Weak convergence of the conditional U-statistics for locally stationary functional time series

Abstract

In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asymptotics created from non-parametric statistics. The present paper aims to develop a framework for inference of locally stationary functional time series based on the so-called conditional U-statistics introduced by Stute (Ann Probab 19:812–825, 1991), and may be viewed as a generalization of the Nadaraya-Watson regression function estimates. In this paper, we introduce an estimator of the conditional U-statistics operator that takes into account the nonstationary behavior of the data-generating process. We are mainly interested in establishing weak convergence of conditional U-processes in the locally stationary functional mixing data framework. More precisely, we investigate the weak convergence of conditional U-processes when the explicative variable is functional. We treat the weak convergence when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are established under fairly general structural conditions on the classes of functions and the underlying models. The theoretical results established in this paper are (or will be) critical tools for further functional data analysis developments.

Appendix

This appendix contains supplementary information that is an essential part of providing a more comprehensive understanding of the paper.

Lemma 8.0.4

Let $I_{h}=\left[ C_{1} h, 1-C_{1} h\right]$. Suppose that kernel $K_{1}$ satisfies Assumption 2 part(i). Then for $q=0,1,2$ and $m>1$:

Proof of Lemma 8.0.4

Notice that

The last inequality can be founded by applying some additional treatment with the help of the mean value theorem and the telescoping arguments. $\square$

Lemma 8.0.5

Suppose that kernel $K_{1}$ satisfies Assumption 2 part (i) and let $g:[0,1] \times {\mathscr {H}} \rightarrow {\mathbb {R}}$, $({\textbf{u}}, {\textbf{x}}) \mapsto g({\textbf{u}}, {\textbf{x}})$ be continuously differentiable with respect to ${\textbf{u}}$. Then,

$$\begin{aligned}{} & {} \sup _{u \in I_{h}}\left|\frac{1}{n^m h^m} \sum _{{\textbf{i}}\in I_n^m} \prod _{k=1}^m K_{1}\left( \frac{u_k-i_k /n}{h}\right) g\left( \frac{i_k}{n}, x_{k}\right) - \prod _{k=1}^m g(u_k, x_k)\right|\nonumber \\{} & {} \quad = O\left( \frac{1}{nh^{m+1}}\right) +o(h). \end{aligned}$$

(8.86)

Proof of Lemma 8.0.5

Remark that

where

This completes the proof. $\square$

The following result is an exponential inequality for strongly mixing sequences given in Liebscher (1996).

Lemma 8.0.6

(Liebscher 1996, Theorem 2.1). Let $\left\{ Z_{i, n}\right\}$ be a zero-mean triangular array such that $\left|Z_{i, n}\right|\le b_{n}$ with $\alpha$-mixing coefficients $\alpha (k)$. Then for any $\varepsilon >0$ and $S_{n} \le n$ with $\epsilon >4 S_{n} b_{n}$,

$$\begin{aligned} {\mathbb {P}}\left( \left|\sum _{i=1}^{n} Z_{i, i}(u, x)\right|\ge \varepsilon \right) \le 4 \exp \left( -\frac{\varepsilon ^{2}}{64 \sigma _{S_{n}, n}^{2} \frac{n}{S_{n}}+\frac{8}{3} \varepsilon b_{n} S_{n}}\right) +4 \frac{n}{S_{n}} \alpha \left( S_{n}\right) . \end{aligned}$$

(8.87)

Lemma 8.0.7

Let $\left\{ Z_{i, n}\right\}$ be a zero-mean triangular array such that $\left|Z_{i, n}\right|\le b_{n}$ with $\beta$-mixing coefficients $\beta (k)$. Then for any $\varepsilon >0$ and $S_{n} \le n$ with $\epsilon >4 S_{n} b_{n}$,

$$\begin{aligned} {\mathbb {P}}\left( \left|\sum _{i=1}^{n} Z_{i, i}(u, x)\right|\ge \varepsilon \right) \le 4 \exp \left( -\frac{\varepsilon ^{2}}{64 \sigma _{S_{n}, n}^{2} \frac{n}{S_{n}}+\frac{8}{3} \varepsilon b_{n} S_{n}}\right) +4 \frac{n}{S_{n}} \beta \left( S_{n}\right) . \end{aligned}$$

(8.88)

Proof of Lemma 8.0.7

Using Lemma 8.0.6 and the fact that for any $\sigma$-algebra ${\mathcal {A}}$ and ${\mathcal {B}}$, $\alpha ({\mathcal {A}},{\mathcal {B}}) \subseteq \beta ({\mathcal {A}},{\mathcal {B}})$, Lemma 8.0.11 holds.

