^{-1} \ge \delta /2$.

To prove the second part of the Lemma, note that from p. 100 of van der Vaart and Wellner (1996) and calculations similar to those presented above,

$$\begin{aligned} \left[ \textrm{E}\left\{ \sup _{{\varvec{\kappa }}\in \varDelta } \left| \mathbb {G}_n^{\varvec{\kappa }}\right| ^6 \right\} \right] ^{1/6}= & {} \left[ \textrm{E}\left\{ \sup _{{\varvec{\kappa }}\in {\widetilde{\varDelta }}} \left| \mathbb {G}_n^{\varvec{\kappa }}\right| ^6 \right\} \right] ^{1/6} \\\le & {} \left[ \textrm{E}\left\{ \left| \mathbb {G}_n({\varvec{\kappa }}_0) \right| ^6 \right\} \right] ^{1/6} + K \int _0^2 \left\{ D(\epsilon ,\sqrt{\Vert \cdot \Vert _1}) \right\} ^{1/6} \, \textrm{d}\epsilon \\\le & {} 8 \times 6! + 2 K \int _0^2 (4\epsilon ^{-4})^{1/6} \, \textrm{d}\epsilon \\\le & {} 8 \times 6! + 48 K \\\le & {} 4 \times 6! (K+1). \end{aligned}$$

The proof concludes by using Markov’s inequality, so that for any $\epsilon >0$,

$$\begin{aligned} \mathbb {P}\left( \sup _{{\varvec{\kappa }}\in \varDelta } \left| \mathbb {G}_n^{\varvec{\kappa }}\right| > \left\{ 8 \times 6!(K+1) \over \epsilon \right\} ^{1/6} \right) \le \epsilon . \end{aligned}$$

Asymptotic covariance structure of $T_n^{\varvec{\kappa }}= (T_{F,n}^{\varvec{\kappa }},T_{G,n}^{\varvec{\kappa }}$)

1.1 The general case

Define $\varepsilon _F({\varvec{\kappa }},{\varvec{\kappa }}_0) = \mu ^{{\varvec{\kappa }}_0} \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) - \mu ^{\varvec{\kappa }}\, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}_0)$, ${\widetilde{\varepsilon }}_F({\varvec{\kappa }},{\varvec{\kappa }}_0) = (1-\mu ^{{\varvec{\kappa }}_0}) \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) + \mu ^{\varvec{\kappa }}\, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}_0)$, $\varepsilon _G({\varvec{\kappa }},{\varvec{\kappa }}_0) = (1-\mu ^{{\varvec{\kappa }}_0}) \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) - (1-\mu ^{\varvec{\kappa }}) \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}_0)$ and ${\widetilde{\varepsilon }}_G({\varvec{\kappa }},{\varvec{\kappa }}_0) = \mu ^{{\varvec{\kappa }}_0} \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) + (1-\mu ^{\varvec{\kappa }}) \, \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}_0)$, where $\mu ^{\varvec{\kappa }}= \lim _{n\rightarrow \infty } {\bar{\omega }}^{\varvec{\kappa }}= (2-\kappa _1-\kappa _2)/2$. Then for $\delta _{1,j}^{\varvec{\kappa }}= \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) - \mu ^{\varvec{\kappa }}(\omega _j^{\varvec{\kappa }}- {\bar{\omega }}^{\varvec{\kappa }})$ and $\delta _{2,j}^{\varvec{\kappa }}= \varLambda ({\varvec{\kappa }},{\varvec{\kappa }}) + (1-\mu ^{\varvec{\kappa }}) (\omega _j^{\varvec{\kappa }}- {\bar{\omega }}^{\varvec{\kappa }})$,

