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Asymptotics for random functions moderated by dependent noise

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Abstract

We study limit theorems for a wide class of multi-parameter stochastic processes which are driven by a noise process which may be weakly or even long-range dependent. The processes under study arise in diverse areas and fields such as functional data analysis, life science, engineering and finance. It turns out that under fairly weak conditions on the underlying noise process the limiting law of the corresponding partial sum process is a consequence of the weak convergence of the sequential empirical Kiefer process. The asymptotic limit theory covers the classical large sample situation as well as a general change-point model which extends the location-scale model often considered in change-point analysis. The scope of the results is illustrated by various applications.

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Acknowledgments

The author thanks Dr. A. Prause for careful proof-reading of a previous and the final version, and, in particular, for a proposal for a simplified argument to show (6.8) under the weak total variation assumption.

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Correspondence to Ansgar Steland.

Appendix: Proofs

Appendix: Proofs

Let \( l \in {\mathbb {N}} \). It is worth recalling the following facts on the relevant Skorohod spaces. For brevity of exposition, we consider the case \( D( [0, 1]^d; {\mathbb {R}} ^l ) \), the required modifications for the spaces \( D( [0,1] \times [a, b]; {\mathbb {R}} ^l ) \), \( -\infty \le a < b \le \infty \) being straightforward. In particular, the case \( [a,b] = [-\infty , \infty ] \) is treated by noting that \( [-\infty , \infty ] \) is a complete, separable and metric space when equipped with the metric \(\tilde{d}(x, y) = | G(x) - G(y) | \), \( x, y, \in [-\infty , \infty ] \), where G is strictly increasing, continuous and bounded, e.g., \( G(x) = \text {atan}(x) \). Let \( \Lambda \) denote the class of homeomorphisms of [0, 1] onto itself with \( \lambda (0) = 0 \). In general, for any homeomorphism \( \lambda \) of, say, \( [0,1]^d \), we may define the slope norm by

$$\begin{aligned} \Vert \lambda \Vert ^0 = \sup _{x, y \in [0,1]^d, x \not = y } \left| \log \frac{ \Vert \lambda (y) - \lambda (x) \Vert }{ \Vert y - x \Vert } \right| < \infty , \end{aligned}$$

where \( \Vert \bullet \Vert \) denotes an arbitrary vector norm on \( {\mathbb {R}} ^d \). We consider the group of homeomorphisms of \( [0, 1]^d \) given by \( \Lambda ^d = \Lambda \times \cdots \times \Lambda \), where \( \lambda (t) = (\lambda _1(t), \dots , \lambda _d(t) ) \), \( t = (t_1, \dots , t_d) \in [0, 1]^d \), for \( \lambda \in \Lambda ^d \). A partition \( {\mathcal {D}}\) being invariant under \( \Lambda ^d \) is given by \( {\mathcal {D}}= \{ \lambda ( \Pi _k ) : \lambda \in \Lambda ^d, k \in {\mathbb {N}} _0 \} \), where \( \Pi _k \) is the partition of cells \( A_{i_1, \dots , i_d} = [ (i_1-1)/2^k, i_1/2^k ) \times \cdots \times [(i_d-1)/2^k, i_d/2^k) \), \( 1 \le i_1, \dots , i_d \le 2^k \), \( k \in {\mathbb {N}} _0 \). The simple functions are those functions assuming a constant value on each cell \( D \in {\mathcal {D}}\) and \( D([0,1]^d; {\mathbb {R}} ^l) \) then consists, by definition, of the uniform limits of sequences of such simple functions. The (finite slope) Skorohod metric on \( D([0,1]^d; {\mathbb {R}} ^l) \) is now defined by

$$\begin{aligned} d( f, g ) = d_{[0,1]}( f, g ) = \inf \{ \varepsilon > 0 : \exists \lambda \in \Lambda ^d : \Vert \lambda \Vert ^0 < \varepsilon , \Vert f - g \circ \lambda \Vert _\infty < \varepsilon \}. \end{aligned}$$

By Straf (1972, Theorems 3.7, 3.10), \( ( D([0,1]^d; {\mathbb {R}} ^l), d ) \) is a complete and separable metric space.

Further, recall that the space \( D( [0,1] \times {\mathbb {T}}; {\mathbb {R}} ^l ) \) of càdlàg functions is defined analogously, since \( E = [0,1] \times {\mathbb {T}}\) is a complete metric space.

1.1 Proofs of Section 3

Proof of Theorem 3.1

Observe that \( W_N( u, t ) \) can be represented via the sequential Kiefer process in the form

$$\begin{aligned} W_N(u,t)&= r_N \sum _{i=1}^{ {\lfloor Nu \rfloor } } Z_{ {\lfloor Nu \rfloor } ,i} [ \phi ( Y_i; t ) - \vartheta ( F; t ) ] \\&= r_N \int \phi ( x; t) \, {\lfloor Nu \rfloor } [\widehat{F}_{ {\lfloor Nu \rfloor } }(dx) - F(dx)] \\&= \int \phi (x;t) \, r_N {\lfloor Nu \rfloor } [ \widehat{F}_{ {\lfloor Nu \rfloor } }(dx) - F(dx)] \\&= \int \phi (x;t) \, {\mathcal {F}}_N(u,dx), \end{aligned}$$

for \( (u,t) \in [0,1] \times {\mathbb {T}}\). By assumption,

$$\begin{aligned} {\mathcal {F}}_N(u,x) \Rightarrow {\mathcal {F}}(u,x), \end{aligned}$$

as \( N \rightarrow \infty \). Since the Skorohod space \(D( [0,1] \times [-\infty ,\infty ]; {\mathbb {R}} )\) is Polish, we may apply the Skorohod/Dudley/Wichura representation theorem in general metric spaces to obtain equivalent versions of \( {\mathcal {F}}, {\mathcal {F}}_N \), \( N \ge 1 \), defined on a new probability space, which converge a.s. in the Skorohod metric. For simplicity of notation, we denote these equivalent versions again by \( {\mathcal {F}}_N \) and \( {\mathcal {F}}\), such that

$$\begin{aligned} P\biggl ( \lim _{N \rightarrow \infty } d( {\mathcal {F}}_N, F ) = 0 \biggr ) = 1. \end{aligned}$$
(6.1)