Lemma 8.0.8

(de la Peña 1992) Let $X_{1}, \ldots , X_{n}$ be a sequence of independent random elements taking values in a Banach space $(B,\Vert \cdot \Vert )$ with ${\mathbb {E}} X_{i}=0$ for all i. Let $\left\{ \varepsilon _{i}\right\}$ be a sequence of independent Bernoulli r.v’s independent of $\left\{ X_{i}\right\} .$ Then, for any convex increasing function $\Phi$,

$$\begin{aligned} {\mathbb {E}} \Phi \left( \frac{1}{2}\left\| \sum _{i=1}^{n} X_{i} \varepsilon _{i}\right\| \right) \le {\mathbb {E}} \Phi \left( \left\| \sum _{i=1}^{n} X_{i}\right\| \right) \le {\mathbb {E}} \Phi \left( 2\left\| \sum _{i=1}^{n} X_{i} \varepsilon _{i}\right\| \right) . \end{aligned}$$

Proposition 8.0.9

(Arcones and Giné 1993, Proposition 3.6) Let $\{{X}_i: i \in {n}\}$ be a process satisfying, for $m \geqslant 1:$

$$\begin{aligned} \left( {\mathbb {E}}\left\| {X}_i-{X}_j\right\| ^{p}\right) ^{1/p} \leqslant \left( \frac{p-1}{q-1}\right) ^{m/2}\left( {\mathbb {E}}\left\| {X}_i-{X}_j\right\| ^{q}\right) ^{1/q}, \quad 1< q<p < \infty , \end{aligned}$$

and the semi-metric:

$$\begin{aligned} \rho (j,i) =\left( {\mathbb {E}}\left\| {X}_i-{X}_j\right\| ^{2}\right) ^{1/2}. \end{aligned}$$

There exists a constant $K=K(m)$ such that:

$$\begin{aligned} {\mathbb {E}}\sup _{i,j \in {n}}\left\| {X}_i-{X}_j\right\| \leqslant K \int _0^{D} [\log {N(\epsilon , {n}, \rho )}]^{m/2}d\epsilon , \end{aligned}$$

where D is the $\rho$-diameter of n.

Corollary 8.0.10

(Hall and Heyde 1980) Suppose that X and Y are random variables which are ${\mathscr {G}}$ and ${\mathscr {H}}$-measurable, respectively, and that ${\mathbb {E}}|X|^{p}<\infty , {\mathbb {E}}|Y|^{q}<\infty$, where p, $q>1, p^{-1}+q^{-1}<1$. Then

$$\begin{aligned} |{\mathbb {E}}X Y- {\mathbb {E}}X {\mathbb {E}}Y|\leqslant 8\Vert X\Vert _{p}\Vert Y \Vert _{q}[\alpha ({\mathscr {G}}, {\mathscr {H}})]^{1-p^{-1}-q^{-1}}. \end{aligned}$$

Lemma 8.0.11

Suppose that X and Y are random variables which are ${\mathscr {G}}$ and ${\mathscr {H}}$-measurable, respectively, and that ${\mathbb {E}}|X|^{p}<\infty , {\mathbb {E}}|Y|^{q}<\infty$, where $p>0$,

$$\begin{aligned} q>1, p^{-1}+q^{-1}<1. \end{aligned}$$

Then

$$\begin{aligned} |{\mathbb {E}}X Y- {\mathbb {E}}X {\mathbb {E}}Y|\leqslant 8\Vert X\Vert _{p}\Vert Y \Vert _{q}[\beta ({\mathscr {G}}, {\mathscr {H}})]^{1-p^{-1}-q^{-1}}. \end{aligned}$$

Proof of Lemma 8.0.11

This Lemma follows directly using Corollary 8.0.10 and the fact that for any $\sigma$-algebra ${\mathcal {A}}$ and ${\mathcal {B}}$, $\alpha ({\mathcal {A}},{\mathcal {B}}) \subseteq \beta ({\mathcal {A}},{\mathcal {B}})$. $\square$

Lemma 8.0.12

Let $V_{1}, \ldots , V_{L}$ be strongly mixing random variables measurable with respect to the $\sigma$-algebras ${\mathscr {F}}_{i_{1}}^{j_{1}}, \ldots , {\mathscr {F}}_{i_{L}}^{j_{L}}$ respectively with $1 \leqslant i_{1}<j_{1}<i_{2}<\cdots <j_{L} \leqslant n, i_{l+1}-j_{l} \geqslant w \geqslant 1$ and $\left|V_{j}\right|\leqslant 1$ for $j=1, \ldots$, L. Then

$$\begin{aligned} \left|{\mathbb {E}}\left( \prod _{j=1}^{L} V_{j}\right) -\prod _{j=1}^{L} {\mathbb {E}}\left( V_{j}\right) \right|\leqslant 16(L-1) \alpha (w), \end{aligned}$$

where $\alpha (w)$ is the strongly mixing coefficient.