2. Notation and results

Since the processes considered in this paper have sample paths in non-separable Banach spaces we use the concept of convergence in outer distribution developed by J. Hoffmann-Jørgensen. Given a probability space $(\Omega,{\mathcal{A}},{\mathrm{P}})$ , let T be a function from $\Omega$ to the extended real line $\bar{\mathbb R}$ . The outer integral of T with respect to ${\mathrm{P}}$ is defined as

\begin{equation*}{\mathrm{E}}^{\ast}(T)\,:\!=\inf\,\{{\mathrm{E}} (U)\colon\mbox{$U\colon\Omega\to\bar{\mathbb R}$ is measurable, ${\mathrm{E}} (U)$ exists, and $U\geq T$}\}.\end{equation*}

The outer probability of a subset B of $\Omega$ is ${\mathrm{P}}^{\ast}(B)\,:\!={\mathrm{E}}^{\ast}(\textbf{1}_B)=\inf\,\{{\mathrm{P}}(A)\colon A\in{\mathcal{A}}, \,A\supset B\}$ ; here and elsewhere, $\textbf{1}_B$ is the indicator function of a set B.

Definition 1. Let E be a metric space. For each $n \in{\mathbb N}$ , let $(\Omega_n, {\mathcal{A}}_n, {\mathrm{P}}_n)$ be a probability space and let $Z_n$ be a function from $\Omega_n$ into E. Suppose that $Z_0$ takes values in some separable subset of E and is measurable for the Borel sets on its range. It is said that the sequence $(Z_n)$ converges in outer distribution to $Z_0$ , denoted $Z_n \xrightarrow{\mathcal{D}^*} Z_0$ , if, for every bounded continuous function $h\,:\,E \to {\mathbb R}$ ,

\begin{equation*}\lim_{n\to\infty}{\mathrm{E}}^*(h(Z_n))={\mathrm{E}} (h(Z_0)).\end{equation*}

Remark 1. If $Z_n$ , $n = 0, 1, \dots$ , are random elements taking values in a separable metric space E endowed with the Borel $\sigma$ -algebra, then the convergence $Z_n \xrightarrow{\mathcal{D}^*} Z_0$ is equivalent to usual convergence in distribution $Z_n \xrightarrow{\mathcal{D}} Z_0$ :

\begin{equation*}\lim_{n\to\infty}{\mathrm{E}} (h(Z_n))={\mathrm{E}} (h(Z_0))\end{equation*}

for every bounded continuous function $h\,:\,E \to {\mathbb R}$ .

To establish convergence in outer distributions on $\ell^{\infty}({\mathcal{F}})$ we need a separable subset for a support of a limit distribution. Given a pseudometric d on ${\mathcal{F}}$ , let $UC({\mathcal{F}},d)$ be a set of all $\nu\in \ell^\infty({\mathcal{F}})$ which are uniformly d-continuous. The set $UC({\mathcal{F}}, d )$ is a separable subspace of $\ell^{\infty}({\mathcal{F}})$ if and only if $({\mathcal{F}},d)$ is totally bounded. As usual, $N(\varepsilon, {\mathcal{F}}, d)$ is the minimal number of open balls of d-radius $\varepsilon$ which are necessary to cover ${\mathcal{F}}$ . The pseudometric space $({\mathcal{F}},d)$ is totally bounded if $N(\varepsilon, {\mathcal{F}}, d)$ is finite for every $\varepsilon > 0$ . This property always holds under the assumptions imposed below.

Let ${\mathcal{L}}^2[0,1]={\mathcal{L}}^2([0,1],\lambda)$ be a set of measurable functions which are square-integrable for Lebesgue measure $\lambda$ on [0, 1] with a pseudometric $\rho_2(\,f, g)=\rho_{2,\lambda}(\,f, g)=(\smallint_{[0,1]}(\,f-g)^2\,d\lambda)^{1/2}$ . Let $L^2[0,1]=L^2([0,1],\lambda)$ be the associated Hilbert space endowed with the inner product $\langle\,f, g\rangle =\int_0^1 f(t)g(t)\lambda(\text{d} t)$ . Given a set ${\mathcal{F}}\subset {\mathcal{L}}^2[0, 1]$ , let $\nu=\{\nu(\,f)\colon f\in {\mathcal{F}}\}$ be a centred Gaussian process such that ${\mathrm{E}}[\nu(\,f)\nu(g)] = \langle\,f, g\rangle$ for all $f, g\in {\mathcal{F}}$ . Such a process exists and provides a linear isometry from $L^2[0, 1]$ to $L^2(\Omega, {\mathcal{F}}, {\mathrm{P}})$ . By [Reference Dudley7] or [Reference Dudley9, Theorems 2.6.1 and 2.8.6], if

(6) \begin{equation}\int_0^1\sqrt{\log N(x,{\mathcal{F}},\rho_2)}\,\text{d} x<\infty\end{equation}

then $\nu=\{\nu(\,f)\colon f\in{\mathcal{F}}\}$ admits a version with almost all sample paths bounded and uniformly continuous on ${\mathcal{F}}$ with respect to $\rho_2$ . In what follows, we denote a suitable version by the same notation $\nu$ , and so $\nu$ itself takes values in $UC({\mathcal{F}}, \rho_2)$ and is measurable for the Borel sets on its range.

In this paper the condition (6) is applied to sets ${\mathcal{F}}$ defined as follows. For $-\infty <a<b<\infty$ and $0 < p < \infty$ , the p-variation of a function $g\colon[a,b]\to{\mathbb R}$ is the supremum

\begin{equation*}v_p(g;\ [a, b])\,:\!=\sup\Big\{\sum^m_{i=1}|g(t_i) - g(t_{i-1})|^p\,:\,a=t_0 < t_1 < \cdots < t_m = b,\ m \in{\mathbb N}_{+}\Big\},\end{equation*}

which can be finite or infinite. If $v_p(g;\ [a, b]) < \infty$ then g is said to have bounded p-variation and the set of all such functions is denoted by ${\mathcal{W}}_p[a, b]$ . We abbreviate $v_p(g)\,:\!= v_p(g;\ [0, 1])$ . For each $g \in {\mathcal{W}}_p[0, 1]$ and $1 \le p < \infty$ , let $\|g\|_{(p)} \,:\!= v^{1/p}_p(g)$ . Then $\|g\|_{(p)}$ is a seminorm equal to zero only for constant functions g. The p-variation norm is

\begin{equation*}\|g\|_{[p]} \,:\!=\|g\|_{\sup} +\|g\|_{(p)} ,\end{equation*}

where $\|g\|_{\sup}\,:\!=\sup_{0\le t\le 1}|g(t)|$ . The set ${\mathcal{W}}_p[0, 1]$ is a non-separable Banach space with the norm $\|\cdot\|_{[p]}$ . If ${\mathcal{F}}$ is a bounded subset of ${\mathcal{W}}_q[0,1]$ with $1\leq q<2$ , then (6) holds by the proof of Theorem 2.1 in [Reference Dudley8] (see also [Reference Norvaiša and Račkauskas14, Theorem 5]).

Now we are prepared to formulate further results. Let $X_1, X_2,\dots$ be a sequence of real-valued random variables. For each positive integer $n\in{\mathbb N}_{+}$ , the nth partial sum process of random variables is defined by

\begin{equation*}S_n(t)\,:\!=\sum_{i=1}^{\lfloor nt\rfloor}X_i=\sum_{i=1}^nX_i\textbf{1}_{[0,t]}\Big (\frac{i}{n}\Big ),\qquad t\in [0,1].\end{equation*}

Here, for a real number $x\geq 0$ , $\lfloor x\rfloor\,:\!=\max\{k\colon k\in{\mathbb N},\,k\leq x\}$ is a value of the floor function. Then the partial sum process is the sequence of nth partial sum processes $S_n=\{S_n(t)\colon t\in [0,1]\}$ , $n\in{\mathbb N}_{+}$ . Let W be a Wiener process on [0, 1]. In [Reference Norvaiša and Račkauskas14], assuming that the random variables $X_1, X_2,\dots$ are independent and identically distributed, it is proved that convergence in outer distribution,

\begin{equation*} n^{-1/2}S_n \xrightarrow{\mathcal{D}^*} \sigma W\quad\mbox{in ${\mathcal{W}}_p[0,1]$ as $n\to\infty$,}\end{equation*}

holds if and only if ${\mathrm{E}} (X_1)=0$ and $\sigma^2={\mathrm{E}} (X_1^2)<\infty$ . The assumption $p>2$ cannot be replaced by $p=2$ since in this case the limiting process W does not belong to ${\mathcal{W}}_2[0,1]$ . The next theorem extends this fact to the case where a sequence of random variables $X_1, X_2,\dots$ is a short-memory linear process.

Theorem 2. Let $X_1, X_2, \dots$ be a short-memory linear process, let $p>2$ , and let W be a Wiener process on [0, 1]. Then

\begin{equation*} n^{-1/2}S_n \xrightarrow{\mathcal{D}^*} \sigma_{\eta} A_{\psi}W\quad\mbox{in ${\mathcal{W}}_p[0, 1]$ as $n\to\infty$.}\end{equation*}

For any $p>0$ , the p-variation of a sample function of the nth partial sum is

(7) \begin{equation}v_p(S_n)=\max\bigg\{\sum_{j=1}^m \, \bigg |\sum_{i=k_{j-1}+1}^{k_j}\!\!\!\!X_i\bigg |^p\colon 0=k_0<\cdots <k_m=n,\,\,1\leq m\leq n\bigg\}.\end{equation}

Theorem 2 and the continuous mapping theorem (e.g. Theorem 1.3.6 in [Reference Van der Vaart and Wellner15]) applied to the p-variation yield the following.

Corollary 1. Under the hypotheses of Theorem 2, we have

\begin{equation*}n^{-\frac{p}{2}}v_p(S_n) \xrightarrow{\mathcal{D}} \sigma_{\eta}^p A_{\psi}^pv_p(W)\quad\mbox{as $n\to\infty$.}\end{equation*}

Suppose that ${\mathcal{F}}$ contains the family of indicator functions of subintervals of [0, 1]. Then the nth partial sum process of a linear process $S_n$ and the nth ${\mathcal{F}}$ -weighted sum of linear process $\nu_n$ are related by the equality

\begin{equation*} S_n(t)=\nu_n(\textbf{1}_{[0,t]})\quad\mbox{ for\ each $t\in [0,1]$.}\end{equation*}

This relation is used in the following theorem to obtain Theorem 2 from a uniform convergence of $n^{-1/2}\nu_n$ over the set ${\mathcal{F}}_q=\{\,f\in{\mathcal{W}}_q[a,b]\colon \|\,f\|_{[q]}\leq 1\}$ , $1\leq q<2$ , which is the unit ball in ${\mathcal{W}}_q[a,b]$ . To this end, the nth ${\mathcal{F}}_q$ -weighted sum of linear process $\nu_n$ is considered as a bounded linear functional over ${\mathcal{W}}_q[a,b]$ .

Theorem 3. Let $1<p<\infty$ and $1<q<\infty$ be such that $p^{-1}+q^{-1}=1$ . For a linear bounded functional $L\colon{\mathcal{W}}_q[a,b]\to{\mathbb R}$ , let $T(L)(t)\,:\!=L(\textbf{1}_{[a,t]})$ for each $t\in [a,b]$ . Then T is a linear mapping from the dual space ${\mathcal{W}}_q^{\ast}[a,b]$ into ${\mathcal{W}}_p[a,b]$ , and

(8) \begin{equation}\|T(L)\|_{[p]}\leq 5\|L\|_{{\mathcal{F}}_q},\qquad L\in{\mathcal{W}}_q^{\ast}[a,b],\end{equation}

where $\|\cdot\|_{{\mathcal{F}}_q}$ is defined by (4).

