1 Introduction and Announcement of Results

Let \(\{Y_j\}_{j\in {\mathbb {Z}}}\) be a sequence of independent and identically distributed random variables. By a linear process built on innovations \(\{Y_j\}\), we mean a stochastic process

$$\begin{aligned} X_i=\sum _{j \in {\mathbb {Z}}}c_j Y_{i-j}, \quad i \in {\mathbb {Z}}, \end{aligned}$$
(1)

where the constants \(\{c_j\}_{j\in {\mathbb {Z}}}\) are such that the above series is \({\mathbb {P}}\)-a.s. convergent. Clearly, in non-trivial cases, such a process is dependent and stationary, and due to the simple linear structure, many of its distributional characteristics can be easily computed (provided they exist). This refers not only to the expectation or the covariances, but also to more involved quantities, like constants for regularly varying tails (see e.g., [21] for discussion) or mixing coefficients (see e.g., [10] for discussion).

There exists a huge literature devoted to applications of linear processes in statistical analysis and modeling of time series. We refer to the popular textbook [6] as an excellent introduction to the topic.

Here, we would like to stress only two particular features of linear processes.

First, linear processes provide a natural illustration for phenomena of local (or weak) dependence and long-range dependence. The most striking results go back to Davydov [9], who obtained a rescaled fractional Brownian motion as a functional weak limit for suitable normalized partial sums of \(\{X_i\}\)’s.

Another important property of linear processes is the propagation of big values. Suppose that some random variable \(Y_{j_0}\) takes a big value, then this big value is propagated along the sequence \(X_i\) (everywhere, where \(Y_{j_0}\) is taken with a big coefficient \(c_{i-j_0}\)). Thus, linear processes form the simplest model for phenomena of clustering of big values, what is important in models of insurance (see e.g., [21]).

In the present paper, we shall deal with heavy-tailed innovations. More precisely, we shall assume that the law of \(Y_i\) belongs to the domain of strict attraction of a non-degenerate strictly \(\alpha \)-stable law \(\mu _{\alpha }\), i.e.,

$$\begin{aligned} Z_n=\frac{1}{a_n}\sum _{i=1}^{n}Y_i \ \mathop {\longrightarrow }_{{\mathcal D}}\ Z, \end{aligned}$$
(2)

where \(Z \sim \mu _{\alpha }\).

Let us observe that by the Skorokhod theorem [25], we also have

$$\begin{aligned} Z_n(t)=\frac{1}{a_n}\sum _{i=1}^{[nt]}Y_i \ \mathop {\longrightarrow }_{{\mathcal D}}\ Z(t), \end{aligned}$$
(3)

where \(\{Z(t)\}\) is the stable Lévy process with \(Z(1) \sim \mu _{\alpha }\), and the convergence holds on the Skorokhod space \({\mathbb {D}}([0,1])\), equipped with the Skorokhod \(J_1\) topology.

Recall, that if the variance of \(Z\) is infinite, then (2) implies the existence of \(\alpha \in (0,2)\) such that

$$\begin{aligned} {\mathbb {P}}(|Y_j| > x) = x^{-\alpha } h(x), \quad x >0, \end{aligned}$$
(4)

where \(h\) is a function that varies slowly at \(x=+\infty \), and also

$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{{\mathbb {P}}(Y_j>x)}{{\mathbb {P}}(|Y_j|>x)}=p \quad \text{ and } \quad \lim _{x \rightarrow \infty }\frac{{\mathbb {P}}(Y_j<-x)}{{\mathbb {P}}(|Y_j|>x)}=q, \quad p+q = 1. \end{aligned}$$
(5)

The norming constants \(a_n\) in (3) must satisfy

$$\begin{aligned} n {\mathbb {P}}(|Y_j| > a_n) = \frac{ n h(a_n)}{a_n^{\alpha }} \rightarrow C >0, \end{aligned}$$
(6)

hence are necessarily of the form \(a_n = n^{1/\alpha } g(n^{1/\alpha })\), where the slowly varying function \(g(x)\) is the de Bruijn conjugate of \(\left( C/h(x)\right) ^{1/\alpha }\) (see [5]). Moreover, if \(\alpha > 1\), then \({\mathbb {E}}Y_j = 0\), and if \(\alpha = 1\), then \(p = q\) in (5).

Conversely, conditions (4), (5) and

$$\begin{aligned} {\mathbb {E}}[Y_j] = 0,&\text { if }&\alpha > 1,\end{aligned}$$
(7)
$$\begin{aligned} \{Y_j\} \text { are symmetric,}&\text { if }&\alpha = 1, \end{aligned}$$
(8)

imply (3).

If \(a_n\) is chosen to satisfy (6) with \(C=1\), then \(\mu _{\alpha }\) is given by the characteristic function

$$\begin{aligned} \hat{\mu }(\theta ) = {\left\{ \begin{array}{ll} \exp \left( \int _{{\mathbb {R}}^1} (e^{i\theta x} - 1) f_{\alpha , p, q} (x) \,\mathrm{d}x\right) &{} \text {if}\;0<\alpha <1,\\ \exp \left( \int _{{\mathbb {R}}^1} (e^{i\theta x} - 1) f_{1, 1/2, 1/2} (x) \,\mathrm{d}x\right) &{} \text {if}\;\alpha =1,\\ \exp \left( \int _{{\mathbb {R}}^1} (e^{i\theta x} - 1 - i\theta x) f_{\alpha , p, q} (x) \,\mathrm{d}x\right) &{}\text {if}\; 1<\alpha <2, \end{array}\right. } \end{aligned}$$
(9)

where

$$\begin{aligned} f_{\alpha , p, q}(x) = \left( p\, {\mathbb {I}}(x>0) + q\, {\mathbb {I}}(x<0)\right) \alpha |x|^{- (1+\alpha )}. \end{aligned}$$

We refer to [12] or any of contemporary monographs on limit theorems for the above basic information.

Suppose that the tails of \(|Y_j|\) are regularly varying, i.e., (4) holds for some \(\alpha \in (0,2)\), and the (usual) regularity conditions (7) and (8) are satisfied. It is an observation due to Astrauskas [1] (in fact: a direct consequence of the Kolmogorov Three Series Theorem—see Proposition 5.4 below) that the series (1) defining the linear process \(X_i\) is \({\mathbb {P}}\)-a.s. convergent if, and only if,

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}} |c_j|^{\alpha } h(|c_j|^{-1}) < +\infty . \end{aligned}$$
(10)

Given the above series is convergent we can define

$$\begin{aligned} S_n(t)=\frac{1}{b_n} \sum _{i=1}^{[nt]}X_i,\quad t\ge 0, \end{aligned}$$
(11)

and it is natural to ask for convergence of \(S_n\)’s, when \(b_n\) is suitably chosen. Astrauskas [1] and Kasahara & Maejima [16] showed that fractional stable Lévy Motions can appear in the limit of \(S_n(t)\)’s, and that some of the limiting processes can have regular or even continuous trajectories, while trajectories of other can be unbounded on every interval.

In the present paper, we consider the important case of summable coefficients:

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}} |c_j| < +\infty . \end{aligned}$$
(12)

In Sect. 2, we give necessary and sufficient conditions for the finite dimensional convergence

$$\begin{aligned} S_n(t)=\frac{1}{a_n} \sum _{i=1}^{[nt]}X_i \ \mathop {\longrightarrow }_{f.d.d.}\ A\cdot Z(t), \end{aligned}$$
(13)

where the constants \(a_n\) are the same as in (2), \(A = \sum _{j\in {\mathbb {Z}}} c_j\) and \(\{Z(t)\}\) is an \(\alpha \)-stable Lévy Motion such that \(Z(1) \sim Z\). The obtained conditions lead to tractable sufficient conditions, which in case \(\alpha < 1\) are new and essentially weaker than condition

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}} |c_j|^{\beta } < +\infty , \quad \text {for some}\;0 < \beta < \alpha , \end{aligned}$$

considered in [1, 8] and [16]. See Sect. 4 for details. Notice that in the case \(A = 0\), another normalization \(b_n\) is possible with a non-degenerate limit. We refer to [22] for comprehensive analysis of dependence structure of infinite variance processes.