By definition of the metric d , there exist sequences \( \{ \lambda _{Nk} : N \in {\mathbb {N}} \} \), \( k = 1,2 \), of strictly increasing continuous mappings on [0, 1] and \( [-\infty , \infty ] \), respectively, such that

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{x \in [-\infty ,\infty ]} | {\mathcal {F}}_N(\lambda _{N1}(u), \lambda _{N2}(x)) - {\mathcal {F}}(u,x) | \rightarrow 0, \qquad a.s., \end{aligned}$$
(6.2)

and

$$\begin{aligned} \Vert \lambda _{N1} \Vert ^0, \ \Vert \lambda _{N2} \Vert ^0 \rightarrow 0, \end{aligned}$$
(6.3)

as \( N \rightarrow \infty \). Notice that for \( {\mathcal {F}}\in C( [0,1] \times [-\infty , \infty ]; {\mathbb {R}} ) \) (6.2) and (6.3) hold true with \( \lambda _{N1} = {\text {id}}_{[0,1]} \) and \( \lambda _{N2} = {\text {id}}_{[-\infty , \infty ]} =: {\text {id}}\), \( N \ge 1 \).

To treat the càdlàg case, we need the following preparation. Since the strictly increasing transformations \( ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1} ) \) converge to the strictly increasing and continuous mapping \( ({\text {id}}_{[0,1]}, {\text {id}}_{[-\infty ,\infty ]} ) \), as \( N \rightarrow \infty \), Whitt (1980, Th. 3.1) yields

$$\begin{aligned} {\mathcal {F}}\circ ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1} ) \Rightarrow {\mathcal {F}}, \end{aligned}$$

in \( D( [0,1] \times [-\infty ,\infty ]; {\mathbb {R}} ) \), as \( N \rightarrow \infty \). (The proof of Whitt carries over with appropriate norms: The mapping \( \mu _n \) constructed there is now constructed for both coordinates). Thus, by invoking again the Skorohod/Dudley/Wichura representation theorem, we may and shall assume in what follows that, additionally,

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{x \in [-\infty ,\infty ]} | {\mathcal {F}}\circ ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1} )(u,x) - {\mathcal {F}}(u,x) | \rightarrow 0, \end{aligned}$$
(6.4)

as \( N \rightarrow \infty \), a.s.

Let us first consider the case \( {\mathcal {F}}\in C( [0,1] \times [-\infty , \infty ]; {\mathbb {R}} ) \) a.s., to prepare the more involved arguments for the càdlàg case. Observe that, by definition of the Stieltjes integral,

$$\begin{aligned} W_N(u,t) - \int \phi ( x; t ) \, d {\mathcal {F}}(u, dx)&= \int \phi ( x; t ) ( {\mathcal {F}}_N - {\mathcal {F}})(u, dx) \\&= \lim _{A \rightarrow -\infty , B \rightarrow \infty } \int _A^B \phi (x; t) ({\mathcal {F}}_N - {\mathcal {F}})(u, dx) \\&= \lim _{A \rightarrow -\infty , B \rightarrow \infty } W_N^{(A,B)}(u,t) - W^{(A,B)}(u,t), \end{aligned}$$

where \( W_N^{(A,B)}(u,t) = \int _A^B \phi (x; t) \, {\mathcal {F}}_N(u,dx) \) and \( W^{(A,B)}(u,t) = \int _A^B \phi (x; t) \, {\mathcal {F}}(u,dx) \), \( -\infty < A < B < \infty \). Observe that

$$\begin{aligned}&\sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \left| W_N(u,t) - \int \phi (x;t) \, d {\mathcal {F}}(u,dx) \right| \\&\qquad =\sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \left| \lim _{A \rightarrow -\infty , B \rightarrow \infty } W_N^{(A,B)}(u,t) - W^{(A,B)}(u,t) \right| \\&\qquad \le \sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \lim _{A \rightarrow -\infty , B \rightarrow \infty } \left| W_N^{(A,B)}(u,t) - W^{(A,B)}(u,t) \right| \\&\qquad \le \lim _{A \rightarrow -\infty , B \rightarrow \infty } \sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \left| W_N^{(A,B)}(u,t) - W^{(A,B)}(u,t) \right| . \end{aligned}$$

Thus, provided we can show that \( \left| W_N^{(A,B)}(u,t) - W^{(A,B)}(u,t) \right| = o(1) \), a.s., uniformly in \( u \in [0,1] \) and \( t \in {\mathbb {T}}\), where in the càdlàg case we shall have to implement appropriately chosen homeomorphisms, the result follows.