To prove Theorem 1 we use the asymptotic equicontinuity criterion for convergence in law in $\ell^{\infty}({\mathcal{F}})$ (see, e.g., [Reference Giné and Nickl12, Theorem 3.7.23] or [Reference Van der Vaart and Wellner15, p. 41]). The conclusion of Theorem 1 holds if and only if (i), (ii), and (iii) hold, where

1. (i) the finite-dimensional distributions of $n^{-1/2}\nu_n$ converge in distribution to the corresponding finite-dimensional distributions of $\nu$ ;

2. (ii) $n^{-1/2}\nu_n$ is asymptotically equicontinuous with respect to $\rho_2$ ;

3. (iii) ${\mathcal{F}}$ is totally bounded for $\rho_2$ .

3. Convergence of finite-dimensional distributions

In this section we establish convergence of finite-dimensional distributions of the processes $n^{-1/2}\nu_n$ . Recall that ${\mathcal{F}}_q=\{\,f\in{\mathcal{W}}_q[0, 1]\colon\|\,f\|_{[q]}\le 1\}$ is endowed with the pseudometric $\rho_2$ . We begin with a one-dimensional case.

We have not seen results in the literature which yield the convergence in distribution of real random variables $n^{-1/2}\nu_n(g)$ when $g\in{\mathcal{W}}_q[0,1]$ for some $q\in [1,2)$ under the hypotheses of Theorem 4 below. The best available related results are due to K. M. Abadir et al. [Reference Abadir, Distaso, Giraitis and Koul1] and give the desired fact when g has bounded total variation. Next is a more general result for a short-memory linear process with independent, identically distributed inovations and weights given by a function g.

Theorem 4. Suppose $(X_i)_{i\in{\mathbb Z}}$ is a linear process defined by (1) and (2), and $\nu$ is the isonormal Gaussian processes on ${\mathcal{L}}_2[0,1]$ . If $g\in{\mathcal{W}}_q[0,1]$ for some $1\leq q<2$ , then

(9) \begin{equation}n^{-1/2}\nu_n(g) \xrightarrow{\mathcal{D}} \sigma_{\eta}A_{\psi}\nu (g),\quad\mbox{as $n\to\infty$.}\end{equation}

Proof. Let $1\leq q<2$ and $g\in{\mathcal{W}}_q[0,1]$ . For each $n\in{\mathbb N}_{+}$ and $k\in{\mathbb N}$ , let

(10) \begin{equation}T_{nk}\,:\!=\sum_{i=1}^n\eta_{i-k}g\Big (\frac{i}{n}\Big ).\end{equation}

By (3) and (1) we have the representation

\begin{align*}\nu_n(g)& = \sum_{i=1}^n\Big (\sum_{k=0}^{\infty}\psi_k\eta_{i-k}\Big )g\Big (\frac{i}{n}\Big )=\sum_{k=0}^{\infty}\psi_kT_{nk}\\ & = \sum_{k=0}^{\infty}\psi_k [T_{nk}-T_{n0}]+ A_{\psi}T_{n0}.\end{align*}

Since the function $g\in{\mathcal{W}}_q[0,1]$ , it is regulated, meaning that it has left limits on (0, 1] and right limits on [0,1) (see, e.g., [Reference Dudley and Norvaiša11, p. 213]). Thus, $g^2$ is Riemann integrable, and so

\begin{equation*}\text{Var}\Big (n^{-1/2}T_{n0}\Big )=\frac{\sigma_{\eta}^2}{n}\sum_{i=1}^ng^2\Big (\frac{i}{n}\Big )\to\sigma_{\eta}^2\int_0^1g^2\,\text{d}\lambda,\quad\mbox{as $n\to\infty$.}\end{equation*}

Since $\nu$ is the isonormal Gaussian processes on ${\mathcal{L}}_2[0,1]$ it follows by the Lindeberg central limit theorem that

\begin{equation*}n^{-1/2}T_{n0}=\frac{1}{\sqrt{n}\,}\sum_{i=1}^n\eta_{i}g\Big (\frac{i}{n}\Big )\xrightarrow{\mathcal{D}} \sigma_{\eta}\nu (g),\quad\mbox{as $n\to\infty$.}\end{equation*}

Therefore, to prove (9), due to the Slutsky theorem, it is enough to show that

(11) \begin{equation}R_n\,:\!=\sum_{k=0}^{\infty}\frac{\psi_k}{\sqrt{n}}[T_{nk}-T_{n0}]\to 0\quad\mbox{in probability ${\mathrm{P}}$ as $n\to\infty$.}\end{equation}

We will show that the following two properties hold true:

(12) \begin{equation}\sup_{n,k}\frac{1}{n}{\mathrm{E}} (T_{nk}^2)<\infty\end{equation}

and

(13) \begin{equation}\mbox{for each $k\in{\mathbb N}$},\quad\lim_{n\to\infty}\frac{1}{\sqrt{n}\,}|T_{nk}-T_{n0}|= 0\quad\mbox{in probability ${\mathrm{P}}$.}\end{equation}

For the moment, suppose that (12) and (13) hold true. Let $\epsilon >0$ and $K\in{\mathbb N}$ . Split the sum $R_n$ given by (11) into the sum with all $k\leq K$ and the sum with all $k>K$ to get the inequality

(14) \begin{eqnarray}\lefteqn{{\mathrm{P}}(\{|R_n|>\epsilon\})}\nonumber\\&\leq&{\mathrm{P}}\bigg (\bigg \{ \sum_{k=0}^K\frac{|\psi_k|}{\sqrt{n}}|T_{nk}-T_{n0}|>\frac{\epsilon}{2}\bigg\}\bigg )+{\mathrm{P}}\bigg (\bigg\{\sum_{k>K}\frac{|\psi_k|}{\sqrt{n}}|T_{nk}-T_{n0}|>\frac{\epsilon}{2}\bigg\}\bigg ).\end{eqnarray}

Clearly we have the bound

\begin{equation*}{\mathrm{P}}\bigg (\bigg\{\sum_{k>K}\frac{|\psi_k|}{\sqrt{n}}|T_{nk}-T_{n0}|>\frac{\epsilon}{2}\bigg\}\bigg )\leq\frac{4}{\epsilon}\sup_{n,k}\bigg (\frac{{\mathrm{E}} (T_{nk}^2)}{n}\bigg )^{1/2}\sum_{k>K}|\psi_k|.\end{equation*}

By (12) and (2), taking $K\in{\mathbb N}$ large enough, one can make the right-hand side of the preceding bound as small as one wishes. Then the first probability on the right-hand side of (14) is as small as one wishes by (13) and taking $n\in{\mathbb N}_{+}$ large enough. Therefore, (11) holds true and we are left to prove (12) and (13).

Recalling the notation $T_{nk}$ given by (10), for each $n\in{\mathbb N}_{+}$ and $k\in{\mathbb N}$ , we have

\begin{equation*}\frac{1}{n}{\mathrm{E}} (T_{nk}^2)=\frac{\sigma_{\eta}^2}{n}\sum_{i=1}^ng^2\Big (\frac{i}{n}\Big )\leq\sigma_{\eta}^2\|g\|_{\sup}^2.\end{equation*}

This proves (12). To prove (13), let $k\in{\mathbb N}_{+}$ . Changing the index i of summation, it follows that the representation

\begin{equation*}T_{nk}-T_{n0}=\sum_{i=1-k}^0\eta_ig\Big (\frac{i+k}{n}\Big )+\sum_{i=1}^{n-k}\eta_i\Big [g\Big (\frac{i+k}{n}\Big ) -g\Big (\frac{i}{n}\Big )\Big ]-\sum_{i=n-k+1}^n\eta_ig\Big (\frac{i}{n}\Big )\end{equation*}

holds for each integer $n>k$ . Since g is bounded and k is fixed, the first and the third sums on the right-hand side divided by $\sqrt{n}$ tend to zero in probability ${\mathrm{P}}$ as $n\to\infty$ . For the second sum divided by $\sqrt{n}$ we apply the Hölder inequality,

\begin{equation*}\bigg |\frac{1}{\sqrt{n}\,}\sum_{i=1}^{n-k}\eta_i\Big [g\Big (\frac{i+k}{n}\Big ) -g\Big (\frac{i}{n}\Big )\Big ]\bigg |\leq \bigg (n^{-\frac{p}{2}}\sum_{i=1}^{n-k}|\eta_i|^p\bigg )^{\frac{1}{p}}\bigg (\sum_{i=1}^{n-k}\Big |g\Big (\frac{i+k}{n}\Big ) -g\Big (\frac{i}{n}\Big )\Big |^q\bigg )^{\frac{1}{q}} ,\end{equation*}

with $p\in{\mathbb R}$ such that $\frac{1}{p}+\frac{1}{q}\geq 1$ . The telescoping sum representation and repeated application of the Minkowski inequality for sums imply that the inequality

\begin{equation*}\bigg (\sum_{i=1}^{n-k}\Big |g\Big (\frac{i+k}{n}\Big ) -g\Big (\frac{i}{n}\Big )\Big |^q\bigg )^{\frac{1}{q}}\leq k\|g\|_{(q)}\end{equation*}

holds for each integer $n>k$ . Since $1\leq q<2$ , then $(2/p)<1$ . Also, since k is fixed and ${\mathrm{E}}(|\eta_1|^p)^{\frac{2}{p}}=\sigma_{\eta}^2<\infty$ , by the Marcinkiewicz–Zygmund strong law of large numbers we have

\begin{equation*}\lim_{n\to\infty}n^{-\frac{p}{2}}\sum_{i=1}^{n-k}|\eta_i|^p=0\quad\mbox{with probability 1}.\end{equation*}

This completes the proof of (13), and thus Theorem 3 is proved. □

By the definition of the Gaussian process $\nu$ , for any $g_1,\dots,g_d\in {\mathcal{W}}_q$ , $(\nu (g_1),\dots, \nu (g_d))$ have a jointly normal distribution with covariance given by the inner products $\int_0^1g_ig_j\,\text{d}\lambda$ , $i,j=1, \dots, d$ .