Section 3 contains strengthening of (13) to a functional convergence in some suitable topology on the Skorokhod space \({\mathbb {D}}([0,1])\). Since the paper [2], it is known that in non-trivial cases (when at least two coefficients are nonzero) the convergence in the Skorokhod \(J_1\) topology cannot hold. In fact, none of Skorokhod’s \(J_1, J_2, M_1\) and \(M_2\) topologies are applicable. This can be seen by analysis of the following simple example ([2], p. 488). Set \(c_0 = 1, c_1 = -1\) and \(c_i = 0\) if \(j\ne 0,1\). Then \(X_i = Y_i - Y_{i-1}\) and (13) holds with \(A = \sum _j c_j = 0\), i.e.,

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0, \quad t\ge 0. \end{aligned}$$

But we see that

$$\begin{aligned} \sup _{t\in [0,1]} S_n(t) = \max _{k\le n} \left( Y_k - Y_0\right) /a_n \end{aligned}$$

converges in law to a Fréchet distribution. This means that supremum is not a continuous (or almost surely continuous) functional, what excludes convergence in Skorokhod’s topologies in the general case.

For linear processes with nonnegative coefficients \(c_i\), partial results were obtained by Avram and Taqqu [2], where convergence in the \(M_1\) topology was considered. Recently, these results have been improved and developed in various directions in [20] and [3]. We use the linear structure of processes and the established convergence in the \(M_1\) topology to show that in the general case, the finite dimensional convergence (13) can be strengthen to convergence in the so-called \(S\) topology, introduced in [13]. This is a sequential and non-metric, but fully operational topology, for which addition is sequentially continuous.

Section 5 is devoted to some consequences of results obtained in previous sections. We provide examples of functionals continuous in the \(S\) topology. In particular, we show that for every \(\gamma >0\)

$$\begin{aligned} \frac{1}{n a_n^{\gamma }} \sum _{k=1}^n \left( \sum _{i=1}^k \left( \sum _j c_{i-j} Y_j\right) - A Y_i\right) ^{\gamma } \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0. \end{aligned}$$

We also discuss possible extensions of the theory to linear sequences built on dependent summands.

The “Appendix” contains technical results of independent interest.

Conventions and notations. Throughout the paper, in order to avoid permanent repetition of standard assumptions and conditions, we adopt the following conventions. We will say that \(\{Y_j\}\)’s satisfy the usual conditions if they are independent identically distributed and (4), (5), (7) and (8) hold. When we write \(X_i\), it is always the linear process given by (1) and is well-defined, i.e., satisfies (10). Similarly, the norming constants \(\{a_n\}\) are defined by (6) and the normalized partial sums \(S_n(t)\) and \(Z_n(t)\) are given by (11) with \(b_n = a_n\) and (3), respectively, where \(Z\) is the limit in (2) and \(Z(t)\) is the stable Lévy Motion such that \(Z(1) \sim Z\).

2 Convergence of Finite Dimensional Distributions for Summable Coefficients

We begin with stating the main result of this section followed by its important consequence.

Theorem 2.1

Let \(\{Y_j\}\) be an i.i.d. sequence satisfying the usual conditions. Suppose that

$$\begin{aligned} \sum _j |c_j| < +\infty . \end{aligned}$$

Then

$$\begin{aligned} S_n(t)=\frac{1}{a_n} \sum _{i=1}^{[nt]}X_i \ \mathop {\longrightarrow }_{f.d.d.}\ A\cdot Z(t),\quad \text {where}\; A = \sum _j c_j, \end{aligned}$$

if, and only if,

$$\begin{aligned} \begin{aligned} \sum _{j=-\infty }^0 \frac{\left| d_{n,j}\right| ^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\left| d_{n,j}\right| }\right)&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty ,\\ \sum _{j=n+1}^{\infty }\frac{\left| d_{n,j}\right| ^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\left| d_{n,j}\right| }\right)&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty , \end{aligned} \end{aligned}$$
(14)

where

$$\begin{aligned} d_{n,j} = \sum _{k=1-j}^{n-j}c_k, \quad n\in {\mathbb {N}}, j\in {\mathbb {Z}}. \end{aligned}$$

Corollary 2.2

Under the assumptions of Theorem 2.1, define

$$\begin{aligned} U_i = \sum _j |c_{i-j}| Y_j,\quad X_i^+ = \sum _j c_{i-j}^+ Y_j,\quad X_i^- = \sum _j c_{i-j}^- Y_j, \end{aligned}$$
(15)

where \(c^+= c\vee 0,\ c^- = (-c) \vee 0,\ c\in {\mathbb {R}}^1\), and set

$$\begin{aligned} T_n(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} U_i, \quad T_n^+(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i^+,\quad T_n^-(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i^-. \end{aligned}$$
(16)

Then

$$\begin{aligned} T_n(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A_{|\cdot |}\cdot Z(t),\quad \text {where}\;A_{|\cdot |} = \sum _j |c_j|, \end{aligned}$$

implies

$$\begin{aligned} T_n^+(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A_{+}\cdot Z(t),&\quad \text { where}\;A_{+} = \sum _j c_j^+,\\ T_n^-(t)\ \mathop {\longrightarrow }_{f.d.d.}\ A_{-}\cdot Z(t),&\quad \text {where}\;A_{-} = \sum _j c_j^-,\\ S_n(t) = T_n^+(t) - T_n^-(t)\ \mathop {\longrightarrow }_{f.d.d.}\ A \cdot Z(t),&\quad \text {where}\;A = \sum _j c_j. \end{aligned}$$

Proof of Corollary 2.2

In view of Theorem 2.1, it is enough to notice that

$$\begin{aligned} \frac{\left| d_{n,j}\right| ^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\left| d_{n,j}\right| }\right) = {\mathbb {P}}\left( \left| \sum _{k=1-j}^{n-j}c_k\right| \cdot |Y_j | > a_n\!\right) \le {\mathbb {P}}\left( \!\left( \sum _{k=1-j}^{n-j}|c_k|\right) \cdot |Y_j|> a_n\!\right) . \end{aligned}$$

Proof of Theorem 2.1

Using Fubini’s theorem, we obtain that

$$\begin{aligned} S_n(t)=\frac{1}{a_n}\sum _{i=1}^{[nt]}\sum _{j \in {\mathbb {Z}}}c_{i-j}Y_j =\sum _{j\in {\mathbb {Z}}} \frac{1}{a_n}\left( \sum _{k=1-j}^{[nt]-j}c_k\right) Y_j =\sum _{j \in {\mathbb {Z}}} \frac{1}{a_n} d_{[nt],j} Y_j. \end{aligned}$$
(17)

Further, we may decompose

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}} \frac{1}{a_n}d_{[nt],j} Y_j&= \sum _{j=-\infty }^0 \frac{1}{a_n}d_{[nt],j} Y_j \nonumber \\&+\sum _{j=1}^{[nt]} \frac{1}{a_n}d_{[nt],j} Y_j\nonumber \\&+ \sum _{j=[nt]+ 1}^{\infty } \frac{1}{a_n}d_{[nt],j} Y_j \nonumber \\&= S_n^-(t) + S_n^0(t) + S_n^+(t). \end{aligned}$$
(18)

Let us consider the partial sum process:

$$\begin{aligned} Z_n(t)=\frac{1}{a_n}\sum _{i=1}^{[nt]}Y_i,\quad t \ge 0. \end{aligned}$$

First we will show

Lemma 2.3

Under the assumptions of Theorem 2.1 we have for each \(t > 0\)

$$\begin{aligned} S_n^0(t) - A \cdot Z_n(t) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0. \end{aligned}$$
(19)