$$\begin{aligned} \sigma _{FF}({\varvec{\kappa }},{\varvec{\kappa }}')= & {} \lim _{n\rightarrow \infty } {1\over n} \sum _{j=1}^n \textrm{Cov}\left\{ {\widetilde{\varepsilon }}_F({\varvec{\kappa }},{\varvec{\kappa }}_0) \, \delta _{1,j}^{\varvec{\kappa }}\, V_{F,j} + \varepsilon _F({\varvec{\kappa }},{\varvec{\kappa }}_0) \, \delta _{1,j}^{\varvec{\kappa }}\, V_{G,j}, \right. \\{} & {} \left. {\widetilde{\varepsilon }}_F({\varvec{\kappa }}',{\varvec{\kappa }}_0) \, \delta _{1,j}^{{\varvec{\kappa }}'} \, V_{F,j} + \varepsilon _F({\varvec{\kappa }}',{\varvec{\kappa }}_0) \, \delta _{1,j}^{{\varvec{\kappa }}'} \, V_{G,j} \right\} , \\ \sigma _{GG}({\varvec{\kappa }},{\varvec{\kappa }}')= & {} \lim _{n\rightarrow \infty } {1\over n} \sum _{j=1}^n \textrm{Cov}\left\{ \varepsilon _G({\varvec{\kappa }},{\varvec{\kappa }}_0) \, \delta _{2,j}^{\varvec{\kappa }}\, V_{F,j} + {\widetilde{\varepsilon }}_G({\varvec{\kappa }},{\varvec{\kappa }}_0) \, \delta _{2,j}^{\varvec{\kappa }}\, V_{G,j}, \right. \\{} & {} \left. \varepsilon _G({\varvec{\kappa }}',{\varvec{\kappa }}_0) \, \delta _{2,j}^{{\varvec{\kappa }}'} \, V_{F,j} + {\widetilde{\varepsilon }}_G({\varvec{\kappa }}',{\varvec{\kappa }}_0) \, \delta _{2,j}^{{\varvec{\kappa }}'} \, V_{G,j} \right\} , \\ \sigma _{FG}({\varvec{\kappa }},{\varvec{\kappa }}')= & {} \lim _{n\rightarrow \infty } {1\over n} \sum _{j=1}^n \textrm{Cov}\left\{ {\widetilde{\varepsilon }}_F({\varvec{\kappa }},{\varvec{\kappa }}_0) \, \delta _{1,j}^{\varvec{\kappa }}\, V_{F,j} + \varepsilon _F({\varvec{\kappa }},{\varvec{\kappa }}_0) \, \delta _{1,j}^{\varvec{\kappa }}\, V_{G,j}, \right. \\{} & {} \left. \varepsilon _G({\varvec{\kappa }}',{\varvec{\kappa }}_0) \, \delta _{2,j}^{{\varvec{\kappa }}'} \, V_{F,j} + {\widetilde{\varepsilon }}_G({\varvec{\kappa }}',{\varvec{\kappa }}_0) \, \delta _{2,j}^{{\varvec{\kappa }}'} \, V_{G,j} \right\} . \end{aligned}$$

1.2 Special case ${\varvec{\kappa }}={\varvec{\kappa }}_0$

When ${\varvec{\kappa }}= {\varvec{\kappa }}_0$, one has $\varepsilon _F({\varvec{\kappa }},{\varvec{\kappa }}_0) = 0$, ${\widetilde{\varepsilon }}_F({\varvec{\kappa }},{\varvec{\kappa }}_0) = \varLambda ({\varvec{\kappa }}_0,{\varvec{\kappa }}_0)$, $\varepsilon _G({\varvec{\kappa }},{\varvec{\kappa }}_0) = 0$ and ${\widetilde{\varepsilon }}_F({\varvec{\kappa }},{\varvec{\kappa }}_0) = \varLambda ({\varvec{\kappa }}_0,{\varvec{\kappa }}_0)$. It follows easily that

$$\begin{aligned} \sigma _{FF}= & {} \{\varLambda ({\varvec{\kappa }}_0,{\varvec{\kappa }}_0)\}^2 \lim _{n\rightarrow \infty } {1\over n} \sum _{j=1}^n (\delta _{1,j}^{{\varvec{\kappa }}_0})^2 \, \textrm{var}(V_{F,j}), \\ \sigma _{GG}= & {} \{\varLambda ({\varvec{\kappa }}_0,{\varvec{\kappa }}_0)\}^2 \lim _{n\rightarrow \infty } {1\over n} \sum _{j=1}^n (\delta _{2,j}^{{\varvec{\kappa }}_0})^2 \, \textrm{var}(V_{G,j}), \\ \sigma _{FG}= & {} \{\varLambda ({\varvec{\kappa }}_0,{\varvec{\kappa }}_0)\}^2 \lim _{n\rightarrow \infty } {1\over n} \sum _{j=1}^n \delta _{1,j}^{{\varvec{\kappa }}_0} \, \delta _{2,j}^{{\varvec{\kappa }}_0} \, \textrm{Cov}(V_{F,j}, V_{G,j}). \end{aligned}$$