Proposition 1. Suppose $(X_i)_{i\in{\mathbb Z}}$ is a short-memory linear process and $\nu$ is the isonormal Gaussian processes on ${\mathcal{L}}_2[0,1]$ . If $g_1,\dots,g_d\in{\mathcal{W}}_q[0,1]$ for some $1\leq q<2$ , then

(15) \begin{equation}n^{-1/2}(\nu_n(g_1),\dots,\nu_n(g_d)) \xrightarrow{\mathcal{D}} \sigma_{\eta}A_{\psi}(\nu (g_1),\dots,\nu (g_d))\quad\mbox{as $n\to\infty$.}\end{equation}

Proof. Let $d\in{\mathbb N}_{+}$ , and let $g_1,\dots,g_d\in{\mathcal{W}}_q[0,1]$ . To prove (15) we use the Cramér–Wold device. Let $a=(a_1,\dots,a_d)\in{\mathbb R}^d$ . Recalling the definition in (3) of $\nu_n$ , we have

\begin{equation*}\sum_{h=1}^da_h\nu_n(g_h)=\nu_n\bigg (\sum_{h=1}^da_hg_h\bigg )\end{equation*}

for each $n\in{\mathbb N}_{+}$ . Since $\sum_{h=1}^da_hg_h\in{\mathcal{W}}_q[0,1]$ , by Theorem 4 it follows that

\begin{equation*}n^{-1/2}\nu_n\bigg (\sum_{h=1}^da_hg_h\bigg ) \xrightarrow{\mathcal{D}} \sigma_{\eta} A_{\psi}\nu\bigg (\sum_{h=1}^da_hg_h\bigg )\quad\mbox{as $n\to\infty$.}\end{equation*}

Due to the linear isometry of $\nu$ , the convergence

\begin{equation*}n^{-1/2}a{\cdot}(\nu_n(g_1),\dots,\nu_n(g_d)) \xrightarrow{\mathcal{D}} \sigma_{\eta}A_{\psi}a{\cdot}(\nu (g_1),\dots,\nu (g_d))\quad\mbox{as $n\to\infty$}\end{equation*}

holds. Since $a\in{\mathbb R}^d$ is arbitrary, (15) holds by the Cramér–Wold device. □

4. Asymptotic equicontinuity

Let $(\mathcal{F},\rho)$ be a pseudometric space. For each $n\in{\mathbb N}_{+}=\{1,2,\dots\}$ , let $Z_{nk}$ , $k\in{\mathbb Z}$ , be independent stochastic processes indexed by $f\in \mathcal{F}$ and defined on the product probability space $(\Omega_n,\mathcal{A}_n,{\mathrm{P}}_n)\,:\!=\bigotimes_{k\in{\mathbb Z}}(\Omega_{nk},\mathcal{A}_{nk},{\mathrm{P}}_{nk})$ so that $Z_{nk}(\,f,\omega)=Z_{nk}(\,f,\omega_k)$ for each $\omega=(\omega_k)_{k\in{\mathbb Z}}$ and $f\in\mathcal{F}$ . For each $n\in{\mathbb N}_{+}$ consider a stochastic process defined as the series

\begin{equation*}\sum_{k\in{\mathbb Z}}Z_{nk}(\,f)\,:\!=\lim_{m\to +\infty}\sum_{k=-m}^mZ_{nk}(\,f),\qquad f\in{\mathcal{F}},\end{equation*}

provided the convergence holds almost surely. We write $(Z_{nk})\in\mathcal{M} (\Omega_n,\mathcal{A}_n, {\mathrm{P}}_n)$ if every one of the functions

(16) \begin{equation}\omega\mapsto\sup\Big\{\Big |\sum_{k\in{\mathbb Z}}e_k[Z_{nk}(\,f,\omega)-Z_{nk}(g,\omega)]\Big | \colon f,g\in\mathcal{F},\,\,\rho (\,f,g)<\delta\Big\} \end{equation}

and

(17) \begin{equation}\omega\mapsto\sup\Big\{\Big |\sum_{k\in{\mathbb Z}}e_k[Z_{nk}(\,f,\omega)-Z_{nk}(g,\omega)]^2\Big | \colon f,g\in\mathcal{F},\,\,\rho (\,f,g)<\delta\Big\} \end{equation}

is measurable for the completion of the probability space $(\Omega_n,\mathcal{A}_n,{\mathrm{P}}_n)$ , for every $\delta >0$ and every tuple $(e_k)_{k\in{\mathbb Z}}$ with $e_k\in\{-1,0,1\}$ .

The following is Theorem 2.11.1 in [Reference Van der Vaart and Wellner15] adopted to the convergence of sums of linear processes.

Theorem 5. Let $(\mathcal{F},\rho)$ be a totally bounded pseudometric space. Under the preceding notation, assume that $(Z_{nk})\in \mathcal{M}(\Omega_n,\mathcal{A}_n,{\mathrm{P}}_n)$ and there is a subsequence of positive integers $(m_n)_{n\in{\mathbb N}_{+}}$ such that

(18) \begin{align}\lim_{n\to\infty}{\mathrm{P}}_n^*\bigg(\bigg\{\bigg\|\sum_{k<-m_n}\!\!\!Z_{nk}+\sum_{k>m_n}\!\!Z_{nk}\bigg\|_{{\mathcal{F}}}>\varepsilon\bigg\}\bigg) & = 0 \quad \mbox{for every $\varepsilon >0$,} \end{align}

(19) \begin{align} \lim_{n\to\infty}\sum_{k=-m_n}^{m_n} \!\!\!{\mathrm{E}}^{\ast}\|Z_{nk}\|_{{{\mathcal{F}}}}^2 \textbf{1}_{\{\| Z_{nk}\|_{{{\mathcal{F}}}}>\epsilon\}} & = 0 \quad \mbox{for every $\epsilon >0$,} \end{align}

(20) \begin{align} \lim_{n\to\infty}\sup_{\rho (\,f,g) < \delta_n}\sum_{k=-m_n}^{m_n}\!\!\! {\mathrm{E}}[Z_{nk}(\,f)-Z_{nk}(g)]^2& =0\quad\mbox{for every $\delta_n\downarrow 0$,} \end{align}

(21) \begin{align} \lim_{n\to\infty}\int_0^{\delta_n} \!\!\!\sqrt{\log N(x, \mathcal{F}, d_n)}\,\text{d} x & = 0 \quad \textrm{in}\ {\mathrm{P}}_n^*\ \textrm{for every} \ \delta_n\downarrow 0, \end{align}

where $d_n$ is a random pseudometric on ${\mathcal{F}}$ defined for each $n\in{\mathbb N}_{+}$ and $f,g\in\mathcal{F}$ by

(22) \begin{equation} d_n(\,f, g)\,:\!=\bigg (\sum_{k=-m_n}^{m_n} [Z_{nk}(\,f)-Z_{nk}(g)]^2 \bigg )^{1/2}.\end{equation}

Then, $Z_n\,:\!=\sum_{k\in{\mathbb Z}}[Z_{nk}-{\mathrm{E}} (Z_{nk})]$ is asymptotically $\rho$ -equicontinuous; that is, for every $\varepsilon> 0$ ,

\begin{equation*}\lim_{\delta \downarrow 0}\limsup_n{\mathrm{P}}_n^*(\{\sup\{|Z_n(\,f) - Z_n(g)|\colon f,g\in{\mathcal{F}},\,\,\rho(\,f,g)<\delta\}> \varepsilon\})=0.\end{equation*}

Proof. Let $(m_n)$ be a subsequence of positive integers $(m_n)_{n\in{\mathbb N}_{+}}$ such that (18) holds. Clearly, $(Z_{nk})_{-m_n\leq k\leq m_n}\in\mathcal{M} (\Omega_n,\mathcal{A}_n, {\mathrm{P}}_n)$ . Using Theorem 2.11.1 in [Reference Van der Vaart and Wellner15] one can show that

\begin{equation*}\lim_{\delta \downarrow 0}\limsup_n{\mathrm{P}}_n^*\bigg(\bigg\{\sup\bigg\{\bigg |\sum_{k=-m_n}^{m_n}[Z_{nk}(\,f) - Z_{nk}(g)]\bigg|\colon f,g\in{\mathcal{F}},\,\,\rho(\,f,g)<\delta\bigg\} > \varepsilon\bigg\}\bigg)=0\end{equation*}

for each $\varepsilon>0$ . For a given $\epsilon >0$ and for each $n\in{\mathbb N}_{+}$ we have

\begin{multline*} {{\mathrm{P}}_n^*(\{\sup\{|Z_n(\,f)- Z_n(g)|\colon f,g\in{\mathcal{F}},\,\,\rho(\,f,g)<\delta\} > \varepsilon\})} \\\leq{\mathrm{P}}_n^*\bigg(\bigg\{ \sup\bigg\{\bigg |\sum_{k=-m_n}^{m_n}[Z_{nk}(\,f) - Z_{nk}(g)]\bigg|\colon f,g\in{\mathcal{F}},\,\,\rho(\,f,g)<\delta\bigg\}>\frac{\varepsilon}{2}\bigg\}\bigg)\\\quad + {\mathrm{P}}_n^*\bigg(\bigg\{\bigg\|\sum_{k<-m_n}\!\!\!Z_{nk}+\sum_{k>m_n}\!\!Z_{nk}\bigg \|_{{\mathcal{F}}}>\frac{\varepsilon}{4}\bigg\}\bigg ).\end{multline*}

By hypothesis (18), the conclusion follows. □

Since the sequence $X_1, X_2,\dots$ is a short-memory linear process, the sequence of real numbers $(\psi_j)_{j\in{\mathbb N}}$ is square summable, and so each series in (1) converges almost surely by Lévy’s equivalence theorem (e.g. [Reference Dudley10, Theorem 9.7.1]). Letting $\psi_k\,:\!=0$ for each $k<0$ , we obtain the representation

(23) \begin{equation}X_i=\sum_{k=-\infty}^i\psi_{i-k}\eta_k=\sum_{k\in{\mathbb Z}} \psi_{i-k}\eta_k,\qquad i\in{\mathbb Z}.\end{equation}

Lemma 1. Suppose $X_1,X_2,\dots$ is a linear process given by (23), $f\colon[0,1]\to{\mathbb R}$ , and $\nu_n(\,f)$ is the nth f-weighted partial sum given by (3). For each $n\in{\mathbb N}_{+}$ and $k\in{\mathbb Z}$ , let

\begin{equation*}a_{nk}(\,f)=\sum_{i=1}^n\psi_{i-k}\,f\Big (\frac{i}{n}\Big ),\end{equation*}

where $\psi_{i-k}=0$ if $i<k$ . Then, for each $n\in {\mathbb N}_{+}$ ,

(24) \begin{equation}{\mathrm{E}}[\nu_n^2(\,f)]=\sigma_{\eta}^2\sum_{k\in{\mathbb Z}}a_{nk}^2(\,f)\quad\mbox{and}\quad\nu_n(\,f)=\sum_{k\in{\mathbb Z}}a_{nk}(\,f)\eta_k,\end{equation}

where the random series converges almost surely.

Proof. Let $n\in{\mathbb N}_{+}$ . For each $i,j\in\{1,\dots,n\}$ , since the filter $(\psi_k)_{k\in{\mathbb N}}$ is square summable, the series representation

\begin{equation*}{\mathrm{E}} (X_iX_j)=\sigma_{\eta}^2\sum_{k\in{\mathbb Z}}\psi_{i-k}\psi_{j-k}\end{equation*}

converges absolutely. Thus, we have

\begin{align*}{\mathrm{E}}[\nu_n^2(\,f)] & = \sum_{i,j=1}^n{\mathrm{E}} (X_iX_j)\,f\Big (\frac{i}{n}\Big )\,f\Big (\frac{j}{n}\Big )\\ & = \sigma_{\eta}^2\sum_{k\in{\mathbb Z}}\sum_{i,j=1}^n\psi_{i-k}\psi_{j-k}f\Big (\frac{i}{n}\Big )f\Big (\frac{j}{n}\Big )=\sigma_{\eta}^2\sum_{k\in{\mathbb Z}}a_{nk}^2(\,f) , \end{align*}

and the series on the right-hand side converges. This proves the first equality in (24). The second one follows next:

\begin{align*}\nu_n(\,f) & = \sum_{i=1}^n\bigg [\sum_{k\in{\mathbb Z}}\psi_{i-k}\eta_k\bigg ]f\Big (\frac{i}{n}\Big )\\ & = \sum_{k\in{\mathbb Z}}\bigg [\sum_{i=1}^n\psi_{i-k}f\Big (\frac{i}{n}\Big ) \bigg ]\eta_k=\sum_{k\in{\mathbb Z}}a_{nk}(\,f)\eta_k.\end{align*}

The series on the right-hand side converges almost surely by Lévy’s equivalence theorem, since $(a_{nk}(\,f))_{k\in{\mathbb Z}}$ is square summable. □

5. Proof of Theorem 1

As shown at the end of this section, Theorem 1 is a simple corollary of the next theorem. Following [Reference Giné and Nickl12, p. 267], we say that a set of functions ${\mathcal{F}}$ satisfies the pointwise countable approximation property provided there exists a countable subset ${\mathcal{F}}_0\subset{\mathcal{F}}$ such that every f in ${\mathcal{F}}$ is a pointwise limit of functions in ${\mathcal{F}}_0$ . Given a probability measure Q on $([0, 1], {\mathcal{B}}_{[0, 1]})$ , let $\rho_{2,Q}$ be a pseudometric on ${\mathcal{F}}$ with values

\begin{equation*}\rho_{2,Q}(\,f,g)=\bigg(\int_{[0,1]}(\,f-g)^2\,\text{d} Q\bigg )^{1/2},\qquad f,g\in{\mathcal{F}} .\end{equation*}

Theorem 6. Let $X_1, X_2,\dots$ be a short-memory linear process given by (1) and let $1 \le q < 2$ . Suppose that a set of functions ${\mathcal{F}} \subset {\mathcal{W}}_q[0, 1]$ is bounded, satisfies the pointwise countable approximation property, and

(25) \begin{equation}\int^1_0\sup_{Q\in{\mathcal{Q}}}\sqrt{\log N(x,{\mathcal{F}}, \rho_{2,Q})}\, \text{d} x < \infty,\end{equation}

where ${\mathcal{Q}}$ is the set of all probability measures on $([0, 1], {\mathcal{B}}_{[0, 1]})$ . There exists a version of the isonormal Gaussian process $\nu$ restricted to ${\mathcal{F}}$ with values in a separable subset of $\ell^{\infty}({\mathcal{F}})$ , it is measurable for the Borel sets on its range, and (5) holds.