In particular,

$$\begin{aligned} S_n^0(t) \ \mathop {\longrightarrow }_{{\mathcal D}}\ A \cdot Z(t). \end{aligned}$$
(20)

Proof of Lemma 2.3

Define

$$\begin{aligned} V_n^0 = \sum _{j=1}^{[nt]} \frac{\left( A - d_{[nt],j}\right) }{a_n} Y_j = A \cdot Z_n(t) - S_n^0(t). \end{aligned}$$
(21)

To prove that \(V^0_n \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0\), we apply Proposition 5.5. We have to show that

$$\begin{aligned} \begin{aligned}&\sum _{j=1}^{[nt]} \frac{\left| A - d_{[nt],j}\right| ^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\left| A -d_{[nt],j}\right| }\right) \\&\quad = \sum _{j=1}^{[nt]} {\mathbb {P}}\left( \left| A - d_{[nt],j} \right| \cdot |Y_j| > a_n\right) \rightarrow 0,\quad \text {as}\; n\rightarrow \infty . \end{aligned} \end{aligned}$$
(22)

Since \(a_n \rightarrow \infty \) and \(\big | A - d_{[nt],j}\big | \le \sum _{k\in {\mathbb {Z}}} |c_k|\), we have

$$\begin{aligned} \max _{1\le j \le [nt]} {\mathbb {P}}\left( \big | A - d_{[nt],j}\big |\cdot |Y_j| > a_n\right) \rightarrow 0. \end{aligned}$$
(23)

We need a simple lemma.

Lemma 2.4

Let \(\{a_{n,j}\,;\, 1 \le j \le n, \ n\in {\mathbb {N}}\}\) be an array of numbers such that

$$\begin{aligned} \max _{1\le j\le n} |a_{n,j}| \rightarrow 0, \quad \text {as}\;n\rightarrow \infty . \end{aligned}$$

Then there exists a sequence \(j_n\rightarrow \infty \), \(j_n = o(n)\), such that

$$\begin{aligned} \sum _{j=1}^{j_n} |a_{n,j}| \rightarrow 0. \end{aligned}$$

Proof of Lemma 2.4

For each \(m\in {\mathbb {N}}\) there exists \(N_m > \max \{N_{m-1}, m^2\}\) such that for \(n \ge N_m\)

$$\begin{aligned} \sum _{j=1}^{m} |a_{n,j}| < \frac{1}{m}. \end{aligned}$$

Set \(j_n = m\), if \(N_m \le n < N_{m+1}\). By the very definition, if \(N_m \le n < N_{m+1}\) then

$$\begin{aligned} \sum _{j=1}^{j_n} |a_{n,j}| < \frac{1}{m}\qquad \text { and }\qquad \frac{j_n}{n} \le \frac{j_n}{N_m} = \frac{m}{N_m} \le \frac{m}{m^2} = \frac{1}{m}. \end{aligned}$$

By the above lemma and (23), we can find a sequence \(j_n \rightarrow \infty \), \(j_n = o(n)\), increasing so slowly that still

$$\begin{aligned} \sum _{j=1}^{j_n} {\mathbb {P}}\left( \left| A - d_{[nt],j}\right| \cdot |Y_j| > a_n\right) + \sum _{j=[nt]-j_n + 1}^{[nt]} {\mathbb {P}}\left( \left| A - d_{[nt],j}\right| \cdot |Y_j| > a_n\right) \rightarrow 0. \end{aligned}$$

For the remaining part we have

$$\begin{aligned} \max _{j_n < j \le [nt]- j_n} \left| A - d_{[nt],j}\right| = \max _{j_n < j \le [nt]- j_n} \left| A - \sum _{k=1-j}^{[nt]-j}c_k\right| = \delta _n \rightarrow 0, \end{aligned}$$

hence for \(\delta \ge \delta _n\)

$$\begin{aligned} \sum _{j=j_n+1}^{[nt] - j_n} {\mathbb {P}}\left( \left| A - d_{[nt],j}\right| \cdot |Y_j| > a_n\right)&\le \sum _{j=j_n+1}^{[nt] - j_n} {\mathbb {P}}\left( |\delta _n| |Y_j| > a_n\right) \\&\le \sum _{j=1}^{[nt]}{\mathbb {P}}\left( |\delta _n| |Y_j| > a_n\right) \\&\le [nt] \frac{\delta ^{\alpha }}{a_n^{\alpha }} h(a_n/\delta ) \\&= [nt] \delta ^{\alpha }\frac{h(a_n)}{a_n^{\alpha }} \frac{h(a_n/\delta )}{h(a_n)}. \end{aligned}$$

Since \( n a_n^{-\alpha } h(a_n) = n {\mathbb {P}}(|Y| > a_n) \rightarrow 1\) and \(h\) varies slowly we have

$$\begin{aligned}{}[nt] \delta ^{\alpha }\frac{h(a_n)}{a_n^{\alpha }} \frac{h(a_n/\delta )}{h(a_n)} \sim [nt] \delta ^{\alpha } \frac{1}{n} \rightarrow t \delta ^{\alpha }, \text { as}\; n\rightarrow \infty . \end{aligned}$$

But \(\delta > 0\) is arbitrary, hence we have proved (22) and

$$\begin{aligned} V_n^0 = A\cdot Z_n(t) - S_n^0(t) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0. \end{aligned}$$

Since

$$\begin{aligned} A \cdot Z_n(t) \ \mathop {\longrightarrow }_{{\mathcal D}}\ A \cdot Z(t), \end{aligned}$$

Lemma 2.3 follows.

In the next step, we shall prove

Lemma 2.5

Under the assumptions of Theorem 2.1, the following items (i)–(iii) are equivalent.

(i):
$$\begin{aligned} S_n(1) \ \mathop {\longrightarrow }_{{\mathcal D}}\ A\cdot Z(1), \end{aligned}$$
(24)
(ii):
$$\begin{aligned} S_n^{-}(1) + S_n^{+}(1) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0. \end{aligned}$$
(25)
(iii):

For every \(t\in [0,1]\)

$$\begin{aligned} S_n(t) - A\cdot Z_n(t) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0. \end{aligned}$$
(26)

Proof of Lemma 2.5

By Lemma 2.3 we know that \(S_n^0(1) - A\cdot Z_n(1) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0\) and \(S_n^0(1) \ \mathop {\longrightarrow }_{{\mathcal D}}\ A\cdot Z(1)\). Since \(S_n(1) = S_n^-(1) + S_n^0(1) + S_n^+(1)\), (26) implies (25) and the latter implies (24).

So let us assume (24). By regular variation of \(a_n\), we have for each \(t\in (0,1]\)

$$\begin{aligned} S_n(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i = \frac{a_{[nt]}}{a_n} \frac{1}{a_{[nt]}} \sum _{i=1}^{[nt]} X_i \ \mathop {\longrightarrow }_{{\mathcal D}}\ t^{1/\alpha }A\cdot Z(1) \sim A\cdot Z(t). \end{aligned}$$

It follows that

$$\begin{aligned} {\mathbb {E}}\big [e^{i\theta S_n(t)}\big ] = {\mathbb {E}}\left[ e^{i\theta S_n^0(t)}\right] {\mathbb {E}}\left[ e^{i\theta \left( S_n^-(t) + S_n^+(t)\right) }\right] \rightarrow {\mathbb {E}}\left[ e^{i\theta A\cdot Z(t)}\right] , \quad \theta \in {\mathbb {R}}^1. \end{aligned}$$

Since also

$$\begin{aligned} {\mathbb {E}}\left[ e^{i\theta S_n^0(t)}\right] \rightarrow {\mathbb {E}}\left[ e^{i\theta A\cdot Z(t)}\right] , \quad \theta \in {\mathbb {R}}^1, \end{aligned}$$

and \({\mathbb {E}}\left[ e^{i\theta A\cdot Z(t)}\right] \ne 0, \ \theta \in {\mathbb {R}}^1\) (for \(Z(t)\) has infinitely divisible law), we conclude that

$$\begin{aligned} {\mathbb {E}}\left[ e^{i\theta \left( S_n^-(t) + S_n^+(t)\right) }\right] \rightarrow 1, \quad \theta \in {\mathbb {R}}^1. \end{aligned}$$

Thus \(S_n^-(t) + S_n^+(t) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0\) and by Lemma 2.3 also \(S_n^0(t) - A\cdot Z(t) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0\). Hence (26) follows.