Thus, fix \( - \infty < A < B < \infty \). Integration by parts yields

$$\begin{aligned}&W_N^{(A,B)}(u,t) - \int _A^B \phi (x; t) \, {\mathcal {F}}(u,dx)\\&\quad = \int _A^B ({\mathcal {F}}_N - {\mathcal {F}})(u, x) \, \phi (dx; t) + ({\mathcal {F}}_N - {\mathcal {F}})(u, x) \phi (x; t) \biggr |_{x=A}^{x=B}. \end{aligned}$$

The right-hand side can be bounded by

$$\begin{aligned}&2 \sup _{u \in [0,1]} \sup _{x \in [A, B]} | {\mathcal {F}}_N(u,x) - {\mathcal {F}}(u,x) | \left( \sup _{t \in {\mathbb {T}}} \Vert \phi ( \bullet ; t ) \Vert _{TV,[A, B|} + \sup _{t \in {\mathbb {T}}} \sup _{x \in [A, B]} | \phi (x; t) | \right) \\&\quad \le 2 (C_{TV} + C_{\sup }) \sup _{u \in [0,1]} \sup _{x \in [-\infty ,\infty ]} | {\mathcal {F}}_N(u,x) - {\mathcal {F}}(u,x) |, \end{aligned}$$

which is o(1) , as \( N \rightarrow \infty \), a.s. Thus the result is shown. Having in mind the corollary, notice that if \( \Vert {\mathcal {F}}_N - {\mathcal {F}}\Vert _\infty \rightarrow 0 \), as \( N \rightarrow \infty \), a.s., on the original probability space, we obtain the stronger result of a.s. uniform convergence, \( \Vert W_N - W \Vert _\infty \rightarrow 0 \), as \( N \rightarrow \infty \), a.s. In the general case, the weak convergence \( W_N \Rightarrow W \), as \( N \rightarrow \infty \), follows by the second half of the Skorohod/Dudley/Wichura representation theorem.

To establish the result for the càdlàg case, let \( \lambda _{N1} : [0,1] \rightarrow [0,1] \) and \( \lambda _{N2} : [-\infty , \infty ] \rightarrow [-\infty , \infty ] \), \( N \ge 1 \), be homeomorphisms such that

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{x \in [-\infty , \infty ]} | {\mathcal {F}}_N \circ (\lambda _{N1}, \lambda _{N2}) (u,x) - {\mathcal {F}}(u,x) | = o(1), \end{aligned}$$

as \( N \rightarrow \infty \), a.s. For \( -\infty < A < B < \infty \) define

$$\begin{aligned} \widetilde{W}_N^{(A,B)}(u,t)&= W_N^{(A,B)} \circ (\lambda _{N1}, {\text {id}}_{{\mathbb {T}}}) (u,t) \\&= \int _A^B \phi ( x; t ) \, {\mathcal {F}}_N \circ (\lambda _{N1},{\text {id}}_{[A,B]}) (u,dx), \end{aligned}$$

\( u \in [0,1] \), \( t \in {\mathbb {T}}\). We shall show that

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \left| \widetilde{W}_N^{(A,B)}(u,t) - W^{(A,B)}(u,t) \right| = O\left( (C_{TV} + C_{\sup } ) \Vert {\mathcal {F}}_N \circ ( \lambda _{N1}, \lambda _{N2} ) - {\mathcal {F}}\Vert _\infty \right) , \end{aligned}$$
(6.5)

as \( N \rightarrow \infty \), a.s., which yields

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \left| W_N(u,t) - \int \phi (x,t) {\mathcal {F}}(u,dx) \right| = o(1), \end{aligned}$$

a.s., as \( N \rightarrow \infty \), i.e., \( W_N \Rightarrow W \) for the original versions.

To verify (6.5) put \( \phi _A^B(x;t) = \phi (x;t) { 1}_{[A,B]}(x) \). Recalling that \( {\mathcal {F}}_N(u,x) = r_N {\lfloor Nu \rfloor } [ F_{ {\lfloor Nu \rfloor } }(x) - F(x) ] \), we have

$$\begin{aligned} \widetilde{ W } _N^{(A, B)}(u,t)&= \int _A^B \phi (x;t) \, d {\mathcal {F}}_N \circ ( \lambda _{N1}, {\text {id}}_{[A,B]} )(u, dx) \\&= \int \phi _A^B(x;t) {\mathcal {F}}_N \circ ( \lambda _{N1}, {\text {id}}_{[A,B]} )( u, dx ). \end{aligned}$$

Recall that \( \lambda _{N1} \) and \( \lambda _{N2} \) are homeomorphisms of [0, 1] and \( [-\infty , \infty ] \), respectively, which ensure (6.2). Since

$$\begin{aligned} F_{ {\lfloor N \lambda _{N1}(u) \rfloor } }( \lambda _{N2}(x) ) = \frac{1}{ {\lfloor N \lambda _{N1}(u) \rfloor } } \sum _{i=1}^{ {\lfloor N \lambda _{N1}(u) \rfloor } } Z_{ {\lfloor N \lambda _{N1}(u) \rfloor } ,i } { 1}( \lambda _{N2}^{-1}( Y_i ) \le x ), \end{aligned}$$

we have

$$\begin{aligned} \widetilde{W}_N^{(A,B)}(u,t)&= r_N \sum _{i=1}^{ {\lfloor N\lambda _{N1}(u) \rfloor } } Z_{ {\lfloor N \lambda _{N1}(u) \rfloor } ,i } \left[ \phi _A^B( Y_i;t) - \int _A^B \phi (x;t) \, d F(x) \right] \\&= r_N \sum _{i=1}^{ {\lfloor N \lambda _{N1}(u) \rfloor } } Z_{ {\lfloor N \lambda _{N1}(u) \rfloor } ,i } \phi _A^B( \lambda _{N2} \circ \lambda _{N2}^{-1}(Y_i) ; t ) \\&\quad \; - {\lfloor N\lambda _{N1}(u) \rfloor } r_N \int _A^B \phi ( \lambda _{N2}(x); t ) \, dF( \lambda _{N2}(x) ) \\&= r_N \int \phi _A^B( \lambda _{N2}(x); t ) d \sum _{j=1}^{ {\lfloor N \lambda _{N1}(u) \rfloor } } Z_{ {\lfloor N \lambda _{N1}(u) \rfloor } ,j } { 1}( Y_j \le \lambda _{N2}(x) ) \\&\quad \; - {\lfloor N\lambda _{N1}(u) \rfloor } r_N \int _A^B \phi ( \lambda _{N2}(x); t ) \, dF( \lambda _{N2}(x) ) \\&= \int \phi _N^{(A,B)}(x;t) {\mathcal {F}}_N \circ \lambda _N (u, dx), \end{aligned}$$

for \( (u,t) \in [0,1] \times [a,b] \), where

$$\begin{aligned} \phi _N^{(A,B)}(x;t) = \phi _A^B( \lambda _{N2}(x); t ) \quad \text {and} \quad \lambda _N = (\lambda _{N1}, \lambda _{N2} ). \end{aligned}$$