Since ${\mathcal{F}}\subset{\mathcal{W}}_q[0,1]$ with $q\in [1,2)$ , the finite-dimensional distributions of $n^{-1/2}\nu_n$ converge in distribution to the corresponding finite-dimensional distributions of $\nu$ by Proposition 1. By hypothesis (25), ${\mathcal{F}}$ is totally bounded with respect to pseudometric $\rho_2$ . Therefore, to prove Theorem 6 we have to show that $n^{-1/2}\nu_n$ is asymptotically equicontinuous with respect to $\rho_2$ . To this end we use Theorem 5.

For each $n\in{\mathbb N}_{+}$ , $k\in{\mathbb Z}$ , and $f\colon[0,1]\to{\mathbb R}$ , let

(26) \begin{equation}u_{nk}(\,f)\,:\!=\frac{1}{\sqrt{n}\,}\sum_{i=1}^n\psi_{i-k}f\Big (\frac{i}{n}\Big )=\frac{a_{nk}(\,f)}{\sqrt{n}},\end{equation}

where $\psi_{i-k}=0$ if $i<k$ . By Lemma 1 we have the useful series representation

(27) \begin{equation}\frac{\nu_n(\,f)}{\sqrt{n}}=\sum_{k\in{\mathbb Z}}u_{nk}(\,f)\eta_k . \end{equation}

We apply Theorem 5 to the sequence of processes

(28) \begin{equation}Z_{nk}=\{Z_{nk}(\,f)\,:\!=u_{nk}(\,f)\eta_k\colon f\in {\mathcal{F}}\},\qquad k\in{\mathbb Z},\,\,n\in{\mathbb N}_{+}.\end{equation}

5.1. Measurability

We can and do assume that $(\eta_k)_{k\in{\mathbb Z}}$ is defined on the product probability space

\begin{equation*}(\Omega,{\mathcal{A}},{\mathrm{P}})=\bigotimes_{k\in{\mathbb Z}}(\Omega_k,{\mathcal{A}}_k,{\mathrm{P}}_k)\end{equation*}

with its joint distribution equal to the product of distributions of $\eta_k$ . We will show that $(Z_{nk})_{k\in{\mathbb Z}}\in\mathcal{M} (\Omega,{\mathcal{A}},{\mathrm{P}})$ using the fact that ${\mathcal{F}}$ satisfies the pointwise countable approximation property.

Given a tuple $e=(e_k)_{k\in{\mathbb Z}}$ with $e_k\in\{-1,0,1\}$ , for each $i\in\{1,\dots,n\}$ and $\omega\in\Omega$ , let

\begin{equation*}X_i^e(\omega)\,:\!=\sum_{k\in{\mathbb Z}}e_k\psi_{i-k}\eta_k(\omega).\end{equation*}

By (28) and (26), for each pair $f,g\in{\mathcal{F}}$ , $n\in{\mathbb N}_{+}$ , and $\omega\in\Omega$ , we have

\begin{equation*}\sum_{k\in{\mathbb Z}}e_k[Z_{nk}(\,f,\omega)-Z_{nk}(g,\omega)]=\frac{1}{\sqrt{n}\,}\sum_{i=1}^nX_i^e(\omega)(\,f-g)\Big (\frac{i}{n}\Big )=\!:\,T_n^e(\,f,g,\omega).\end{equation*}

For each $\delta >0$ , let ${\mathcal{F}}^{\delta}\,:\!=\{(\,f,g)\in{\mathcal{F}}\times{\mathcal{F}}\colon\rho_2(\,f,g)<\delta\}$ . Let ${\mathcal{F}}_0\subset{\mathcal{F}}$ be a countable set such that every $f\in{\mathcal{F}}$ is a pointwise limit of functions in ${\mathcal{F}}_0$ . Then (16) with ${\mathcal{F}}_0$ in place of ${\mathcal{F}}$ is measurable, and

\begin{equation*}{\mathrm{P}}^{\ast}\left(\left\{\sup\{|T_n^e(\,f,g,\cdot)|\colon(\,f,g)\in{\mathcal{F}}^{\delta}\}\not =\sup\{|T_n^e(\,f,g,\cdot)|\colon(\,f,g)\in{\mathcal{F}}_0^{\delta}\}\right\}\right)=0\end{equation*}

for each $\delta >0$ , each $e=(e_k)_{k\in{\mathbb Z}}$ , and each $n\in{\mathbb N}_{+}$ . Therefore the function (16) is measurable.

Measurability of (17) follows similarly once we show that the series

\begin{equation*}\omega\mapsto\sum_{k\in{\mathbb Z}}[Z_{nk}(\,f,\omega)]^2=\sum_{k\in{\mathbb Z}}u_{nk}^2(\,f)\eta_k^2(\omega)\end{equation*}

converges for each $f\in{\mathcal{F}}$ and $n\in{\mathbb N}_{+}$ . But this is true due to Lemma 1 and to the fact that

\begin{equation*}{\mathrm{E}}\left [\sum_{k\in{\mathbb Z}}u_{nk}^2(\,f)\eta_k^2\right ] = \frac{\sigma_{\eta}^2}{n}\sum_{k\in{\mathbb Z}}a_{nk}^2(\,f)<\infty.\end{equation*}

Therefore, $(Z_{nk})_{k\in{\mathbb Z}}\in\mathcal{M} (\Omega,{\mathcal{A}},{\mathrm{P}})$ .

5.2. Hypothesis (18)

By definition, for each $n\in{\mathbb N}_{+}$ we have $u_{nk}=0$ for each $k>n$ . Therefore, $\sum_{k>n}\|u_{nk}\|_{{\mathcal{F}}}=0$ for each $n\in{\mathbb N}_{+}$ . We will choose a subsequence of positive integers $(m_n)_{n\in{\mathbb N}_{+}}$ such that

(29) \begin{equation}\lim_{n\to\infty}\sum_{-\infty<k < -m_n}\|u_{nk}\|_{{\mathcal{F}}}=0.\end{equation}

Let ${\mathbb F}_{{\mathcal{F}}}$ be the function on [0, 1] with values

\begin{equation*}{\mathbb F}_{{\mathcal{F}}}(x)\,:\!= \sup\{|\,f(x)|\,:\,f \in {\mathcal{F}}\},\quad x\in [0, 1].\end{equation*}

Since ${\mathcal{F}}$ is bounded in ${\mathcal{W}}_q[0,1]$ , then $\|{\mathbb F}_{{\mathcal{F}}}\|_{\sup}<\infty$ . By (26), for each $n\in{\mathbb N}_{+}$ , $k\in{\mathbb Z}$ , and $f\in{\mathcal{F}}$ we have

(30) \begin{equation}|u_{nk}(\,f)|\leq\frac{1}{\sqrt{n}\,}\sum_{i=1}^n|\psi_{i-k}|\Big |f\Big (\frac{i}{n}\Big)\Big |\leq\frac{\|{\mathbb F}_{\mathcal{F}}\|_{\sup}}{\sqrt{n}}\sum_{i=1}^n|\psi_{i-k}|.\end{equation}

Let $0\leq m<M$ . Then

\begin{align*}\sum_{-M\leq k\leq -m}\!\!\!\|u_{nk}\|_{{\mathcal{F}}} & \leq \frac{\|{\mathbb F}_{{\mathcal{F}}}\|_{\sup}}{\sqrt{n}}\sum_{-M\leq k\leq -m}\sum_{i=1}^n|\psi_{i-k}|\\ & =\frac{\|{\mathbb F}_{{\mathcal{F}}}\|_{\sup}}{\sqrt{n}}\sum_{i=1}^n\sum_{-M\leq k\leq -m}\!\!\!|\psi_{i-k}|\\ &\leq \frac{\|{\mathbb F}_{{\mathcal{F}}}\|_{\sup}}{\sqrt{n}}\sum_{i=1}^n\sum_{j\geq i+m}\!\!|\psi_j|\leq\frac{\|{\mathbb F}_{{\mathcal{F}}}\|_{\sup}}{\sqrt{n}}n\sum_{j\geq 1+m}\!\!|\psi_j|.\end{align*}

Now, one can choose a subsequence $(m_n)_{n\in{\mathbb N}_{+}}$ such that $\sum_{j\geq 1+m_n}|\psi_j|\leq n^{-1}$ for each $n\in{\mathbb N}_{+}$ . Hence,

\begin{equation*}\sum_{-\infty<k\leq -m_n}\!\!\!\|u_{nk}\|_{{\mathcal{F}}}\leq \frac{\|{\mathbb F}_{{\mathcal{F}}}\|_{\sup}}{\sqrt{n}}\end{equation*}

for each $n\in{\mathbb N}_{+}$ , and so (29) holds. One can assume that $m_n>n$ for each $n\in{\mathbb N}_{+}$ , and so (18) holds with the subsequence $(m_n)$ .

5.3. Hypothesis (19)

To establish hypothesis (19) it is enough to prove that

(31) \begin{equation}U\,:\!=\sup_{n\ge 1}\bigg (\sum_{k\in{\mathbb Z}}\|u_{nk}\|_{{\mathcal{F}}}^2\bigg )<\infty.\end{equation}

Indeed, suppose it is true. By (30) and assumption (2) we have

\begin{equation*}\|u_{nk}\|_{{\mathcal{F}}}\leq\frac{\|{\mathbb F}_{\mathcal{F}}\|_{\sup}}{\sqrt{n}}\sum_{j\in{\mathbb N}}|\psi_{j}|=\!:\,\frac{c}{\sqrt{n}\,}.\end{equation*}

By (28), for each $m,n\in{\mathbb N}_{+}$ and $\varepsilon>0$ we have

\begin{align*}\sum_{k=-m}^m{\mathrm{E}}^{\ast}(\|Z_{nk}\|_{{{\mathcal{F}}}}^2) \textbf{1}_{\{\| Z_{nk}\|_{{{\mathcal{F}}}}>\varepsilon\}}& \leq \sum_{k=-m}^m\|u_{nk}\|_{{\mathcal{F}}}^2{\mathrm{E}}(\eta_k^2) \textbf{1}_{\{\| u_{nk}\|_{{{\mathcal{F}}}}|\eta_k|>\varepsilon\}}\\ &\leq U{\mathrm{E}}(\eta_0^2) \textbf{1}_{\{|\eta_0|>c^{-1}\varepsilon\sqrt{n}\}}.\end{align*}

This yields (19). We are left to prove (31).