Let us observe that by Proposition 5.5 (25) holds if, and only if,

$$\begin{aligned} \sum _{j=-\infty }^0 \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) +\ \sum _{j=n+ 1}^{\infty } \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \rightarrow 0,\quad \text {as}\;n\rightarrow \infty , \end{aligned}$$
(27)

i.e., relation (14) holds. Therefore the Proof of Theorem 2.1 will be complete, if we can show that convergence of one-dimensional distributions implies the finite dimensional convergence. But this is obvious in view of (26):

$$\begin{aligned} \left( S_n(t_1), S_n(t_2), \ldots , S_n(t_m)\right) - A\cdot \left( Z_n(t_1), Z_n(t_2), \ldots , Z_n(t_m)\right) \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0, \end{aligned}$$

and the finite dimensional distributions of stochastic processes \(A\cdot Z_n(t)\) are convergent to those of \(A\cdot Z(t)\).

Remark 2.6

Observe that for one-sided moving averages, the two conditions in (14) reduce to one (the expression in the other equals \(0\)). This is the reason we use in Theorem 2.1 two conditions replacing the single statement (27).

Remark 2.7

In the Proof of Proposition 5.5, we used the Three Series Theorem with the level of truncation \(1\). It is well-known that any \(r \in (0,+\infty )\) can be chosen as the truncation level. Hence, conditions (14) admit an equivalent reformulation in the “\(r\)-form”

$$\begin{aligned} \begin{aligned} \sum _{j=-\infty }^0 \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{r \cdot a_n}{\big |d_{n,j}\big |}\right)&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty .\\ \sum _{j=n+ 1}^{\infty } \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{r \cdot a_n}{\big |d_{n,j}\big |}\right)&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty . \end{aligned} \end{aligned}$$

3 Functional Convergence

3.1 Convergence in the \(M_1\) Topology

As outlined in Introduction (see also Sect. 5.2 below), the convergence of finite dimensional distributions of linear processes built on heavy-tailed innovations cannot be, in general, strengthened to functional convergence in any of Skorokhod’s topologies \(J_1, J_2, M_1, M_2\).

The general linear process \(\{X_i\}\) can be, however, represented as a difference of linear processes with nonnegative coefficients. Let us recall the notation introduced in Corollary 2.2:

$$\begin{aligned} X_i^+ =&\sum _j c_{i-j}^+ Y_j,\qquad T_n^+(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i^+,\\ \quad X_i^- =&\sum _j c_{i-j}^- Y_j,\qquad T_n^-(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i^-. \end{aligned}$$

Notice, that in general \(X_i^{\pm }(\omega )\) is not equal to \(\left( X_i(\omega )\right) ^{\pm }\) and that we have

$$\begin{aligned} S_n(t) = T_n^+(t) - T_n^-(t). \end{aligned}$$
(28)

The point is that both \(T_n^+(t)\) and \(T_n^-(t)\) are partial sums of associated sequences in the sense of [11] (see e.g., [7] for the contemporary theory) and thus exhibit much more regularity.

Theorem 1 of Louhichi and Rio [20] can be specified to the case of linear processes considered in our paper in the following way.

Proposition 3.1

Let the innovation sequence \(\{Y_j\}\) satisfies the usual conditions. Let

$$\begin{aligned} c_j \ge 0, j\in {\mathbb {Z}}, \text { and } \sum _j c_j < +\infty . \end{aligned}$$
(29)

If the linear process \(\{X_i\}\) is well-defined and

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A\cdot Z(t), \end{aligned}$$

then also functionally

$$\begin{aligned} S_n \ \mathop {\longrightarrow }_{{\mathcal D}}\ A\cdot Z \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the \(M_1\) topology.

Remark 3.2

The first result of this type was obtained by Avram and Taqqu [2]. They required however more regularity on coefficients (e.g., monotonicity of \(\{c_j\}_{j\ge 1}\) and \(\{c_{-j}\}_{j\ge 1}\)).

3.2 \(M_1\)-Convergence Implies \(S\)-Convergence

Let us turn to linear processes with coefficients of arbitrary sign. Given decomposition (28) and Proposition 3.1, the strategy is now clear: Choose any linear topology \(\tau \) on \({\mathbb {D}}([0,1])\) which is coarser than \(M_1\), then

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A\cdot Z(t), \end{aligned}$$

should imply

$$\begin{aligned} S_n \ \mathop {\longrightarrow }_{{\mathcal D}}\ A\cdot Z \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the topology \(\tau \). Since convergence of càdlàg functions in the \(M_1\) topology is bounded and implies pointwise convergence outside of a countable set, there are plenty of such topologies. For instance, any space of the form \(L^{p}\left( [0,1],\mu \right) \), where \(p\in [0,\infty )\) and \(\mu \) is an atomless finite measure on \([0,1]\), is suitable. The point is to choose the finest among linear topologies with required properties, for we want to have the maximal family of continuous functionals on \({\mathbb {D}}([0,1])\).

Although we are not able to identify such an “ideal” topology, we believe that this distinguished position belongs to the \(S\) topology, introduced in [13]. This is a non-metric sequential topology, with sequentially continuous addition, which is stronger than any of mentioned above \(L^p(\mu )\) spaces and is functional in the sense it has the following classic property (see Theorem 3.5 of [13]).

Proposition 3.3

Let \({\mathbb {Q}}\subset [0,1]\) be dense, \(1 \in {\mathbb {Q}}\). Suppose that for each finite subset \({\mathbb {Q}}_0 = \{ q_1 < q_2 < \cdots < q_m \} \subset {\mathbb {Q}}\) we have as \(n\rightarrow \infty \)

$$\begin{aligned} (X_n(q_1), X_n(q_2), \ldots , X_n(q_m)) \ \mathop {\longrightarrow }_{{\mathcal D}}\ (X_0(q_1), X_0(q_2), \ldots , X_0(q_m)), \end{aligned}$$

where \(X_0\) is a stochastic process with trajectories in \({\mathbb {D}}[0,1])\). If \(\{X_n\}\) is uniformly \(S\)-tight, then

$$\begin{aligned} X_n \ \mathop {\longrightarrow }_{{\mathcal D}}\ X_0, \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the \(S\) topology.

For readers familiar with the limit theory for stochastic processes, the above property may seem obvious. But it is trivial only for processes with continuous trajectories. It is not trivial even in the case of the Skorokhod \(J_1\) topology, since the point evaluations

$$\begin{aligned} \pi _t : {\mathbb {D}}([0,1]) \rightarrow {\mathbb {R}}^1,\quad \pi _t(x) = x(t), \end{aligned}$$

can be \(J_1\)-discontinuous at some \(x\in {\mathbb {D}}([0,1])\) (see [26] for the result corresponding to Proposition 3.3). In the \(S\) topology, the point evaluations are nowhere continuous (see [13], p. 11). Nevertheless, Proposition 3.3 holds for the \(S\) topology, while it does not hold for the linear metric spaces \(L^p(\mu )\) considered above. It follows that the \(S\) topology is suitable for the needs of limit theory for stochastic processes. It admits even such efficient tools like the a.s Skorokhod representation for subsequences [14]. On the other hand, since \({\mathbb {D}}([0,1])\) equipped with \(S\) is non-metric and sequential, many of apparently standard reasonings require special tools and careful analysis. This will be seen below.