Notice that \( \phi _N^{(A,B)} = \phi _N \) on \( [A,B] \times {\mathbb {T}}\), where

$$\begin{aligned} \phi _N(x;t) = \phi ( \lambda _{N2}(x);t), \qquad \text {for } (x,t) \in [-\infty , \infty ] \times {\mathbb {T}}, \end{aligned}$$

since \( \phi _A^B = \phi \) on \( [A,B] \times {\mathbb {T}}\) and \( \lambda _{N2} \), \( N \ge 1 \), are homeomorphisms of \( [-\infty ,\infty ] \). It follows that

$$\begin{aligned} \widetilde{W}_N^{(A,B)}(u,t) = \int _A^B \phi _N(x;t) \, {\mathcal {F}}_N \circ \lambda _N(u,dx), \qquad u \in [0,1], \quad t \in {\mathbb {T}}. \end{aligned}$$

We will now show that

$$\begin{aligned}&\sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} \left| \int _A^B \phi _N(x;t) \, {\mathcal {F}}_N \circ \lambda _N( u, dx ) - W^{(A,B)}(u,t) \right| \nonumber \\&\qquad = O\left( (C_{TV} + C_{\sup } ) \Vert {\mathcal {F}}_N \circ (\lambda _{N1}, \lambda _{N2} ) - {\mathcal {F}}\Vert _\infty \right) , \end{aligned}$$
(6.6)

as \( N \rightarrow \infty \), a.s. To do so, we employ the decomposition

$$\begin{aligned} \int _A^B \phi _N(x;t) {\mathcal {F}}_N \circ \lambda _N( u, dx ) - \int _A^B \phi (x;t) \, {\mathcal {F}}( u, dx ) = I_{N1}(u,t) + I_{N2}(u,t) \rightarrow 0,\quad \end{aligned}$$
(6.7)

as \( N \rightarrow \infty \), a.s., where

$$\begin{aligned} I_{N1}&= I_{N1}(u,t) = \int _A^B \phi _N(x;t) \, ({\mathcal {F}}_N \circ \lambda _N - {\mathcal {F}})(u,dx), \\ I_{N2}&= I_{N2}(u,t) = \int _A^B [\phi _N (x;t) - \phi (x;t)] \, {\mathcal {F}}(u,dx), \end{aligned}$$

as \( N \rightarrow \infty \). Let us first consider \( I_{N2} \). Recalling that \( \phi _N(x;t) = \phi ( \lambda _{N2}(x); t ) \), \( (x,t) \in [A,B] \times {\mathbb {T}}\) and the change of variable formula for Stieltjes integrals,

$$\begin{aligned} \int _{\lambda (a)}^{\lambda (b)} f(t) \, d g(t) = \int _a^b f( \lambda (x) ) \, d g( \lambda (x) ), \end{aligned}$$

see e.g., Apostol (1974, Th. 7.4), we obtain

$$\begin{aligned} \int _A^B \phi \circ ( \lambda _{N2}, {\text {id}}_{{\mathbb {T}}} )(x;t) {\mathcal {F}}(u,dz)&= \int _{\lambda _{N2}(A)}^{\lambda _{N2}(B)} \phi (z;t) {\mathcal {F}}\circ ({\text {id}}_{[0,1]}, \lambda _{N2}^{-1})( u, dz ) \\&= \int _A^B \phi (x; t ) {\mathcal {F}}\circ ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1})( u, dx ). \end{aligned}$$

Thus,

$$\begin{aligned} I_{N2}&= \int _A^B \phi (x;t) \, [{\mathcal {F}}\circ ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1}) - {\mathcal {F}}] (u,dx) \\&= \phi (x;t) [{\mathcal {F}}\circ ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1}) - {\mathcal {F}}] \biggr |_{x=A}^{x=B} - \int _A^B [{\mathcal {F}}\circ ( {\text {id}}_{[0,1]}, \lambda _{N2}^{-1}) - {\mathcal {F}}] \, \phi ( dx; t ), \end{aligned}$$

such that, by virtue of (6.4), a.s.

$$\begin{aligned}&\sup _{u \in [0,1]} \sup _{x \in [A, B]} | I_{N2}(u,x) | \\&\quad \le C \sup _{u \in [0,1]} \sup _{x \in [-\infty ,\infty ]} | {\mathcal {F}}\circ ({\text {id}}_{[0,1]}, \lambda _{N2}^{-1})(u,x) - {\mathcal {F}}(u,x) | { 1}_{[A,B]}(x) \\&\quad = o(1), \end{aligned}$$

as \( N \rightarrow \infty \), where \( C = 2 ( C_{TV} + C_{\sup } ) \).