By (30) it is enough to prove that

(32) \begin{equation}\sup_{n\ge 1}\frac{1}{n}\sum_{k\in {\mathbb Z}}\bigg(\sum_{i=1}^n |\psi_{i-k}|\bigg)^2<\infty.\end{equation}

For each $i\in{\mathbb Z}$ let

\begin{equation*}\widetilde{X}_i=\sum_{k\in{\mathbb N}}|\psi_{k}|\eta_{i-k}.\end{equation*}

Then

\begin{align*}\sum_{k\in {\mathbb Z}}\bigg(\sum_{i=1}^n |\psi_{i-k}|\bigg)^2&={\mathrm{E}}\bigg(\sum_{k\in {\mathbb Z}}\bigg(\sum_{i=1}^n|\psi_{i-k}|\bigg)\eta_k\bigg)^2\\&={\mathrm{E}}\bigg(\sum_{i=1}^n\sum_{k\in {\mathbb Z}}|\psi_{i-k}|\eta_k\bigg)^2={\mathrm{E}}\bigg(\sum_{i=1}^n \widetilde{X}_i\bigg)^2.\end{align*}

Since the linear process $(\widetilde{X}_i)$ is covariance stationary, we have

\begin{align*}{\mathrm{E}}\bigg(\sum_{i=1}^n \widetilde{X}_i\bigg)^2&=\sum_{i,j=1}^n {\mathrm{E}}(\widetilde{X}_i\widetilde{X}_j)=n\sum_{j=-(n-1)}^{n-1}\Big (1-\frac{|\,j|}{n}\Big ){\mathrm{E}}(\widetilde{X}_0\widetilde{X}_j)\\ &\le n\sum_{j=0}^n |{\mathrm{E}}(\widetilde{X}_j\widetilde{X}_0)|=n\sigma^2\sum_{j=0}^n\sum_{k=0}^\infty|\psi_{k+j}|\cdot|\psi_k|\\&\le \sigma^2\bigg(\sum_{k=0}^\infty|\psi_k|\bigg)^2 n.\end{align*}

Due to assumption (2), this completes the proof of (32).

5.4. Hypothesis (20)

To prove hypotheses (20) and (21) we use the following representation of the series (27). For a sequence $(t_k)_{k\in{\mathbb Z}}$ of real numbers such that $\sum_{k\geq 0}\psi_kt_{i-k}$ converges for each $i\in{\mathbb N}_{+}$ , the series $\sum_{k\in{\mathbb Z}}\psi_{i-k}t_k$ also converges (here, $\psi_k=0$ for $k<0$ ), and for each $n\in{\mathbb N}_{+}$ we have

(33) \begin{align}\sum_{k\in{\mathbb Z}}\bigg [\sum_{i=1}^n\psi_{i-k}\,f\Big (\frac{i}{n}\Big )\bigg ]t_k&= \sum_{i=1}^n\bigg [\sum_{k\in{\mathbb Z}}\psi_{i-k}t_k\bigg ]\,f\Big (\frac{i}{n}\Big )\nonumber\\ &= \sum_{i=1}^n\bigg [\sum_{k=0}^{\infty}\psi_{k}t_{i-k}\bigg ]\,f\Big (\frac{i}{n}\Big )=\sum_{k=0}^{\infty}\psi_k\bigg [\sum_{i=1}^nt_{i-k}f\Big (\frac{i}{n}\Big )\bigg ].\end{align}

Now, to establish hypothesis (20), recall [(26) and (28)] that

\begin{equation*}{\mathrm{E}}[Z_{nk}(\,f)-Z_{nk}(g)]^2={\mathrm{E}}[Z_{nk}(\,f-g)]^2=\frac{\sigma^2}{n}\bigg(\sum_{i=1}^n\psi_{i-k}(\,f-g)\Big(\frac{i}{n}\Big )\bigg)^2 \end{equation*}

for all $f,g\in {\mathcal{F}}$ , $n\in{\mathbb N}_{+}$ , and $k\in {\mathbb Z}$ . Let $(r_k)_{k\in{\mathbb Z}}$ be a Rademacher sequence, $h\in{\mathcal{F}}$ , and $n\in{\mathbb N}_{+}$ . By the Khinchin–Kahane inequality with the constant K, we have

\begin{align*}\sum_{k\in{\mathbb Z}}\bigg(\sum_{i=1}^n\psi_{i-k}h\Big(\frac{i}{n}\Big )\bigg)^2 & = {\mathrm{E}}\bigg (\sum_{k\in{\mathbb Z}}\bigg(\sum_{i=1}^n\psi_{i-k}h\Big(\frac{i}{n}\Big )\bigg )r_k\bigg )^2\\ &\leq K^2\bigg ({\mathrm{E}}\bigg |\sum_{k\in{\mathbb Z}}\bigg(\sum_{i=1}^n\psi_{i-k}h\Big(\frac{i}{n}\Big )\bigg )r_k\bigg |\bigg )^2.\end{align*}

The series on the right-hand side converges and has representation (33) with $t_k=r_k(\omega)$ . Therefore,

\begin{align*}{\mathrm{E}}\bigg |\sum_{k\in{\mathbb Z}}\bigg(\sum_{i=1}^n\psi_{i-k}h\Big(\frac{i}{n}\Big )\bigg )r_k\bigg | & = {\mathrm{E}}\bigg |\sum_{k=0}^{\infty}\psi_k\bigg (\sum_{i=1}^nr_{i-k}h\Big(\frac{i}{n}\Big )\bigg )\bigg | \\ &\leq \sum_{k=0}^{\infty}|\psi_k|{\mathrm{E}}\bigg |\sum_{i=1}^nr_{i-k}h\Big(\frac{i}{n}\Big )\bigg | \\&\leq \sum_{k=0}^{\infty}|\psi_k|\bigg (\sum_{i=1}^nh^2\Big(\frac{i}{n}\Big )\bigg )^{1/2}.\end{align*}

Using the Minkowski inequality for integrals and then the Minkowski inequality for sums, we obtain

(34) \begin{align}\lefteqn{\bigg(\frac{1}{n}\sum^n_{i=1}h^2\Big(\frac{i}{n} \Big )\bigg)^{1/2}=\bigg(\sum^n_{i=1}\int^{i/n}_{(i-1)/n}\Big[h\Big (\frac{i}{n}\Big )- h(t) + h(t) \Big]^2\text{d} t\bigg)^{1/2}}\nonumber\\&\le \bigg(\sum^n_{i=1}\bigg \{\bigg (\int^{i/n}_{(i-1)/n}\Big[h\Big (\frac{i}{n}\Big )- h(t)\Big ]^2\,\text{d} t\bigg )^{1/2}+\bigg (\int_{(i-1)/n}^{i/n}h^2(t)\,dt\bigg )^{1/2}\bigg \}^2\bigg )^{1/2}\nonumber\\&\le\bigg(\frac{1}{n}\sum^n_{i=1}\sup\bigg\{\Big [h\Big (\frac{i}{n}\Big )-h(t)\Big ]^2\,:\,t \in \Big [\frac{i-1}{n},\frac{i}{n} \Big ]\bigg\}\bigg)^{1/2}+\bigg(\int^1_0 h^2(t) \,\text{d} t\bigg)^{1/2}.\end{align}

For each $i\in\{1,\dots,n\}$ , we have the bound

\begin{equation*}\sup\bigg\{\Big [h\Big (\frac{i}{n}\Big )-h(t)\Big ]^2\colon\, t \in \Big [\frac{i-1}{n},\frac{i}{n} \Big ]\bigg\}\le v_2\Big(h;\Big [\frac{i-1}{n},\frac{i}{n}\Big ]\Big ).\end{equation*}

Summing the bounds over i, and continuing to bound the right-hand side of (34), it follows that

\begin{equation*}\bigg(\frac{1}{n}\sum^n_{i=1}h^2\Big(\frac{i}{n} \Big )\bigg)^{1/2}\le n^{-1/2}||h||_{(2)} + \rho_2(h, 0).\end{equation*}

Summing up the preceding inequalities and replacing h by $f-g$ , it follows that

\begin{equation*}\bigg (\sum_{k\in{\mathbb Z}}{\mathrm{E}}[Z_{nk}(\,f)-Z_{nk}(g)]^2\bigg)^{1/2}\le \sigma K\bigg(\sum_{k=0}^{\infty}|\psi_k|\bigg)\bigg [\frac{||f-g||_{(2)}}{\sqrt{n}}+\rho_2(\,f,g)\bigg].\end{equation*}

Since ${\mathcal{F}}\subset{\mathcal{W}}_q[0,1]\subset{\mathcal{W}}_2[0,1]$ , this proves hypothesis (20).

5.5. Hypothesis (21)

To establish hypothesis (21), recall the random pseudometric $d_n(\,f,g)$ defined by (22). Since the function $f\mapsto Z_{nk}(\,f)$ defined by (28) is linear, for simplicity consider instead $p_n(\,f)\,:\!=d_n(\,f,0)$ for each $f\in{\mathcal{F}}$ . Let $n\in{\mathbb N}_{+}$ , $f\in{\mathcal{F}}$ , and let $(r_k)_{k\in{\mathbb Z}}$ be a Rademacher sequence. By the Khinchin–Kahane inequality with the constant K again, we have

(35) \begin{equation}p_n^2(\,f)=\sum_{k\in{\mathbb Z}}u^2_{nk}(\,f)\eta_k^2={\mathrm{E}}_r\bigg(\sum_{k\in{\mathbb Z}}u_{nk}(\,f)\eta_kr_k\bigg)^2\le K^2\bigg({\mathrm{E}}_r\bigg|\sum_{k\in{\mathbb Z}}u_{nk}(\,f)\eta_kr_k\bigg|\bigg)^2.\end{equation}

Now recall the notation in (26) for $u_{nk}(\,f)$ . The expression in (33) with $t_k=\eta_k(\omega_1)r_k(\omega_2)$ for the series on the right-hand side gives the equality

\begin{equation*}\sum_{k\in{\mathbb Z}}u_{nk}(\,f)\eta_kr_k=\frac{1}{\sqrt{n}\,}\sum_{k=0}^{\infty}\psi_k\sum_{i=1}^nf\Big (\frac{i}{n}\Big )\eta_{i-k}r_{i-k}.\end{equation*}

Continuing (35) with this representation, we obtain

(36) \begin{align}\lefteqn{\frac{\sqrt{n}}{K}p_n(\,f)}\nonumber\\&\le {\mathrm{E}}_r\bigg|\sum_{k=0}^\infty \psi_k\sum_{i=1}^nf\Big (\frac{i}{n}\Big )\eta_{i-k}r_{i-k}\bigg|\le \sum_{k=0}^\infty |\psi_k| {\mathrm{E}}_r\bigg|\sum_{i=1}^n f\Big (\frac{i}{n}\Big )\eta_{i-k}r_{i-k}\bigg|\end{align}

\begin{align}&\le\sum_{k=0}^{\infty}|\psi_k|\bigg(\sum_{i=1}^nf^2\Big (\frac{i}{n}\Big )\eta_{i-k}^2\bigg)^{1/2}\leq \bigg (\sum_{k=0}^{\infty}|\psi_k|\bigg )^{1/2}\bigg (\sum_{k=0}^{\infty}|\psi_k|\sum_{i=1}^nf^2\Big (\frac{i}{n}\Big )\eta_{i-k}^2\bigg)^{1/2}.\nonumber\end{align}