Before we define the \(S\) topology, we need some notation. Let \({\mathbb {V}}([0,1]) \subset {\mathbb {D}}([0,1])\) be the space of (regularized) functions of finite variation on \([0,1]\), equipped with the norm of total variation \(\Vert v\Vert = \Vert v\Vert (1)\), where

$$\begin{aligned} \Vert v\Vert (t) = \sup \left\{ |v(0)| + \sum _{i=1}^{m} |v(t_i) - v(t_{i-1})|\right\} , \end{aligned}$$

and the supremum is taken over all finite partitions \(0=t_0 < t_1 < \cdots < t_m = t\). Since \({\mathbb {V}}([0,1])\) can be identified with a dual of \(({\mathbb {C}}([0,1]),\Vert \cdot \Vert _{\infty })\), we have on it the weak-\(*\) topology. We shall write \(v_n \Rightarrow v_0\) if for every \(f \in {\mathbb {C}}([0,1])\)

$$\begin{aligned} \int _{[0,1]} f(t) \mathrm{d}v_n(t) \rightarrow \int _{[0,1]} f(t) \mathrm{d}v_0(t). \end{aligned}$$

Definition 3.4

(\(S\)-convergence and the \(S\) topology) We shall say that \(x_n\) \(S\)-converges to \(x_0\) (in short \(x_n \rightarrow _S x_0\)) if for every \(\varepsilon > 0\) one can find elements \(v_{n,\varepsilon }\in {\mathbb {V}}([0,1])\), \(n=0,1,2,\ldots \) which are \(\varepsilon \)-uniformly close to \(x_n\)’s and weakly-\(*\) convergent:

$$\begin{aligned} \Vert x_n - v_{n,\varepsilon }\Vert _{\infty } \le \varepsilon ,&\ n = 0, 1, 2, \ldots ,\end{aligned}$$
(30)
$$\begin{aligned} v_{n,\varepsilon } \Rightarrow v_{0,\varepsilon },&\text {as}\; n\rightarrow \infty . \end{aligned}$$
(31)

The \(S\) topology is the sequential topology determined by the \(S\)-convergence.

Remark 3.5

This definition was given in [13], and we refer to this paper for detailed derivation of basic properties of \(S\)-convergence and construction of the \(S\) topology, as well as for instruction how to effectively operate with \(S\). Here, we shall stress only that the \(S\) topology emerges naturally in the context of the following criteria of compactness, which will be used in the sequel.

Proposition 3.6

(2.7 in [13]) For \(\eta > 0\), let \(N_{\eta }(x)\) be the number of \(\eta \) -oscillations of the function \(x \in {\mathbb {D}}([0,1])\), i.e., the largest integer \(N \ge 1\), for which there exist some points

$$\begin{aligned} 0 \le t_1<t_2 \le t_3<t_4 \le \cdots \le t_{2N-1}<t_{2N} \le 1, \end{aligned}$$

such that

$$\begin{aligned} |x(t_{2k})-x(t_{2k-1})|>\eta \quad \text{ for } \text{ all } \ k=1,\ldots ,N. \end{aligned}$$

Let \({\mathcal K}\subset {\mathbb {D}}\). Assume that

$$\begin{aligned} \sup _{x\in {\mathcal K}} \Vert x\Vert _{\infty }&< +\infty ,\end{aligned}$$
(32)
$$\begin{aligned} \sup _{x\in {\mathcal K}} N_{\eta }(x)&< +\infty , \text { for each}\;\eta > 0. \end{aligned}$$
(33)

Then from any sequence \(\{x_n\} \subset {\mathcal K}\) one can extract a subsequence \(\{x_{n_k}\}\) and find \(x_0 \in {\mathbb {D}}([0,1])\) such that \(x_{n_k} \ \mathop {\longrightarrow }_{S}\ x_0\).

Conversely, if \({\mathcal K}\subset {\mathbb {D}}([0,1])\) is relatively compact with respect to \(\mathop {\longrightarrow }_{S}\), then it satisfies both (32) and (33).

Corollary 3.7

(2.14 in [13]) Let \({\mathbb {Q}}\subset [0,1]\), \(1\in {\mathbb {Q}}\), be dense. Suppose that \(\{x_n\}\subset {\mathbb {D}}([0,1])\) is relatively \(S\)-compact and as \(n\rightarrow \infty \)

$$\begin{aligned} x_n(q) \rightarrow x_0(q),\quad q \in {\mathbb {Q}}. \end{aligned}$$

Then \(x_n \rightarrow x_0\) in \(S\).

Remark 3.8

The \(S\) topology is sequential, i.e., it is generated by the convergence \(\mathop {\longrightarrow }_{S}\). By the Kantorovich–Kisyński recipe [17] \(x_n \rightarrow x_0\) in \(S\) topology if, and only if, in each subsequence \(\{x_{n_k}\}\) one can find a further subsequence \(x_{n_{k_l}} \ \mathop {\longrightarrow }_{S}\ x_0\). This is the same story as with a.s. convergence and convergence in probability of random variables.

According to our strategy, we are going to prove that Skorokhod’s \(M_1\)-topology is stronger than the \(S\) topology or, equivalently, that \(x_n \ \mathop {\longrightarrow }_{M_1}\ x_0\) implies \(x_n \ \mathop {\longrightarrow }_{S}\ x_0\). We refer the reader to the original Skorohod’s article [24] for the definition of the \(M_1\) topology, as well as to Chapter 12 of [28] for a comprehensive account of properties of this topology.

The \(M_1\) convergence can be described using a suitable modulus of continuity. We define for \(x\in {\mathbb {D}}([0,1])\) and \(\delta > 0\)

$$\begin{aligned} w^{M_1}(x,\delta ):=\sup _{0 \vee (t_2-\delta ) \le t_1<t_2<t_3 \le 1 \wedge (t_2+\delta )}H\left( x(t_1),x(t_2),x(t_3)\right) , \end{aligned}$$
(34)

where \(H(a,b,c)\) is the distance between \(b\) and the interval with endpoints \(a\) and \(c\):

$$\begin{aligned} H(a,b,c)=(a\wedge c-a \wedge c \wedge b) \vee (a \vee c \vee b-a \vee c). \end{aligned}$$

Proposition 3.9

(2.4.1 of [24]) Let \((x_n)_{n \ge 1}\) and \(x_0\) be arbitrary elements in \({\mathbb {D}}([0,1])\). Then

$$\begin{aligned} x_n \ \mathop {\longrightarrow }_{M_1}\ x_0 \end{aligned}$$

if, and only if, for some dense subset \({\mathbb {Q}}\subset [0,1]\) containing \(0\) and \(1\),

$$\begin{aligned} x_n(t) \rightarrow x(t), \quad t\in {\mathbb {Q}}, \end{aligned}$$
(35)

and

$$\begin{aligned} \lim _{\delta \rightarrow 0}\limsup _{n \rightarrow \infty }w^{M_1}(x_n,\delta )=0. \end{aligned}$$
(36)

In particular, if \(x_n \ \mathop {\longrightarrow }_{M_1}\ x_0\), then

$$\begin{aligned} x_n(t) \rightarrow x_0(t) \end{aligned}$$

for \(t=1\) and at every point of continuity of \(x_0\).

Lemma 3.10

For any \(a,b,c,d \in {\mathbb {R}}^1\)

$$\begin{aligned} |a - b| \le |c-d| + H(c,a,d) + H(c,b,d). \end{aligned}$$

Proof

If \(c \le a \le b \le d\), then \(b-a \le d - c = d-c + H(c,a,d) + H(c,b,d)\). If \(a \le c \le b \le d\) then \(b - a = b-c + c - a \le d- c + H(c,a,d) = d - c + H(c,a,d) + H(c,b,d)\). If \(a \le c \le d \le b\) then \(b - a = b - d + d - c + c -a = H(c,b,d) + d -c + H(c,a,b)\). If \( a \le b \le c \le d\), then \(b - a \le |b - c| + |c-a| = H(c,b,d) + H(c,a,d) \le H(c,b,d) + H(c,a,d) + d - c\). The other cases can be reduced to the considered above.