To handle \( I_{N1} \), let us verify that \( x \mapsto \phi _N(x;t) \) is of uniformly bounded variation on [AB] . Let \( A = x_0 < \cdots < x_n = B \) be an arbitrary partition of [AB] . Notice that \( \{ \lambda _{N2}( x_i ) : i = 1, \dots , n \} \) forms again a partition of [AB] (that depends on N), since the \( \lambda _{N2} \) are increasing bijections. But then we have

$$\begin{aligned}&\sup _{N \ge 1} \sum _{i=0}^{n-1} | \phi ( \lambda _{N2}(x_{i+1});t) - \phi ( \lambda _{N2}(x_i); t ) | \nonumber \\&\qquad \le \sup _{N \ge 1} \sup _{t \in {\mathbb {T}}} \sup _{A = z_0 < \cdots < z_L = B} \sum _{i=0}^{L-1} | \phi ( z_{i+1};t) - \phi ( z_i;t) | \nonumber \\&\qquad \le \sup _{N \ge 1} \sup _{t \in {\mathbb {T}}} \Vert \phi ( \bullet ; t) \Vert _{TV,[A,B]} \le C_{TV}, \end{aligned}$$
(6.8)

a.s., as \( N \rightarrow \infty \), which verifies that, a.s.,

$$\begin{aligned} \sup _{A, B \in {\mathbb {R}} , A \le B} \sup _{N \ge 1} \sup _{t \in {\mathbb {T}}} \Vert \phi \circ (\lambda _{N2}, {\text {id}}_{{\mathbb {T}}} )(\bullet ;t) \Vert _{TV,[A,B]} \le C_{TV}. \end{aligned}$$

But now the required estimate of \( I_{N1} \) follows easily again by integration by parts.

$$\begin{aligned}&\sup _{u \in [0,1]} \sup _{t \in {\mathbb {T}}} | I_{N1}(u,t) | \\&\qquad = \sup _{u\in [0,1]} \sup _{t \in {\mathbb {T}}} \left| \int ({\mathcal {F}}_N \circ \lambda _N - {\mathcal {F}})(u,x) { 1}_{[A,B]}( x ) \, \phi _N( dx; t ) \right| \\&\qquad \le C^{\prime } \sup _{s \in [0,1]} \sup _{x \in [-\infty ,\infty ]} | {\mathcal {F}}_N \circ \lambda _N(u,x) - {\mathcal {F}}(u,x) | { 1}_{[A,B]}(x), \end{aligned}$$

which is \(o_P(1)\) as \( N \rightarrow \infty \), where \( C^{\prime } = 2( C_{TV} + C_{\sup } ) \).

This proves (6.6) which in turn implies that

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{x \in [A, B]} \sup _{t \in {\mathbb {T}}} \left| \widetilde{W}_N^{(A,B)}(u,x) - \int _A^B \phi (x;t) {\mathcal {F}}(u,dx) \right| = o(1), \end{aligned}$$

as \( N \rightarrow \infty \), a.s., i.e., (6.5) is established. \(\square \)

Proof of Corollary 3.2

By periodicity, we have

$$\begin{aligned} \int \phi (y; t) \, d F(y)&= \sum _{k \in {\mathbb {Z}} } \int _{k\kappa /t}^{(k+1) \kappa /t} \varphi (yt) f(y) \, d y \\&= \sum _{k \in {\mathbb {Z}} } \int _0^{\kappa /t} \varphi ( z ) f( z + k\kappa / t ) \, dz. \end{aligned}$$

Let \( \varepsilon > 0 \). Since \( f(x) = O (|x|^{-\gamma }) \), if \( |x| \rightarrow \infty \), we may find \( k_0 \) such that

$$\begin{aligned} \int \varphi (yt) \, d F(y)&= \sum _{|k| \le k_0} \int _0^{\kappa /t} \varphi (z) f( z + k\kappa /t ) \, dz + O\left( \frac{\kappa }{t} \sum _{|k| > k_0} \left( \frac{k \kappa }{t} \right) ^{-\gamma } \right) \\&= \int \varphi ( yt ) { 1}_{[-k_0 \kappa /t, k_0 \kappa /t]}(y) \, d F(y) + \varepsilon . \end{aligned}$$

Obviously, if \( t \ge t_0 > 0 \) for some \( t_0 \), the O term is uniform in \( t \in {\mathbb {T}} \cap [t_0,\infty ) \). \( \Box \)

Proof of Lemma 3.1

By boundedness of \( \phi \) and Davydov’s inequality, we have

$$\begin{aligned}&E\left( \int _{[-a,a]^c} \phi (y;t) {\mathcal {F}}_N(u,dy) \right) ^2 \\&\qquad = \frac{1}{N} \sum _{i,j=1}^{ {\lfloor Nu \rfloor } } \mathrm{Cov}\,( \phi (Y_i;t) { 1}( |Y_i | > a), \phi ( Y_j; t ) { 1}( | Y_j | > a ) ) \\&\qquad \le \Vert \phi (Y_1;t) { 1}( |Y_1| > a ) \Vert _{2+\delta }^2 \frac{8}{N} \sum _{i, j = 1}^N \alpha ( |i-j| )^{\delta /(2+\delta )}. \end{aligned}$$

Hölder’s inequality and dominated convergence thus imply

$$\begin{aligned}&E\left( \int _{[-a,a]^c} \phi (y;t) {\mathcal {F}}_N(u,dy) \right) ^2 \\&\qquad = O\left( \left( \int |\phi ( y;t )|^{2+\delta } { 1}( |y| > a ) \, d F(y) \right) ^{2/(2+\delta )} \sum _k \alpha (k)^{2/(2+\delta )} \right) \\&\qquad = O\left( \Vert \phi ( \bullet ; t ) \Vert _{(2+\delta )p}^2 \left( \int { 1}( |y| > a ) d F(y) \right) ^{\frac{p-1}{p}} \right) \\&\qquad = o(1), \end{aligned}$$

as \( a \rightarrow \infty \). \(\square \)