The last inequality is Hölder’s inequality. On $([0,1],{\mathcal{B}})$ , define random measures $\mu_n$ by

\begin{equation*}\mu_n(B)\,:\!=\sum_{k=0}^{\infty}|\psi_k|\frac{1}{n}\sum_{i=1}^n\eta_{i-k}^2\delta_{i/n}(B),\qquad B\in{\mathcal{B}},\,\,n\in{\mathbb N}_{+}.\end{equation*}

Since $\sigma_{\eta}\not =0$ , given $\epsilon >0$ one can find $\Omega_{\epsilon}\subset\Omega$ and $n_{\epsilon}\in{\mathbb N}$ such that ${\mathrm{P}} (\Omega_{\epsilon})<\epsilon$ and $\mu_n([0,1])>0$ for each $\omega\not\in\Omega_{\epsilon}$ and $n\geq n_{\epsilon}$ . Thus, without loss of generality we assume that $\mu_n([0,1])>0$ almost surely. Then $Q_n\,:\!=\mu_n/\mu_n([0,1])$ , $n\in{\mathbb N}_{+}$ , are random probability measures on $([0,1],{\mathcal{B}})$ . For each $n\in{\mathbb N}_{+}$ let

\begin{equation*}\xi_n\,:\!=K\bigg (\sum_{k=0}^{\infty}|\psi_k|\bigg )^{1/2}\sqrt{\mu_n([0,1])}=K\bigg (\sum_{k=0}^{\infty}|\psi_k|\bigg)^{1/2}\bigg (\sum_{k=0}^{\infty}|\psi_k|\frac{1}{n}\sum_{i=1}^n\eta_{i-k}^2\bigg )^{1/2}.\end{equation*}

By (36), it then follows that

\begin{equation*}d_n(\,f,g)=p_n(\,f-g)\leq\xi_n\rho_{2,Q_n}(\,f,g).\end{equation*}

By hypothesis (25), the set ${\mathcal{F}}$ is totally bounded with respect to pseudometric $\rho_{2,Q_n}$ . Given $x>0$ , since each $\rho_{2,Q_n}$ -ball of radius $x/\xi_n$ is contained in a $d_n$ -ball of radius x, we have

\begin{equation*}N(x,{\mathcal{F}},d_n)\leq N(x/\xi_n,{\mathcal{F}},\rho_{2,Q_n}).\end{equation*}

Then, by a change of variables it follows that, for each $\delta > 0$ ,

\begin{equation*}I(\delta)\,:\!=\int^\delta_0\sqrt{\log N(x, {\mathcal{F}}, d_n)}\, \text{d} x \leq \xi_n\int^{\delta/\xi_n}_0\sqrt{\log N(x, {\mathcal{F}}, \rho_{2,Q_n})}\, \text{d} x.\end{equation*}

For each $\delta >0$ , let

\begin{equation*}J(\delta)\,:\!=\int^\delta_0\sup_{Q\in{\mathcal{Q}}}\sqrt{\log N(x, {\mathcal{F}}, \rho_{2,Q})}\, \text{d} x.\end{equation*}

Let $\varepsilon>0$ and let $\delta_n \downarrow 0$ . For each $0 < m <M < \infty$ and $n \in {\mathbb N}_{+}$ , we have

(37) \begin{equation}{\mathrm{P}}(I(\delta_n) > \varepsilon) \le{\mathrm{P}}(MJ(\delta_n/m) > \varepsilon)+{\mathrm{P}}(\xi_n>M)+{\mathrm{P}}(\xi_n <m).\end{equation}

Taking $m>0$ small enough, the rightmost probability tends to zero with $n\to\infty$ since $\liminf_{n\to\infty}\xi_n \geq c\sigma_{\eta}>0$ almost surely. For the next-to-rightmost probability we have

\begin{equation*}\sup_{n\ge 1}{\mathrm{P}}(\xi_n>M) \le M^{-2}\sup_{n\ge 1} {\mathrm{E}}(\xi_n^2)=\frac{K^2\sigma_{\eta}^2}{M^2}\bigg (\sum_{k=0}^{\infty}|\psi_k|\bigg )^2\to 0\end{equation*}

as $M \to\infty$ . Since $J(\delta_n/m) \to 0$ as $n \to\infty$ by condition (25), the first probability on the right-hand side of (37) is zero for sufficiently large n. It then follows that hypothesis (21) holds.

Summing up, by Theorem 5, $n^{-1/2}\nu_n$ is asymptotically equicontinuous with respect to $\rho_2$ . By Proposition 1, the finite-dimensional distributions of $n^{-1/2}\nu_n$ converge to finite-dimensional distributions of $\sigma_{\eta}A_{\psi}\nu$ . Thus, by Theorem 3.7.23 in [Reference Giné and Nickl12], the conclusion of Theorem 6 is proved.

6. Proofs of Theorems 3 and 2

We begin with the proof of Theorem 3. Let $G\,:\!=T(L)$ , $(t_i)_{i=0}^m$ be a partition of [a, b], and let $b=(b_1,\dots,b_m)\in{\mathbb R}^m$ . Then $f_b\,:\!=\sum_{i=1}^mb_i\textbf{1}_{(t_{i-1},t_i]}\in{\mathcal{W}}_q[a,b]$ and

\begin{equation*}\bigg |\sum_{i=1}^mb_i[G(t_i)-G(t_{i-1})]\bigg |=\bigg |L\bigg (\sum_{i=1}^mb_i\textbf{1}_{(t_{i-1},t_i]}\bigg )\bigg |\leq \|L\|_{{\mathcal{F}}_q}\|\,f_b\|_{[q]}.\end{equation*}

Let $\|b\|_q\,:\!=\big(\sum_{i=1}^m|b_i|^q\big)^{1/q}$ . Then $\|\,f_b\|_{\sup}=\max_i|b_i|\leq\|b\|_q$ and $\|\,f_b\|_{(q)}\leq 2\|b\|_q$ due to the Minkowski inequality. Using the extremal Hölder equality we obtain the bound

\begin{equation*}\bigg (\sum_{i=1}^m|G(t_i)-G(t_{i-1})|^p\bigg )^{1/p}=\sup\bigg\{\bigg |\sum_{i=1}^mb_i[G(t_i)-G(t_{i-1})]\bigg |\colon\|b\|_q\leq 1\bigg \}\leq 3\|L\|_{{\mathcal{F}}_q}.\end{equation*}

Since the partition $(t_i)_{i=1}^m$ of [a, b] is arbitrary, it follows that $\|G\|_{(q)}\leq 3\|L\|_{{\mathcal{F}}_q}$ . Since $\|\textbf{1}_{[a,(\cdot)]}\|_{[q]}=2$ , we have the bound $\|G\|_{\sup}\leq2\|L\|_{{\mathcal{F}}_q}$ and so (8) holds. The proof of Theorem 3 is complete.

To prove Theorem 2, for $p\in (2,\infty)$ given as the hypothesis, let $q\,:\!=(p-1)/p$ . Then $p^{-1}+q^{-1}=1$ and $1<q<2$ . By Theorem 1, the isonormal Gaussian process $\nu$ restricted to ${\mathcal{F}}_q=\{\,f\in{\mathcal{W}}_q[0,1]\colon\|\,f\|_{[q]}\leq 1\}$ takes values in a separable subset of $\ell^{\infty}({\mathcal{F}}_q)$ , it is measurable for Borel sets on its range, and

(38) \begin{equation}(\sigma_{\eta}|A_{\psi}|\sqrt{n})^{-1}\nu_n {\stackrel{\mathcal{D}} \Longrightarrow} \nu \qquad\mbox{in $\ell^\infty({\mathcal{F}}_q)$.}\end{equation}

By the Skorokhod–Dudley–Wichura representation theorem (Theorem 3.5.1 in [Reference Dudley9]), there exist a probability space $(S,{\mathcal{S}},Q)$ and perfect measurable functions $g_n\colon\,S\to\Omega$ such that $Q{\circ}g_n^{-1}={\mathrm{P}}$ on ${\mathcal{A}}$ for each $n\in{\mathbb N}$ and

(39) \begin{equation}\lim_{n\to\infty}\big\|(\sigma_{\eta}|A_{\psi}|\sqrt{n})^{-1}\nu_n{\circ}g_n-\nu{\circ}g_0\big\|_{{\mathcal{F}}_q}^{\ast}=0\qquad\mbox{almost surely.}\end{equation}

Here, as for any real-valued function $\phi$ on a probability space, $\phi^{\ast}$ is its measurable cover, which always exists (e.g. Theorem 3.2.1 in [Reference Dudley9]). For each $n\in{\mathbb N}_{+}$ and $s\in S$ , let

\begin{equation*}\mu_n(\,f,s)\,:\!=\frac{\nu_n(\,f,g_n(s))}{\sigma_{\eta}|A_{\psi}|\sqrt{n}}, \,\,f\in{\mathcal{F}}_q,\quad\mbox{and}\quad W_n(t,s)\,:\!=\mu_n(\textbf{1}_{[0,t]},s),\,\,t\in [0,1].\end{equation*}

Also, for each $n\in{\mathbb N}_{+}$ , $s\in S$ , and $f\in{\mathcal{F}}_q$ ,

\begin{equation*}|\mu_n(\,f,s)|\leq\frac{\sum_{i=1}^n|X_i(g_n(s))|}{\sigma_{\eta}|A_{\psi}|\sqrt{n}}\|\,f\|_{\sup} .\end{equation*}

Hence, for each $n\in{\mathbb N}_{+}$ and $s\in S$ , $\mu_n(\cdot,s)$ is a linear bounded functional on ${\mathcal{W}}_q[0,1]$ . Given $n,m\in{\mathbb N}_{+}$ , let $L\,:\!=\mu_n-\mu_m$ . Then $T(L)(t)=L(\textbf{1}_{[0,t]})=W_n(t)-W_m(t)$ for each $t\in [0,1]$ , and

\begin{equation*}\|W_n-W_m\|_{[p]}\leq 5\|\mu_n-\mu_m\|_{{\mathcal{F}}_q}\end{equation*}

for each $s\in S$ , by Theorem 3. For any functions $\phi,\xi\colon S\to{\mathbb R}$ , we have $(\phi+\xi)^{\ast}\leq\phi^{\ast}+\xi^{\ast}$ almost surely (e.g. Lemma 3.2.2 in [Reference Dudley9]). By (39), it then follows that

\begin{equation*}\lim_{m, n\to\infty}\|\mu_n-\mu_m\|_{{\mathcal{F}}_q}^{\ast}=0\qquad\mbox{almost surely.}\end{equation*}

Therefore, for each $s\in S$ , $(W_n(\cdot,s))$ is a Cauchy sequence in the Banach space ${\mathcal{W}}_p[0,1]$ .

For each $s\in S$ , let $W(s)\,:\!=\{W_t(s)\colon t\in [0,1]\}\in{\mathcal{W}}_p[0,1]$ be a function such that $\|W_n(\cdot,s)-W(s)\|_{[p]}\to 0$ almost surely as $n\to\infty$ . For each $t\in [0,1]$ , since $|W_t-W_n(t)|\leq \|W-W_n\|_{\sup}\to 0$ as $n\to\infty$ , $W_t$ is measurable, and so W is a stochastic process. For a Borel set $B\in{\mathbb R}^k$ and $t_1,\dots,t_k\in [0,1]$ , we have

\begin{equation*}Q(\{(W_n(t_1),\dots,W_n(t_k))\in B\})={\mathrm{P}}(\{(\sigma_{\eta}|A_{\psi}|\sqrt{n})^{-1}(\nu_n(\textbf{1}_{[0,t_1]}),\dots,\nu_n(\textbf{1}_{[0,t_k]})\in B\} ).\end{equation*}

By (38), $(\sigma_{\eta}|A_{\psi}|\sqrt{n})^{-1}(\nu_n(\textbf{1}_{[0,t_1]}),\dots,\nu_n(\textbf{1}_{[0,t_k]}))$ converge in distribution as $n\to\infty$ to $(\nu(\textbf{1}_{[0,t_1]},\dots,\nu (\textbf{1}_{[0,t_k]}))$ for the isonormal Gaussian process $\nu$ on ${\mathcal{L}}^2([0,1])$ . Also, $(W_n(t_1),\dots, W_n(t_k))$ converge in distribution as $n\to\infty$ to $(W_{t_1},\dots,W_{t_k})$ . It then follows that W is a Gaussian process with the covariance of a Wiener process. Since sample paths of $\nu$ are uniformly continuous with respect to the pseudometric $\rho_2$ , W has almost all sample paths continuous, and so W is a standard Wiener process on [0, 1].