Proof of inequality (4)

First note that

$$\begin{aligned} \langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle = {1 \over n} \sum _{j=1}^n (\omega _j^{\varvec{\kappa }})^2 - \left( {1 \over n} \sum _{j=1}^n \omega _i^{\varvec{\kappa }}\right) . \end{aligned}$$

From the definition of $\omega _j^{\varvec{\kappa }}$,

$$\begin{aligned} \sum _{j=1}^n (\omega _j^{\varvec{\kappa }})^2= & {} \left\{ \begin{array}{ll} n - \lfloor n\kappa _2 \rfloor , &{} \lfloor n\kappa _1 \rfloor = \lfloor n\kappa _2 \rfloor ; \\ \\ n - \lfloor n\kappa _2 \rfloor + \displaystyle \frac{\left( \lfloor n\kappa _2 \rfloor - \lfloor n\kappa _1 \rfloor +1 \right) \left( 2\lfloor n\kappa _2 \rfloor - 2\lfloor n\kappa _1 \rfloor + 1 \right) }{6 \left( \lfloor n\kappa _2 \rfloor - \lfloor n\kappa _1 \rfloor \right) } \,,&{} \lfloor n\kappa _1 \rfloor \le \lfloor n\kappa _2 \rfloor + 1. \end{array} \right. \\\ge & {} \left\{ \begin{array}{ll} \displaystyle 1 - \kappa _2 - {1 \over n} \,, &{} \lfloor n\kappa _1 \rfloor = \lfloor n\kappa _2 \rfloor ; \\ \\ \displaystyle 1 - \kappa _2 + \frac{\kappa _2-\kappa _1}{3} - {1 \over n} + A_n, &{} \lfloor n\kappa _1 \rfloor \le \lfloor n\kappa _2 \rfloor + 1, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} A_n= & {} { (\lfloor n\kappa _2 \rfloor -\lfloor n\kappa _1 \rfloor +1) (2\lfloor n\kappa _2 \rfloor -2\lfloor n\kappa _1 \rfloor +1) \over 6n (\lfloor n\kappa _2 \rfloor -\lfloor n\kappa _1 \rfloor ) } - { \lfloor n\kappa _2 \rfloor -\lfloor n\kappa _1 \rfloor \over 3} \\\ge & {} \left( 2\lfloor n\kappa _2 \rfloor - 2\lfloor n\kappa _1 \rfloor + 1 \over 6n \right) - \left( \lfloor n\kappa _2 \rfloor - \lfloor n\kappa _1 \rfloor \over 3 \right) \\\ge & {} \frac{1}{6n} . \end{aligned}$$

Also,

$$\begin{aligned} \sum _{j=1}^n \omega _j^{\varvec{\kappa }}= & {} \left\{ \begin{array}{ll} n-\lfloor n\kappa _2 \rfloor , &{} \lfloor n\kappa _1 \rfloor = \lfloor n\kappa _2 \rfloor ; \\ \\ n - \lfloor n\kappa _2 \rfloor + \displaystyle { \lfloor n\kappa _2 \rfloor - \lfloor n\kappa _1 \rfloor + 1 \over 2 } , &{} \lfloor n\kappa _1 \rfloor \le \lfloor n\kappa _2 \rfloor + 1 \end{array} \right. \\\le & {} \left\{ \begin{array}{ll} (1-\kappa _2)^2, &{} \lfloor n\kappa _1 \rfloor = \lfloor n\kappa _2 \rfloor ; \\ \\ \displaystyle \left( 1 + \frac{ \kappa _1 + \kappa _2 }{2} \right) ^2 + \frac{3}{n} , &{} \lfloor n\kappa _1 \rfloor \le \lfloor n\kappa _2 \rfloor + 1 \end{array} \right. \end{aligned}$$