Corollary 3.11

Let \(x \in {\mathbb {D}}([0,1])\). For any \(0 \le s \le u <v \le t \le 1\),

$$\begin{aligned} |x(u)-x(v)| \le |x(s)-x(t)|+H(x(s),x(u),x(t))+H(x(s),x(v),x(t)). \end{aligned}$$

Lemma 3.12

Let \(x \in {\mathbb {D}}([0,1])\). For \(0 \le s<t \le 1\), define

$$\begin{aligned} \beta =\sup _{s \le u<v<w \le t}H(x(u),x(v),x(w)). \end{aligned}$$

If \(\eta >2\beta \) then

$$\begin{aligned} N_{\eta }(x;[s,t]) \le \frac{|x(t)-x(s)|+\beta }{\eta -\beta }, \end{aligned}$$

where \(N_{\eta }(x;[s,t])\) denotes the number of \(\eta \)-oscillations of \(x\) in the interval \([s,t]\).

Proof

Let \(s \le t_1<t_2 \le t_3<t_4 \le \cdots \le t_{2N-1}<t_{2N} \le t\) be such that

$$\begin{aligned} |x(t_{2k})-x(t_{2k-1})|>\eta \quad \text {for all } \ k=1,\ldots ,N. \end{aligned}$$

Assume first that \(x(t_2)-x(t_1)>\eta \). We claim that

$$\begin{aligned} x(t_3)\ge x(t_2)-\beta \quad \text{ and } \quad \quad x(t_4)-x(t_3)>\eta . \end{aligned}$$

To see this, suppose that \(x(t_3)<x(t_2)-\beta \). Then the distance between \(x(t_2)\) and the interval with endpoints \(x(t_1)\) and \(x(t_3)\) is greater than \(\beta \), which is a contradiction. Hence \(x(t_3)\ge x(t_2)-\beta \). On the other hand, if we assume that \(x(t_4)-x(t_3)<-\eta \), we obtain that

$$\begin{aligned} x(t_1)=x(t_1)-x(t_2)+x(t_2)-x(t_3)+x(t_3)<-\eta +\beta +x(t_3)< x(t_3)-\beta , \end{aligned}$$

which means that the distance between \(x(t_3)\) and the interval with endpoints \(x(t_1)\) and \(x(t_4)\) is greater than \(\beta \), again a contradiction.

Repeating this argument, we infer that:

$$\begin{aligned} x(t_{2k})-x(t_{2k-1})>\eta , \quad \text {for all } k=1, \ldots ,N \end{aligned}$$

and

$$\begin{aligned} x(t_{2k+1})-x(t_{2k})>-\beta \quad \text {for all } k=1, \ldots , N-1. \end{aligned}$$

Taking the sum of these inequalities, we conclude that:

$$\begin{aligned} x(t_{2N})-x(t_1)>N\eta -(N-1)\beta =N(\eta -\beta )+\beta . \end{aligned}$$
(37)

On the other hand, by Corollary 3.11, we have:

$$\begin{aligned} |x(t_{2N})-x(t_1)|\le |x(t)-x(s)|+2\beta . \end{aligned}$$
(38)

Combining (37) and (38), we obtain that

$$\begin{aligned} N \le \frac{|x(t)-x(s)|+\beta }{\eta -\beta }, \end{aligned}$$

which is the desired upper bound.

Assuming that \(x(t_2)-x(t_1) < -\eta \), we come in a similar way to the inequality

$$\begin{aligned} x(t_{2N})-x(t_1) < -N\eta +(N-1)\beta =- N(\eta -\beta )- \beta \end{aligned}$$

or

$$\begin{aligned} |x(t_{2N})-x(t_1)| \> N(\eta -\beta ) + \beta . \end{aligned}$$

This again allows us to use Corollary 3.11 and gives the desired bound for \(N\)

The following result was stated without proof in [13]. A short proof can be given using Skorohod’s criterion 2.2.11 (page 267 of [24]) for the \(M_1\)-convergence, expressed in terms of the number of upcrossings. This proof has a clear disadvantage: It refers to an equivalent definition of the \(M_1\)-convergence, but the equivalence of both definitions was not proved in Skorokhod’s paper. In the present article, we give a complete proof.

Theorem 3.13

The \(S\) topology is weaker than the \(M_1\) topology (and hence, weaker than the \(J_1\) topology). Consequently, a set \(A \subset {\mathbb {D}}([0,1])\) which is relatively \(M_1\)-compact is also relatively \(S\)-compact.

Proof

Let \(x_n \ \mathop {\longrightarrow }_{M_1}\ x_0\). By Proposition 3.9

$$\begin{aligned} x_n(t) \rightarrow x_0(t), \end{aligned}$$

on the dense set of points of continuity of \(x_0\) and for \(t=1\). Suppose, we know that \({\mathcal K}=\{x_n\}\) satisfies conditions (32) and (33). Then by Proposition 3.6 \(\{x_n\}\) is relatively \(S\)-compact and by Corollary 3.7 \(x_n \rightarrow x_0\) in \(S\). Thus, it remains to check conditions

$$\begin{aligned} K_\mathrm{sup} =\sup _n \Vert x_n\Vert _{\infty }&< +\infty ,\end{aligned}$$
(39)
$$\begin{aligned} K_{\eta } = \sup _n N_{\eta }(x_n)&< \infty , \quad \eta > 0. \end{aligned}$$
(40)

First suppose that \(x_0(1-) = x_0(1)\). Then, \({\mathbb {D}}([0,1]) \ni x \mapsto \Vert x\Vert _{\infty }\) is \(M_1\)-continuous at \(x_0\). Consequently, \(x_n \ \mathop {\longrightarrow }_{M_1}\ x_0\) implies \(\Vert x_n\Vert _{\infty } \rightarrow \Vert x_0\Vert _{\infty }\) and (39) follows.

If \(x_0(1-) \ne x_0(1)\), we have to proceed a bit more carefully. Consider (36) and take \(\delta > 0\) and \(n_0\) such that \(w(x_n,\delta ) \le 1, n \ge n_0\). Find \( t_0 \in (1-\delta , 1)\) which is a point of continuity of \(x_0\). Then,

$$\begin{aligned} \sup _{t\in [0,t_0]} |x_n(t)| \rightarrow \sup _{t\in [0,t_0]} |x_0(t)|, \end{aligned}$$

hence \(\sup _n \sup _{t\in [0,t_0]} |x_n(t)| < +\infty \). We also know that \(x_n(t_0) \rightarrow x_0(t_0)\) and \(x_n(1) \rightarrow x_0(1)\). Choose \(n\in {\mathbb {N}}\) and \(u\in (t_0, 1)\). By the very definition of the modulus \(H\)

$$\begin{aligned} \begin{aligned} |x_n(u)|&\le |x_n(t_0)| + |x_n(1)| + H\left( x_n(t_0), x_n(u), x_n(1)\right) \\&\le \sup _n |x_n(t_0)| + \sup _n|x_n(1)| + 1, \quad n \ge n_0. \end{aligned} \end{aligned}$$

It follows that also

$$\begin{aligned} \sup _n \sup _{t\in (t_0,1]} |x_n(t)| < +\infty , \end{aligned}$$

and so (39) holds.