Let ${\mathcal{C}}{\mathcal{W}}_p^{\ast}[0,1]$ be the set of all $f\in {\mathcal{W}}_p[0,1]$ such that

\begin{equation*}\lim_{\epsilon\downarrow 0}\sup\bigg\{\sum_{k=1}^m|f(t_k)-f(t_{k-1})|^p\colon\,0=t_0<t_1<\cdots<t_m=1,\,\max_k(t_k-t_{k-1})\leq\epsilon\bigg\}=0.\end{equation*}

Then ${\mathcal{C}}{\mathcal{W}}_p^{\ast}[0,1]$ is a separable, closed subspace of ${\mathcal{W}}_p[0,1]$ ([Reference Kisliakov13]). Since for each $p'>2$ , almost all sample functions of a Wiener process are of bounded p’-variation on [0, 1], by Lemma 2.14 in [Reference Dudley and Norvaiša11, Part II] it follows that almost all sample functions of W are in ${\mathcal{C}}{\mathcal{W}}_p^{\ast}[0,1]$ . Therefore, $W_n$ converges in law to W in ${\mathcal{W}}_p[0,1]$ by Corollary 3.3.5 in [Reference Dudley9]). The proof of Theorem 2 is complete.

7. Applications

In this section we apply the preceding results to prove the uniform asymptotic normality of least squares estimators in parametric regression models, and to detect change points in trends of a short-memory linear process. Throughout this section, $X_1,X_2,\dots$ is again a short-memory linear process given by (1) with innovations $(\eta_j)$ and summable filter $(\psi_j)$ such that (2) holds.

7.1. Simple regression model

We start with a simple parametric regression model $Y_j=\beta Z_{nj}+X_j$ , $j=1, \dots, n$ , where $\beta\in {\mathbb R}$ is an unknown parameter and $Z_{nj}$ are explanatory variables for the process $(Y_j)$ . We assume that $Z_{nj}=f(j/n)$ for some function f on [0, 1]. Then the least squares estimator of $\beta$ is

\begin{equation*}{\widehat{\beta}}_n={\widehat{\beta}}_n(\,f)\,:\!=\bigg(\sum_{j=1}^nf^2\Big (\frac{j}{n}\Big)\bigg)^{-1}\sum_{j=1}^n Y_j\, f\Big(\frac{j}{n}\Big).\end{equation*}

As a choice of the function f in the representation of $Z_{nj}$ is not unique, finding an admissible class of functions becomes an important task. In response to this question we present what follows from our main result.

Note that the equality

(40) \begin{equation}\sum_{j=1}^n f^2\Big (\frac{j}{n}\Big )[{\widehat{\beta}}_n(\,f)-\beta]=\sum_{j=1}^n X_jf\Big (\frac{j}{n}\Big ) \end{equation}

holds for each real-valued function f on [0, 1] and each $n\in{\mathbb N}_{+}$ . By Theorem 4, it then follows that

\begin{equation*}W_n(\,f)\,:\!=\frac{1}{\sqrt{n}}\sum_{j=1}^n f^2\Big (\frac{j}{n}\Big )[{\widehat{\beta}}_n(\,f)-\beta] \xrightarrow{\mathcal{D}} \sigma_{\eta}A_{\psi}\nu (\,f)\qquad\mbox{as $n\to\infty$,}\end{equation*}

for each $f\in{\mathcal{W}}_q[0,1]$ with $q\in [1,2)$ , where $\nu$ is the isonormal Gaussian process on ${\mathcal{L}}_2[0,1]$ .

As a straightforward consequence of Theorem 1 and the equality in (40), we obtain a weighted asymptotic normality of the estimator ${\widehat{\beta}}_n(\,f)$ uniformly over the set of functions ${\mathcal{F}}_q=\{\,f\in{\mathcal{W}}_q[0,1]\colon\|\,f\|_{[q]}\leq 1\}$ , $1\leq q< 2$ .

Corollary 2. Let $1\le q<2$ . There exists a version of the isonormal Gaussian process $\nu$ restricted to ${\mathcal{F}}_q$ with values in a separable subset of $\ell^{\infty}({\mathcal{F}}_q)$ , it is measurable for the Borel sets on its range, and

\begin{equation*}W_n \xrightarrow{\mathcal{D}^*} \sigma_{\eta}A_{\psi}\nu\quad\mbox{in $\ell^{\infty}({\mathcal{F}}_q)$ as $n\to\infty$}.\end{equation*}

Next, we establish the (unweighted) asymptotic normality of ${\widehat{\beta}}_n(\,f)$ uniformly over a subset of ${\mathcal{F}}_q$ . Since each regulated function is a Riemann function, we have

(41) \begin{equation}I_n(\,f^2)\,:\!=\frac{1}{n}\sum_{j=1}^nf^2\Big (\frac{j}{n}\Big )\to\int_0^1f^2(x)\,\text{d} x=\!:\,I(\,f^2)\qquad\mbox{as $n\to\infty$},\end{equation}

for any regulated function f. Each function having bounded p-variation is regulated (see, e.g., [Reference Dudley and Norvaiša11, p. 213]). Therefore, by (40), Theorem 4, and Slutsky’s lemma, if $f\in{\mathcal{W}}_q[0,1]$ for some $1\leq q<2$ and $I(\,f^2)\not =0$ , then

\begin{equation*}n^{1/2}({\widehat{\beta}}_n(\,f)-\beta) \xrightarrow{\mathcal{D}}\sigma_{\eta}A_{\psi}N(0,v^2)\qquad\mbox{as $n\to\infty$},\end{equation*}

where $N(0,v^2)$ is a mean-zero Gaussian random variable with variance $v^2=(I(\,f^2))^{-1}$ . For each $\delta >0$ and $q\in [1,2)$ , let

(42) \begin{equation}{\mathcal{F}}_{q,\delta}\,:\!=\{\,f\in{\mathcal{W}}_q[0,1]:\|\,f\|_{[q]}\leq 1,\,\,\,I(\,f^2)>\delta\}.\end{equation}

Theorem 7. Let $q\in [1,2)$ and $\delta \in (0,1)$ . There exists a version of a Gaussian process $\nu_1$ indexed by ${\mathcal{F}}_{q,\delta}$ with values in a separable subset of $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ , it is measurable for the Borel sets on its range, and

\begin{equation*}\sqrt{n}({\widehat{\beta}}_n-\beta)\xrightarrow{\mathcal{D}^*} \sigma_{\eta}A_{\psi}\nu_1\quad\mbox{in $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ as $n\to\infty$}.\end{equation*}

Proof. Let $\nu$ be the isonormal Gaussian process from Theorem 1. For each $f\in{\mathcal{F}}_{q,\delta}$ , let $\nu_1(\,f)\,:\!=\nu(\,f)/I(\,f^2)$ . Let $z_1,\dots,z_k\in{\mathbb R}$ and let $f_1,\dots,f_k\in{\mathcal{F}}_{q,\delta}$ . Since $\nu$ is linear, we have

\begin{equation*}\sum_{i=1}^k\sum_{j=1}^kz_iz_j{\mathrm{E}}[\nu_1(\,f_i)\nu_1(\,f_j)]={\mathrm{E}}\bigg[\nu^2\bigg (\sum_{i=1}^kz_i\frac{f_i}{I(\,f_i^2)}\bigg )\bigg]=\int_0^1\bigg (\sum_{i=1}^kz_i\frac{f_i}{I(\,f_i^2)}\bigg )^2\text{d}\lambda\geq 0.\end{equation*}

Therefore, $\{\nu_1(\,f)\colon f\in{\mathcal{F}}_{q,\delta}\}$ is a Gaussian process with mean zero and variance ${\mathrm{E}}[\nu_1^2(\,f)]=(I(\,f^2))^{-1}$ . Since $\nu$ is linear we can consider $\nu_1$ to be $\nu$ restricted to the set $\{\,f/I(\,f^2)\colon f\in{\mathcal{F}}_{q,\delta}\}$ , which is a subset of $\{\,f\in{\mathcal{W}}_q[0,1]\colon\|\,f\|_{[q]}\leq \delta^{-1}\}$ . Therefore, the Gaussian process $\nu_1$ on ${\mathcal{F}}_{q,\delta}$ has values in a separable subset of $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ and it is measurable for the Borel sets on its range.

Using the notation of (15), (3), and the equality in (40), for each $f\in{\mathcal{F}}_{q,\delta}$ and $n\in{\mathbb N}_{+}$ , we have

(43) \begin{equation}\sqrt{n}({\widehat{\beta}}_n(\,f)-\beta)=\bigg (\frac{1}{I_n(\,f^2)}-\frac{1}{I(\,f^2)}\bigg )\frac{\nu_n(\,f)}{\sqrt{n}}+\frac{1}{\sqrt{n}}\nu_n\bigg (\frac{f}{I(\,f^2)}\bigg ).\end{equation}

We show that the second term on the right-hand side converges in outer distribution to $\sigma_{\eta}A_{\psi}\nu_1$ . Let $C(\,f)\,:\!=1/I(\,f^2)$ for each $f\in{\mathcal{F}}_{q,\delta}$ , and let $g\colon\ell^{\infty}({\mathcal{F}}_{q,\delta})\to\ell^{\infty}({\mathcal{F}}_{q,\delta})$ be a multiplication function with values $g(F)\,:\!=CF$ for each $F\in\ell^{\infty}({\mathcal{F}}_{q,\delta})$ . One can check that g is bounded and continuous. Let $h\colon\ell^{\infty}({\mathcal{F}}_{q,\delta})\to{\mathbb R}$ be a bounded and continuous function. Then the composition $h{\circ}g$ is also a bounded and continuous function from $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ to ${\mathbb R}$ . Thus, by (5) it follows that

\begin{equation*}{\mathrm{E}}^{\ast}[h(n^{-1/2}C\nu_n)]={\mathrm{E}}^{\ast}[h{\circ}g(n^{-1/2}\nu_n)]\to{\mathrm{E}} [h{\circ}g(\nu)]={\mathrm{E}}[ h(\sigma_{\eta}A_{\psi}\nu_1)]\end{equation*}

as $n\to\infty$ . Thus, the second term on the right-hand side of (43) converges in outer distribution to $\sigma_{\eta}A_{\psi}\nu_1$ . Finally, we will show that the first term on the right-hand side of (43) converges to zero as $n\to\infty$ uniformly in f and in outer probability (Definition 1.9.1 in [Reference Van der Vaart and Wellner15]).