Proof Theorem 3.2

We may assume that \( \{ X_n : n \in {\mathbb {Z}} \} \) is a strictly stationary sequence of random curves satisfying the assumptions, from which we observe \( X_1, \dots , X_N \). We equip \(L_2[a,b] \) with the Borel \( \sigma \)-field associated to the metric induced by the inner product. Recall that the \( \alpha \)-mixing coefficients are given by \( \alpha (k) = \sup _{A \in {\mathcal {F}}_{-\infty }^0, B \in {\mathcal {F}}_k^\infty } | P(A \cap B) - P(A) P(B) | \), \( k \in {\mathbb {N}} _0 \), where \( {\mathcal {F}}_{-\infty }^0 = \sigma ( X_i : i \le 0 ) \) and \( {\mathcal {F}}_k^{\infty } = \sigma ( X_i : i \ge k ) \). Clearly, the strictly stationary time series \( Y_{i1} = \int _a^b X_i(u) \varphi _1(u) \, du, i \ge 1 \), inherits those mixing coefficients. We have

$$\begin{aligned} | Y_{i1} | \le \Vert X_i \Vert _{L_2[a,b]} \Vert \varphi _1 \Vert _{L_2[a,b]} \le K \Vert \varphi _1 \Vert _{L_2[a,b]}, \qquad a.s., i \ge 1, \end{aligned}$$

such that Assumption (A) holds true by virtue of Yoshihara (1979). We may put \( {\mathbb {Y}} = [-L,L] \) with \(L = K \Vert \varphi _1 \Vert _{L_2[a,b]} \Vert \varphi _1 \Vert _\infty \). By continuity of the covariance operator, \( \Vert \varphi _1 \Vert _\infty < \infty \) holds true. Further, \( \sup _{t \in {\mathbb {T}}} \widetilde{ \phi } ( \bullet ; t ) \) is bounded (on \({\mathbb {Y}}\)) and \( \Vert \widetilde{ \phi } ( \bullet ; t ) \Vert _{TV,{\mathbb {Y}}} \le 2 K \Vert \varphi _1 \Vert _{L_2[a,b]} \Vert \varphi _1 \Vert _\infty \), such that \( C_{TV} < \infty \), a.s. This completes the proof. \(\square \)

1.2 Proofs of Section 4

We show Theorem 4.1, which provides the asymptotic null distribution of \( T_N(u;t) \), and Theorem 4.3, which shows that under the general transformation change-point model the underlying process \( {\mathcal {F}}_N(s,x) \), which induces \( T_N(u;t) \) via the representation

$$\begin{aligned} T_N(u;t) = \sup _{0 \le s \le {\lfloor Nu \rfloor } / {N} } \left| \int {\phi }(x;t) \, {\mathcal {F}}_N(s, dx) - \frac{ {\lfloor Ns \rfloor } }{ {\lfloor Nu \rfloor } } \int \limits {\phi }( x; t ) \, {\mathcal {F}}_N( u, dx ) \right| , \end{aligned}$$
(5.9)

is affected by a drift which yields the detection power.

Proof of Theorem 4.1

Again we apply the Skorohod/Dudley/Wichura representation theorem, assume w.l.o.g. that \( {\mathcal {F}}_N \) converges to \( {\mathcal {F}}\) in the Skorohod metric and fix \( - \infty < a < b < \infty \). Notice that for \( u \in [1/N,1] \) and \( t \in {\mathbb {T}}\)

$$\begin{aligned} T_N(u;t)&= \max _{1 \le k \le {\lfloor Nu \rfloor } } r_N \left| \sum _{i=1}^k [ \phi (Y_i;t) - \overline{\phi }_{ {\lfloor Nu \rfloor } } (t) ] \right| \\&= \max _{1 \le k \le {\lfloor Nu \rfloor } } r_N \left| \sum _{i=1}^k [ \widetilde{\phi }(Y_i;t) - \overline{\widetilde{\phi }}_{ {\lfloor Nu \rfloor } } (t) ] \right| \\&= \max _{0 \le k \le {\lfloor Nu \rfloor } } \left| r_N \sum _{i=1}^k \widetilde{\phi }(Y_i; t) - \frac{k}{ {\lfloor Nu \rfloor } } r_N \sum _{i=1}^{ {\lfloor Nu \rfloor } } \widetilde{\phi }(Y_i;t) \right| \\&= \sup _{0 \le r \le {\lfloor Nu \rfloor } /N} r_N \left| {\lfloor Nr \rfloor } \int \widetilde{ \phi } (x;t) \, d F_{ {\lfloor Nr \rfloor } } - {\lfloor Nr \rfloor } \int \widetilde{ \phi } (x;t) \, d F_{ {\lfloor Nu \rfloor } } \right| , \end{aligned}$$

where

$$\begin{aligned} \widetilde{\phi }(Y_i ; t )&= \phi ( Y_i; t ) - \vartheta ( F_0 ; t ), \qquad 1 \le i \le N, \\ \overline{\widetilde{\phi }}_N(t)&= N^{-1} \sum _{i=1}^N \widetilde{\phi }(Y_i; t ), \\ \overline{\phi }_N(t)&= N^{-1} \sum _{i=1}^N \phi (Y_i; t ), \end{aligned}$$

for \( N \ge 1 \). Since under the assumptions of the theorem \( {\mathcal {F}}_N = r_N {\lfloor Nr \rfloor } [ F_{ {\lfloor Nr \rfloor } }(x) - F_0(x) ] \), by using the fact that

$$\begin{aligned} r_N {\lfloor Nr \rfloor } \int \widetilde{ \phi } (x;t) d F_{ {\lfloor Nr \rfloor } }(x)&= r_N {\lfloor Nr \rfloor } \left\{ \int \phi (x;t) F_{ {\lfloor Nr \rfloor } }(dx) - \vartheta (F_0;t) \int 1 F_{ {\lfloor Nr \rfloor } }(dx) \right\} \\&= \int \phi (x;t) \, {\mathcal {F}}_N(r,dx), \end{aligned}$$

we obtain the representation (5.9). Define

$$\begin{aligned} T(u;t)&= \sup _{0 \le r < u} \left| \int {\phi }(x;t) \, {\mathcal {F}}(r,dx) - \frac{r}{u} \int {\phi }(x;t) \, {\mathcal {F}}(u,dx) \right| , \\ T_{N1}(u;t)&= \sup _{0 \le r < u} \left| \int {\phi }(x;t) \, {\mathcal {F}}(r,dx) - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, {\mathcal {F}}(u,dx) \right| , \\ T_{N2}(u;t)&= \sup _{0 \le r \le {\lfloor Nu \rfloor } /{N}} \left| \int {\phi }(x;t) \, {\mathcal {F}}(r,dx) - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, {\mathcal {F}}(u,dx) \right| , \\ \end{aligned}$$