As a consequence,

$$\begin{aligned} {1\over n} \sum _{j=1}^n (\omega _i^{\varvec{\kappa }})^2 - \left( {1 \over n} \sum _{j=1}^n \omega _j^{\varvec{\kappa }}\right) ^2 \ge \left\{ \begin{array}{ll} \displaystyle \kappa _2(1-\kappa _2) - {1\over n} , &{} \lfloor n\kappa _1 \rfloor = \lfloor n\kappa _2 \rfloor ; \\ \\ \displaystyle { 2\kappa _1 + \kappa _2 \over 3 } - \left( \kappa _1 + \kappa _2 \over 2 \right) ^2 -{5\over n} , &{} \lfloor n\kappa _1 \rfloor \le \lfloor n\kappa _2 \rfloor + 1. \end{array} \right. \end{aligned}$$

Since $\langle {\varvec{\omega }}^{\varvec{\kappa }}, {\varvec{\omega }}^{\varvec{\kappa }}\rangle \ge 0$ and because $(2\kappa _1+\kappa _2 )/3 - \{(\kappa _1+\kappa _2) / 2\}^2 = \kappa _2 (1-\kappa _2)$ when $\kappa _1=\kappa _2$, it follows that

$$\begin{aligned} \langle {\varvec{\omega }}^{\varvec{\kappa }},{\varvec{\omega }}^{\varvec{\kappa }}\rangle \ge \max \left\{ { 2\kappa _1+\kappa _2 \over 3} - \left( \kappa _1+\kappa _2 \over 2 \right) ^2 - {5 \over 6n}, \,0 \right\} . \end{aligned}$$

Useful identities

The following identities will be established:

$$\begin{aligned}{} & {} \displaystyle \left( {\mathcal {I}}_1\right) ~ {1\over n} \sum _{k=1}^n \omega _k^{{\varvec{\kappa }}_0} \, \theta _k^{\varvec{\kappa }}= \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0};\\{} & {} \displaystyle \left( {\mathcal {I}}_2\right) ~ {1\over n} \sum _{k=1}^n (1-\omega _k^{{\varvec{\kappa }}_0}) \theta _k^{\varvec{\kappa }}= 1 - \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0};\\{} & {} \displaystyle \left( {\mathcal {I}}_3\right) ~ {1\over n} \sum _{k=1}^n \omega _k^{{\varvec{\kappa }}_0} \, \gamma _k^{\varvec{\kappa }}= \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0};\\{} & {} \displaystyle \left( {\mathcal {I}}_4\right) ~ {1\over n} \sum _{k=1}^n (1-\omega _k^{{\varvec{\kappa }}_0}) \gamma _k^{\varvec{\kappa }}= 1 - \beta _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}. \end{aligned}$$

To show Identity ${\mathcal {I}}_1$,

$$\begin{aligned} \sum _{k=1}^n \omega _k^{{\varvec{\kappa }}_0} \, \theta _k^{\varvec{\kappa }}= \sum _{k=1}^n \omega _k^{{\varvec{\kappa }}_0} \left\{ 1 - { {\bar{\omega }}^{\varvec{\kappa }}\over \langle \omega ^{\varvec{\kappa }}, \omega ^{\varvec{\kappa }}\rangle } \left( \omega _k^{\varvec{\kappa }}- {\bar{\omega }}^{\varvec{\kappa }}\right) \right\} = n \left( {\bar{\omega }}^{{\varvec{\kappa }}_0} - {\bar{\omega }}^{\varvec{\kappa }}\, { \langle \omega ^{{\varvec{\kappa }}_0}, \omega ^{\varvec{\kappa }}\rangle \over \langle \omega ^{\varvec{\kappa }}, \omega ^{\varvec{\kappa }}\rangle } \right) = n \, \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0}. \end{aligned}$$

Identity ${\mathcal {I}}_2$ follows from

$$\begin{aligned} \sum _{k=1}^n \left( 1-\omega _k^{{\varvec{\kappa }}_0} \right) \theta _k^{\varvec{\kappa }}= \sum _{k=1}^n \theta _k^{\varvec{\kappa }}- \sum _{k=1}^n \omega _k^{{\varvec{\kappa }}_0} \, \theta _k^{\varvec{\kappa }}= n \left( 1 - \alpha _n^{{\varvec{\kappa }},{\varvec{\kappa }}_0} \right) . \end{aligned}$$

Identities ${\mathcal {I}}_3$ and ${\mathcal {I}}_4$ are established in a similar way.