Let $f\in{\mathcal{F}}_{q,\delta}$ and $n\in{\mathbb N}_{+}$ . Then

\begin{eqnarray*}\lefteqn{I_n(\,f^2)-I(\,f^2)=\sum_{j=1}^n\int_{(j-1)/n}^{j/n}\!\Big(\,f^2\Big(\frac{j}{n}\Big)-f^2(x)\Big)\text{d} x}\\&=&{}\,2\sum_{j=1}^n\int_{(j-1)/n}^{j/n}\!\!\!f(x)\Big(\,f\Big(\frac{j}{n}\Big )-f(x)\Big)\text{d} x +\sum_{j=1}^n\int_{(j-1)/n}^{j/n}\!\Big(\,f\Big(\frac{j}{n}\Big )-f(x)\Big)^2\text{d} x.\end{eqnarray*}

Since $v_2(\,f;\ [a,c])+v_2(\,f;\ [c,b])\leq v_2(\,f;\ [a,b])$ for any $0\leq a<c<b\leq 1$ , it follows that

\begin{equation*}\sum_{j=1}^n\int_{(j-1)/n}^{j/n}\!\Big(\,f\Big(\frac{j}{n}\Big )-f(x)\Big)^2\text{d} x\leq\frac{v_2(\,f)}{n}.\end{equation*}

Then, using Hölder’s inequality, we obtain the bound

\begin{equation*}K_n(\,f)\,:\!=|I_n(\,f^2)-I(\,f^2)|\leq 2(I(\,f^2))^{1/2}\Big (\frac{v_2(\,f)}{n}\Big )^{1/2}+\frac{v_2(\,f)}{n}.\end{equation*}

For each $f\in{\mathcal{F}}_{q,\delta}$ , $I(\,f^2)\leq\|\,f\|_{\sup}^2\leq 1$ and $v_2(\,f)\leq \|\,f\|_{[q]}^2\leq 1$ since $q<2$ , and so $K_n(\,f)\break <3/\sqrt{n}$ . We observe that, for each $f\in{\mathcal{F}}_{q,\delta}$ and $n\in{\mathbb N}_{+}$ ,

\begin{equation*}D_n(\,f)\,:\!=\bigg |\frac{1}{I_n(\,f^2)}-\frac{1}{I(\,f^2)}\bigg |\leq\frac{K_n(\,f)}{I^2(\,f^2)(1-I^{-1}(\,f^2)K_n(\,f))} ,\end{equation*}

provided $I^{-1}(\,f^2)K_n(\,f)<1$ . Therefore, for each $n>36/\delta^2$ we have the bound

(44) \begin{equation}\|D_n\|_{{\mathcal{F}}_{q,\delta}}<\frac{6}{\sqrt{n}\delta^2}.\end{equation}

To bound the first term on the right-hand side of (43), note that, for each $\epsilon >0$ and for each $A\in{\mathbb R}$ ,

\begin{equation*}{\mathrm{P}}^{\ast}\bigg (\bigg\{\sup_{f\in{\mathcal{F}}_{q,\delta}}\bigg |\frac{1}{I_n(\,f^2)}-\frac{1}{I(\,f^2)}\bigg |\frac{|\nu_n(\,f)|}{\sqrt{n}}>\epsilon\bigg \}\bigg )\leq{\mathrm{P}}^{\ast}(\{\|D_n\|_{{\mathcal{F}}_{q,\delta}}\|\nu_n\|_{{\mathcal{F}}_{q,\delta}}>\epsilon\sqrt{n}\})\end{equation*}

(45) \begin{equation}\leq{\mathrm{P}}(\{\|D_n\|_{{\mathcal{F}}_{q,\delta}}>\epsilon A\})+{\mathrm{P}}^{\ast}(\{\|\nu_n\|_{{\mathcal{F}}_{q,\delta}}>A\sqrt{n}\}).\end{equation}\vskip-\lastskip\pagebreak

By Theorem 1, the isonormal Gaussian process $\nu$ restricted to ${\mathcal{F}}_{q,\delta}$ has values in separable subset of $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ and is measurable for the Borel sets on its range. Therefore, its law ${\mathrm{P}}{\circ}\nu^{-1}$ is tight. By Lemma 1.3.8 in [Reference Van der Vaart and Wellner15], $n^{-1/2}\nu_n$ is asymptotically tight. It follows that for each $\epsilon >0$ there is $A\in{\mathbb R}$ such that

\begin{equation*}\limsup_{n\to\infty}{\mathrm{P}}^{\ast}(\{\|\nu_n\|_{{\mathcal{F}}_{q,\delta}}>A\sqrt{n}\})<\epsilon .\end{equation*}

By (44), the first probability on the right-hand side of (45) is zero for all sufficiently large n. The first term on the right-hand side of (43) converges to zero as $n\to\infty$ uniformly in f and in outer probability. Thereforem the right-hand side of (43) converges in outer distribution to $\sigma_{\eta}A_{\psi}\nu_1$ , and so does the left-hand side of (43) by Lemma 1.10.2 in [Reference Van der Vaart and Wellner15]. The proof is complete. □

For each real-valued function f on [0, 1] and for each $n\in{\mathbb N}_{+}$ let

\begin{equation*}Q_n(\,f)\,:\!=\bigg [\sum_{j=1}^n f^2\Big (\frac{j}{n}\Big )\bigg ]^{1/2}[{\widehat{\beta}}_n(\,f)-\beta].\end{equation*}

Recalling definition (42) of the class of functions ${\mathcal{F}}_{q,\delta}$ , we have the following result.

Theorem 8. Let $q\in [1,2)$ and $\delta \in(0,1)$ . There exists a version of a Gaussian process $\nu_2$ indexed by ${\mathcal{F}}_{q,\delta}$ with values in a separable subset of $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ , it is measurable for the Borel sets on its range, and

\begin{equation*}Q_n \xrightarrow{\mathcal{D}^*} \sigma_{\eta}A_{\psi}\nu_2\quad\mbox{in $\ell^{\infty}({\mathcal{F}}_{q,\delta})$ as $n\to\infty$}.\end{equation*}

Proof. The proof is similar to that of Theorem 7. Indeed, using equality (40), the notation in (41), and (3), we have the representation

\begin{align*} Q_n(\,f) & = \frac{1}{(I_n(\,f^2))^{1/2}} \frac{\nu_n(\,f)}{\sqrt{n}} \\ & = \left [\frac{1}{(I_n(\,f^2))^{1/2}}-\frac{1}{(I(\,f^2))^{1/2}}\right ]\frac {\nu_n(\,f)}{\sqrt{n}}+\frac{1}{\sqrt{n}\,}\nu_n\bigg (\frac{f}{(I(\,f^2))^{1/2}}\bigg )\end{align*}

for each $f\in{\mathcal{F}}_{q,\delta}$ and each $n\in{\mathbb N}_{+}$ . Note that $|\sqrt{u}-\sqrt{v}|\leq\sqrt{|u-v|}$ for any $u\geq 0$ and $v\geq 0$ . Therefore, the above term in the square brackets approaches zero as $n\to\infty$ uniformly for $f\in{\mathcal{F}}_{q,\delta}$ due to the bound (44) given in the proof of Theorem 7. The conclusion of Theorem 8 then follows as in the preceding proof. □

7.2. Multiple change point model

Consider a time-series model

\begin{equation*}Y_{nj}=\mu_{nj}+X_j,\qquad j=1, \dots, n, \end{equation*}

which is subject to unknown multiple change points $(\tau_1^*,\dots,\tau_{d^*-1}^*)$ . We wish to test the null hypothesis

\begin{equation*}H_0:\quad \mu_{n1}=\cdots=\mu_{nn}=0\end{equation*}

against the multiple change alternative model

\begin{equation*} H_{\text{A}}\,:\,\quad \mu_{nj}= \sum_{k=1}^{d^*}\beta_k\textbf{1}_{I^*_k}(j/n), \qquad j=1, \dots, n,\end{equation*}

with $d^*\in{\mathbb N}_{+}$ , $\beta_1,\dots,\beta_{d^*}$ , and $I^*_k=(\tau^*_{k-1}, \tau^*_k]$ , $0=\tau^*_0<\tau^*_1<\cdots<\tau^*_d=1$ , being unknown parameters.

There is an enormous amount of literature where change detection problems have been studied. The books by Basseville and Nikiforov [Reference Basseville and Nikiforov2], Csörgo and Horváth [Reference Csörgo and Horváth6], Brodsky and Darkhovskay [Reference Brodsky and Darkhovsky3], and Chen and Gupta [Reference Chen and Gupta5] introduce the basics of various methods. We suggest a testing procedure based on uniform asymptotic normality of the partial sum processes obtained in the present paper.

For each $d\in{\mathbb N}_{+}$ , let ${\mathcal T}_d$ be a set of all partitions $(\tau_k)$ of the interval [0, 1] such that $0=\tau_0<\tau_1<\cdots<\tau_d=1$ . For a partition $\tau=(\tau_k)\in {\mathcal T}_d$ one fits the regression model

\begin{equation*} Y_{nj}=\sum_{k=1}^{d}\beta_k\textbf{1}_{I_k}(j/n)+X_j,\qquad j=1, \dots, n,\end{equation*}

where $I_k=(\tau_{k-1}, \tau_k]$ . The parameter $\boldsymbol{\beta}=[\beta_1, \dots, \beta_{d}]'$ is obtained by the least squares estimator

\begin{equation*}{\widehat{\boldsymbol{\beta}}}={\widehat{\boldsymbol{\beta}}}(\boldsymbol{\tau})={\textbf{\textit{Q}}}_n^{-1}\bigg[\sum_{j=1}^nY_{nj}\textbf{1}_{I_1}(j/n), \dots, \sum_{j=1}^nY_{nj}\textbf{1}_{I_d}(j/n)\bigg]',\end{equation*}

where

\begin{equation*}{\textbf{\textit{Q}}}_n={\mathrm{diag}}\{[\tau_kn]-[\tau_{k-1}n],\ k=1, \dots, d\}.\end{equation*}

For $p\geq 1$ let $\|x\|_p$ be the $\ell_p$ -norm of a vector $x\in{\mathbb R}^d$ . Then

\begin{equation*}\|{\textbf{\textit{Q}}}_n{\widehat{\boldsymbol{\beta}}}\|_p=\bigg(\sum_{k=1}^d\bigg|\sum_{j=1}^nY_{nj}\textbf{1}_{I_k}(j/n)\bigg|^p\bigg)^{1/p}.\end{equation*}

Let

\begin{equation*}T_n=T_n(Y_{n1}, \dots, Y_{nn})\,:\!=\sup_{d\in{\mathbb N}_{+}}\sup_{(\tau_k)\in {\mathcal T}_d}\|{\textbf{\textit{Q}}}_n{\widehat{\boldsymbol{\beta}}}\|_p.\end{equation*}

Under the null hypothesis $H_0$ , by (7), the statistic $T_n=\|S_n\|_{(p)}$ , where $S_n$ is the partial sum process of a linear process $(X_k)$ . The following fact is based on Corollary 1.

Theorem 9. Let $p>2$ . Under the null hypothesis $H_0$ we have

\begin{equation*}n^{-1/2}A_{\psi}^{-1}\sigma_{\eta}^{-1}T_n\xrightarrow{\mathcal{D}} \|W\|_{(p)}\qquad\mbox{as $n\to\infty$}.\end{equation*}

Under the alternative $H_{\text{A}}$ we have $n^{-1/2}T_n \xrightarrow[n\to\infty]{\mathrm{P}} \infty$ provided

\begin{equation*}\sqrt{n}\bigg(\sum_{k=1}^{d^*}(\tau_k^*-\tau^*_{k-1})^p|\beta_k|^p\bigg)^{1/p}\to \infty\qquad\mbox{as $n\to\infty$.}\end{equation*}

Calculations of p-variation of piecewise functions are available in the R environment package under the name pvar developed by Butkus and Norvaiša [Reference Butkus and Norvaiša4]. Simulation analysis of the statistic $T_n$ is outside the scope of this paper and will appear elsewhere.