In order to prove (40), choose \(\eta >0\) and \(0<{\varepsilon }< \eta /2\). By Proposition 3.9, there exist some \(\delta > 0\) and an integer \(n_0 \ge 1\) such that \(w^{M_1}(x_n,\delta ) < {\varepsilon }, \ n \ge n_0\). Next, we find a partition \(0 = t_0 < t_1 < \cdots < t_{M} = 1\) consisting of points of continuity of \(x_0\) and such that

$$\begin{aligned} t_{j+1} - t_j < \delta , \quad j= 0, 1,\ldots ,M-1. \end{aligned}$$

Again by Proposition 3.9, there exists an integer \(n_1 \ge n_0\) such that for any \(n \ge n_1\)

$$\begin{aligned} |x_n(t_j) - x(t_j)| < {\varepsilon }, \quad j = 0,1,\ldots , M. \end{aligned}$$
(41)

Fix an integer \(n \ge n_1\). Suppose that \(N_{\eta }(x_n) \ge N\), i.e., there exist some points

$$\begin{aligned} 0\le s_1 < s_2 \le s_3 < s_4 \le \cdots \le s_{2N-1} < s_{2N} \le 1, \end{aligned}$$
(42)

such that

$$\begin{aligned} |x_n(s_{2k}) - x_n(s_{2k-1})| > \eta , \quad \text{ for } \text{ all } \quad k=1,2, \ldots , N. \end{aligned}$$
(43)

The Proof of (40) will be complete once we estimate the number \(N\) by a constant independent of \(n\).

The \(\eta \)-oscillations of \(x_n\) determined by (42) can be divided into two (disjoint) groups. The first group (Group 1) contains the oscillations for which the corresponding interval \([s_{2k-1},s_{2k})\) contains at least one point \(t_{j'}\). Since the number of points \(t_j\) is \(M\),

$$\begin{aligned} \text{ the } \text{ number } \text{ of } \text{ oscillations } \text{ in } \text{ Group } \text{1 } \text{ is } \text{ at } \text{ most }\;M. \end{aligned}$$
(44)

In the second group (Group 2), we have those oscillations for which the corresponding interval \([s_{2k-1},s_{2k})\) contains no point \(t_{j}\), i.e.,

$$\begin{aligned} t_j \le s_{2k-1} < s_{2k} \le t_{j+1} \quad \text{ for } \text{ some } \ j=0,1, \ldots , M-1. \end{aligned}$$
(45)

We now use Lemma 3.12 in each of intervals \([t_j, t_{j+1}]\), \(j=0,1,\ldots , m\). Note that

$$\begin{aligned} \beta _{n,j}:=\sup _{t_{j} \le u<v<w \le t_{j+1}}H\left( x_n(u), x_n(v),x_n(w) \right) \le w^{M_1}(x_n,\delta )<{\varepsilon }, \end{aligned}$$

hence,

$$\begin{aligned} N_{\eta }(x_n,[t_j,t_{j+1}])\le \frac{|x_n(t_{j+1})-x_n(t_{j})| +\beta _{n,j}}{\eta -\beta _{n,j}}<\frac{2 K_\mathrm{sup} + \varepsilon }{\eta -\varepsilon }. \end{aligned}$$

Since there are \(M\) intervals of the form \([t_j,t_{j+1}]\), we conclude that

$$\begin{aligned} \text{ the } \text{ number } \text{ of } \text{ oscillations } \text{ in } \text{ Group } \text{2 } \text{ is } \text{ at } \text{ most } \ M \cdot \frac{2 K_\mathrm{sup} + \varepsilon }{\eta -\varepsilon } \end{aligned}$$
(46)

Summing (44) and (46), we obtain that

$$\begin{aligned} N \le M\left( 1+ \frac{2 K_\mathrm{sup} + \varepsilon }{\eta -\varepsilon }\right) =M\frac{2K_\mathrm{sup}+\eta }{\eta -{\varepsilon }}, \end{aligned}$$

which does not depend on \(n\). Theorem 3.13 follows.

For the sake of completeness, we provide also a typical example of a sequence \((x_n)_{n \ge 1}\) in \({\mathbb {D}}[0,1]\) which is \(S\)-convergent, but does not converge in the \(M_1\) topology.

Example 3.14

Let \(x=0\) and

$$\begin{aligned} x_n(t)=1_{[1/2-1/n,1]}(t)-1_{[1/2+1/n,1]}(t)= \left\{ \begin{array}{l@{\quad }l} 1 &{} \text{ if }\;\frac{1}{2}-\frac{1}{n} \le t <\frac{1}{2}+\frac{1}{n} \\ 0 &{} \text{ otherwise } \end{array} \right. \end{aligned}$$

Then \(x_n \ \mathop {\longrightarrow }_{S}\ x\). To see this, we take \(v_{n,{\varepsilon }}=x_{n}\). Then \(v_{n,{\varepsilon }} \Rightarrow v_{{\varepsilon }}=0\) since for any \(f \in C[0,1]\),

$$\begin{aligned} \int _{0}^{1}f(t)\mathrm{d}v_n(t) =f\left( \frac{1}{2}-\frac{1}{n}\right) -f\left( \frac{1}{2}+\frac{1}{n}\right) \rightarrow 0. \end{aligned}$$

The fact that \((x_n)_{n \ge 1}\) cannot converge in \(M_1\) follows by Proposition 3.9 since if \(t_1<\frac{1}{2}-\frac{1}{n}<t_2<\frac{1}{2}+\frac{1}{n}<t_3\), then \(H\left( x_n(t_1),x_n(t_2), x_n(t_3)\right) =1\).

3.3 Convergence in Distribution in the \(S\) Topology

Now, we are ready to specify results on functional convergence of stochastic processes in the \(S\) topology, which are suitable for needs of linear processes. They follow directly from Proposition 3.6 and Proposition 3.3.

Proposition 3.15

(3.1 in [13]) A family \(\{X_{\gamma }\}_{\gamma \in \Gamma }\) of stochastic processes with trajectories in \({\mathbb {D}}([0,1])\) is uniformly \(S\)-tight if, and only if, the families of random variables \(\{\Vert X_{\gamma }\Vert _{\infty }\}_{\gamma \in \Gamma }\) and \(\{N_{\eta }(X_{\gamma })\}_{\gamma \in \Gamma }\), \(\eta > 0\), are uniformly tight.

Proposition 3.16

Let \(\{X_n\}_{n\ge 0}\) and \(\{Y_n\}_{n\ge 0}\) be two sequences of stochastic processes with trajectories in \({\mathbb {D}}([0,1])\) such that as \(n\rightarrow \infty \)

$$\begin{aligned}&\left( X_n(q_1) + Y_n(q_1), X_n(q_2) + Y_n(q_2), \ldots , X_n(q_k) +Y_n(q_k)\right) \\&\quad \ \mathop {\longrightarrow }_{{\mathcal D}}\ \left( X_0(q_1) + Y_0(q_1), X_0(q_2) + Y_0(q_2)\ldots , X_0(q_k) +Y_0(q_k)\right) , \end{aligned}$$

for each subset \({\mathbb {Q}}_0 = \{ 0 \le q_1 < q_2 < \cdots < q_k \}\) of a dense set \({\mathbb {Q}}\subset [0,1]\), \(1 \in {\mathbb {Q}}\).

If \(\{X_n\}\) and \(\{Y_n\}\) are uniformly \(S\)-tight, then

$$\begin{aligned} X_n+Y_n \ \mathop {\longrightarrow }_{{\mathcal D}}\ X_0+Y_0 \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the \(S\) topology.

Proof of Proposition 3.16

According to Proposition 3.3, it is enough to establish the uniform \(S\)-tightness of \(X_n + Y_n\). This follows immediately from Proposition 3.15 and from the inequalities \(\Vert x+y\Vert _{\infty } \le \Vert x\Vert _{\infty }+\Vert y\Vert _{\infty }\) and

$$\begin{aligned} N_{\eta }(x+y)\le N_{\eta /2}(x)+N_{\eta /2}(y), \end{aligned}$$

valid for arbitrary functions \(x,y \in {\mathbb {D}}[0,1]\) and \(\eta > 0\).