Note that for any functions f and g

$$\begin{aligned} \sup _s | f(s) | - \sup _{s^{\prime }} | g(s^{\prime }) |&= \sup _s \left( | f(s) | - \sup _{s^{\prime }} | g(s^{\prime }) | \right) \nonumber \\&\le \sup _s \left( | f(s) | - | g(s) | \right) . \end{aligned}$$
(5.10)

Combining (5.10) with the reverse triangle inequality yields

$$\begin{aligned} \pm \,(T_{N2} - T_N)(u; t)&\le \biggl | \sup _{0 \le r \le \frac{ {\lfloor Nu \rfloor } }{N}} \biggl ( \left| \int {\phi }(x;t) \, {\mathcal {F}}_N(r,dx) - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, {\mathcal {F}}_N(u,dx) \right| \\&\quad \, - \left| \int {\phi }(x;t) \, {\mathcal {F}}(r,dx) - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, {\mathcal {F}}(u,dx) \right| \biggr ) \biggr | \\&\le \sup _{0 \le r \le \frac{ {\lfloor Nu \rfloor } }{N}} \left| \left| \int {\phi }(x;t) \, {\mathcal {F}}_N(r,dx) - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, {\mathcal {F}}_N(u,dx) \right| \right. \\&\quad \,\left. - \left| \int {\phi }(x;t) \, {\mathcal {F}}(r,dx) - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, {\mathcal {F}}(u,dx) \right| \right| \\&\le \sup _{0 \le r \le \frac{ {\lfloor Nu \rfloor } }{N}} \biggl | \int {\phi }(x;t) \, ({\mathcal {F}}_N-{\mathcal {F}})(r,dx) \\&\quad \, - \frac{ {\lfloor Nr \rfloor } }{ {\lfloor Nu \rfloor } } \int {\phi }(x;t) \, ({\mathcal {F}}_N-{\mathcal {F}})(u,dx) \biggr | = o(1), \end{aligned}$$

a.s., as \( N \rightarrow \infty \), uniformly in \( u \in (0,1] \) and \( t \in [a,b]\), again using the integration by parts techniques elaborated in the previous proofs. Using similar arguments, it follows that

$$\begin{aligned} | T_{N1} - T |(u;t) = o_P(1) \end{aligned}$$

a.s, as \( N \rightarrow \infty \), uniformly in \(u \in (0,1] \) and \( t \in [a,b] \). Lastly, we show that \( | T_{N2} - T_{N1} | \) is negligible. Notice that

$$\begin{aligned} T_{N2}(u;t) = \sup _{0 \le s \le \frac{ {\lfloor Nu \rfloor } }{N} } | A(s; u, t) |, \qquad T_{N1}(u;t) = \sup _{0 \le s < u} | A( s; u, t ) | \, \end{aligned}$$

where

$$\begin{aligned} A(s; u, t ) = \int {\phi }( x; t ) \, {\mathcal {F}}( s, dx ) - \frac{s}{u} \int {\phi }( x; t ) \, {\mathcal {F}}( u, dx ), \end{aligned}$$

for \( 0 \le s \le u \). By virtue of (5.10) it follows that

$$\begin{aligned} | T_{N2} - T_{N1} |(u;t)&\le \left| \sup _{0 \le s < u} | A(s; u, t) | - | A(s; u, t) | { 1}_{ \left[ 0, \frac{ {\lfloor Nu \rfloor } }{N} \right] }(s) \right| \\&= \sup _{0 \le s < u} | A(s; u, t ) | { 1}_{ \left( \frac{ {\lfloor Nu \rfloor } }{N}, u \right) }(s) \\&= o(1), \end{aligned}$$

as \( N \rightarrow \infty \), again uniformly in \( u \in (0,1] \) and \( t \in [a,b] \). This verifies

$$\begin{aligned} \sup _{t \in [a,b]} \sup _{0 < u \le 1} | T_N(u;t) - T(u;t) | = o(1), \end{aligned}$$

a.s., as \( N \rightarrow \infty \). Hence the result follows. \(\square \)

Proof of Theorem 4.3

We argue similarly as in the proof of Theorem 3.1 and make use of the Skorohod/Dudley/Wichura representation theorem. Thus, by defining equivalent versions on a new probability space, we can assume that under \( P_0 \)

$$\begin{aligned} \sup _{u \in [0,1]} \sup _{x \in [a,b]} | {\mathcal {F}}_N(u,x) - {\mathcal {F}}(u,x) | \mathop {\rightarrow }\limits ^{a.s.} 0, \end{aligned}$$

as \( N \rightarrow \infty \). Let us also introduce an additional time series \( \{ Z_t \} \) satisfying

$$\begin{aligned} {\mathcal {L}}( \{ Z_t \}; P_\vartheta ) \mathop {=}\limits ^{d} {\mathcal {L}}( \{ Y_t \} ; P_0 ), \end{aligned}$$

where \( {\mathcal {L}}( \{ \xi _t \}; Q ) \) denotes the law of a time series \( \{ \xi _t \} \) under the probability Q. This means, under \( P_\vartheta \), when there is a change-point, \( \{ Z_t \} \) is distributed as \( \{ Y_t \} \) under the no-change probability measure \( P_0 \). Also define the associated empirical sequential Kiefer process

$$\begin{aligned} \widetilde{ {\mathcal {F}} } _N(u, x) = r_N \sum _{i=1}^{ {\lfloor Nu \rfloor } } [ { 1}( Z_i \le x ) - F_0( x ) ], \qquad u \in [0,1], \quad \ x \in [a,b]. \end{aligned}$$

Clearly, under \( P_\vartheta \) we have \( \widetilde{ {\mathcal {F}} } _N \Rightarrow \widetilde{ {\mathcal {F}} } \), as \( N \rightarrow \infty \), for some stochastic process \( \{ {\mathcal {F}}(u,x) : u \in [0,1], x \in [-\infty , \infty ] \} \) with \( {\mathcal {L}}( \widetilde{ {\mathcal {F}} } ; P_\vartheta ) = {\mathcal {L}}( {\mathcal {F}}; P_0 ) \), and w.l.o.g. we again may assume that the convergence is a.s. w.r.t. the supnorm. By assumption, under \( P_\vartheta \) the law of

$$\begin{aligned} {\mathcal {F}}_N(u,x)&= r_N \sum _{i=1}^{ {\lfloor N u \rfloor } } [ { 1}( Y_i \le x ) - F_0(x) ], \qquad u \in [\vartheta ,1], \quad x \in [a,b], \end{aligned}$$
(5.11)

can be obtained by transforming the after-change observations, i.e., the law of

$$\begin{aligned} {\mathcal {F}}_N(u,x), \qquad u \in [0,1], \quad x \in [-\infty , \infty ], \end{aligned}$$

under \( P_\vartheta \) equals the law of

$$\begin{aligned} r_N \sum _{i=1}^{ {\lfloor N \vartheta \rfloor } -1} [ { 1}( Y_i \le x ) - F_0(x) ] + r_N \sum _{i= {\lfloor N \vartheta \rfloor } }^{ {\lfloor N u \rfloor } } [ { 1}( g( Y_i ) \le g(x) ) - F_0( x ) ], \quad u \in [0,1], \quad x \in [a,b], \end{aligned}$$

under \( P_0 \). Using

$$\begin{aligned} {\mathcal {L}}( \{ Y_1, \dots , Y_{q-1}, g( Y_q ), \dots \}; P_\vartheta ) = {\mathcal {L}}( \{ Y_1, Y_2, \dots \}; P_0 ) = {\mathcal {L}}( \{ Z_1, Z_2, \dots \}; P_\vartheta ), \end{aligned}$$

we therefore obtain, for those parts,

$$\begin{aligned} {\mathcal {F}}_N(u,x)&\mathop {=}\limits ^{d, P_\vartheta } \widetilde{ {\mathcal {F}} } _N(u,x), \qquad u \in [0, \vartheta ), \quad x \in [a,b], \end{aligned}$$

and

$$\begin{aligned}&{\mathcal {F}}_N(u,x) \mathop {=}\limits ^{d, P_\vartheta } r_N \sum _{i= {\lfloor N \vartheta \rfloor } }^{ {\lfloor N u \rfloor } } [ { 1}( Z_i \le g \circ x ) - F_0 \circ g( x ) ]\\&\qquad \qquad \qquad \quad +\, r_N \sum _{i= {\lfloor N \vartheta \rfloor } }^{ {\lfloor N u \rfloor } } [ F_0 \circ g( x ) - F_0(x) ], \quad u \in [\vartheta ,1], \quad x \in [a,b], \end{aligned}$$

as well as jointly. Notice that we centered the terms corresponding to time instants after the change at their expectations which introduces a non-random correction term. For large enough N that result can be written as

$$\begin{aligned} {\mathcal {F}}_N(u,x) \mathop {=}\limits ^{d,P_\vartheta } { 1}( u < \vartheta ) \widetilde{ {\mathcal {F}} } _N(u,x) + { 1}( u \ge \vartheta ) \widetilde{ {\mathcal {F}} } _N^{(\vartheta )}(u,x) \end{aligned}$$

where \( \widetilde{ {\mathcal {F}} } _N^{(\vartheta )}(u,x) \) is given by

$$\begin{aligned}&\widetilde{ {\mathcal {F}} } _N( \vartheta - 1/N, x ) + \widetilde{ {\mathcal {F}} } _N( u, g(x) ) - \widetilde{ {\mathcal {F}} } _N( \vartheta - 1/N, g(x) )\\&\quad +\, r_N( {\lfloor Nu \rfloor } - {\lfloor N \vartheta \rfloor } +1 ) [ F_0 \circ g(x) - F_0(x)] \end{aligned}$$

for \( u \in [\vartheta , 1] \) and \( x \in [a,b] \). Obviously, under \( P_\vartheta \), the process

$$\begin{aligned} \widetilde{ {\mathcal {F}} } _N(u,x) - { 1}( u \ge \vartheta ) r_N( {\lfloor Nu \rfloor } - {\lfloor N \vartheta \rfloor } +1 ) [ F_0 \circ g(x) - F_0(x)] \end{aligned}$$

converges weakly to

$$\begin{aligned} { 1}( u < \vartheta ) \widetilde{ {\mathcal {F}} } (u,x) - { 1}( u \ge \vartheta ) [ \widetilde{ {\mathcal {F}} } ( \vartheta , x ) + \widetilde{ {\mathcal {F}} } ( u, g(x) ) - \widetilde{ {\mathcal {F}} } ( \vartheta , g(x) ) ], \end{aligned}$$

as \( N \rightarrow \infty \), which shows the first assertion. The second assertion now follows easily. \(\square \)

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Steland, A. Asymptotics for random functions moderated by dependent noise. Stat Inference Stoch Process 19, 363–387 (2016). https://doi.org/10.1007/s11203-015-9130-0

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