Remark 3.17

In linear topological spaces, the algebraic sum \({\mathcal K}_1 + {\mathcal K}_2 = \{x_1 + x_2 \,;\, x_1 \in {\mathcal K}_1, x_2 \in {\mathcal K}_2\}\) of compact sets \({\mathcal K}_1\) and \({\mathcal K}_2\) is compact. It follows directly from the continuity of the operation of addition and trivializes the proof of uniform tightness of sum of uniformly tight random elements. In \({\mathbb {D}}([0,1])\) equipped with \(S\), we are, however, able to prove that the addition is only sequentially continuous, i.e., if \(x_n \ \mathop {\longrightarrow }_{S}\ x_0\) and \(y_n \ \mathop {\longrightarrow }_{S}\ y_0\), then \(x_n + y_n \ \mathop {\longrightarrow }_{S}\ x_0 + y_0\). In general, it does not imply continuity (see [13], p. 18, for detailed discussion). Sequential continuity gives a weaker property: the sum \({\mathcal K}_1 + {\mathcal K}_2\) of relatively \(S\)-compact \({\mathcal K}_1\) and \({\mathcal K}_2\) is relatively \(S\)-compact. For the uniform tightness purposes, we also need that the \(S\)-closure of \({\mathcal K}_1 + {\mathcal K}_2\) is again relatively \(S\)-compact. This is guaranteed by the lower-semicontinuity in \(S\) of \(\Vert \cdot \Vert _{\infty }\) and \(N_{\eta }\) (see [13], Corollary 2.10).

3.4 The Main Result

Theorem 3.18

Let \(\{Y_j\}\) be an i.i.d. sequence satisfying the usual conditions and \(\sum _j|c_j| < +\infty \). Let \(S_n(t)\) be defined by (11) and \(T_n(t)\) by (16). Then

$$\begin{aligned} T_n(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A_{|\cdot |}\cdot Z(t),\quad \text {where}\;A_{|\cdot |} = \sum _j |c_j|, \end{aligned}$$

implies

$$\begin{aligned} S_n \ \mathop {\longrightarrow }_{{\mathcal D}}\ A\cdot Z, \quad \text {where}\;A = \sum _j c_j, \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the \(S\) topology.

Proof

By Corollary 2.2

$$\begin{aligned} T_n^+(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i^+ \ \mathop {\longrightarrow }_{f.d.d.}\ A_+ \cdot Z(t),\quad T_n^-(t) = \frac{1}{a_n} \sum _{i=1}^{[nt]} X_i^- \ \mathop {\longrightarrow }_{f.d.d.}\ A_-\cdot Z(t), \end{aligned}$$

where \(A_+ = \sum _{i\in {\mathbb {Z}}} c_i^+\) and \(A_- = \sum _{i\in {\mathbb {Z}}} c_i^-\). It follows from Proposition 3.1 that \(T_n^+ \ \mathop {\longrightarrow }_{{\mathcal D}}\ A_+\cdot Z\) on \({\mathbb {D}}([0,1])\) equipped with the \(M_1\) topology. A similar result holds for \(T_n^-\). Since the law of every càdlàg process is \(M_1\)-tight, Le Cam’s theorem [19] (see also Theorem 8 in Appendix III of [4]) guarantees that both sequences \(\{T_n^+\}\) and \(\{T_n^-\}\) are uniformly \(M_1\)-tight. By Theorem 3.13 we obtain the uniform \(S\)-tightness of both \(\{T_n^+\}\) and \(\{T_n^-\}\). Again by Corollary 2.2

$$\begin{aligned} S_n(t) = T_n^+(t) - T_n^-(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A \cdot Z(t). \end{aligned}$$

Now a direct application of Proposition 3.16 completes the proof of the theorem.

4 Discussion of Sufficient Conditions

Conditions (14) do not look tractable. In what follows, we shall provide three types of checkable sufficient conditions. In both cases, the following slight simplification (47) of (14) will be useful. As in Proof of Lemma 2.3, we can find a sequence \(j_n \rightarrow \infty \), \(j_n = o(n)\), such that

$$\begin{aligned}&\sum _{j=-j_n + 1}^0 \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \rightarrow 0,\quad \text {as}\; n\rightarrow \infty .\\&\quad \sum _{j=n+ 1}^{n+j_n -1}\frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \rightarrow 0,\quad \text {as}\;n\rightarrow \infty . \end{aligned}$$

Hence, it is enough to check

$$\begin{aligned} \begin{aligned} \sum _{j=-\infty }^{-j_n} \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right)&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty .\\ \sum _{j=n+ j_n}^{+\infty }\frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right)&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty . \end{aligned} \end{aligned}$$
(47)

The advantage of this form of the conditions consists in the fact that

$$\begin{aligned} \begin{aligned} \sup _{j \le - j_n} \big |d_{n,j}\big |&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty ,\\ \sup _{j \ge n + j_n} \big |d_{n,j}\big |&\rightarrow 0,\quad \text {as}\; n\rightarrow \infty . \end{aligned} \end{aligned}$$
(48)

We will write \(\ \mathop {\longrightarrow }_{{\mathcal {D}(S)}\ }\) when convergence in distribution with respect to the \(S\) topology takes place.

Corollary 4.1

Under the assumptions of Theorem 2.1, if there exists \(0 < \beta < \alpha \), \(\beta \le 1\) such that

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}} |c_j|^{\beta } < +\infty , \end{aligned}$$
(49)

then

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{{\mathcal {D}(S)}\ }A\cdot Z(t). \end{aligned}$$

Proof

We have to check (47). By simple manipulations and taking into account that due to (6) \(K =\sup _n n a_n^{-\alpha } h(a_n) < +\infty \) we obtain

$$\begin{aligned}&\sum _{j=-\infty }^{-j_n}\frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \\&\quad =\frac{1}{n} \sum _{j=-\infty }^{-j_n}\big | \sum _{k=1-j}^{n-j}c_k\big |^{\beta }\frac{n h(a_n)}{a_n^{\alpha }} \big |d_{n,j}\big |^{\alpha -\beta }\frac{1}{h(a_n)} h \left( \frac{a_n}{\big |d_{n,j}\big |}\right) \\&\quad \le K \frac{1}{n} \sum _{j=-\infty }^{-j_n} \sum _{k=1-j}^{n-j}|c_k|^{\beta }\Psi _{\alpha - \beta }\left( a_n, \frac{a_n}{\big |d_{n,j}\big |}\right) , \end{aligned}$$

where

$$\begin{aligned} \Psi _{\alpha - \beta }(x,y) = \left( \frac{x}{y}\right) ^{\alpha -\beta } \frac{h(y)}{h(x)}. \end{aligned}$$

Let

$$\begin{aligned} h(x) = c(x) \exp \left( \int _a^x \frac{\epsilon (u)}{u} \,\mathrm{d}u\right) , \end{aligned}$$

where \(\lim _{x\rightarrow \infty } c(x) = c\in (0,\infty )\) and \(\lim _{x\rightarrow \infty } \epsilon (x) = 0\), be the Karamata representation of the slowly varying function \(h(x)\) (see e.g., Theorem 1.3.1 in [5]). Take \(0 < \gamma < \min \{\alpha - \beta , c\}\) and let \(L >a\) be such that for \(x > L\)

$$\begin{aligned} \epsilon (x) \le \gamma \text { and } c -\gamma < c(x) < c + \gamma . \end{aligned}$$

Then, we have for \(x \ge y \ge L\)

$$\begin{aligned} \frac{h(y)}{h(x)} = \frac{c(y)}{c(x)} \exp \left( \int _y^x \frac{\epsilon (u)}{u}\,\mathrm{d}u\right) \le \frac{c + \gamma }{c - \gamma } \exp \left( \gamma \log \left( \frac{x}{y}\right) \right) = \frac{c + \gamma }{c - \gamma } \left( \frac{x}{y}\right) ^{\gamma }, \end{aligned}$$

and so

$$\begin{aligned} \Psi _{\alpha -\beta }(x,y) \le K \left( \frac{y}{x}\right) ^{\alpha - \beta -\gamma },\quad x \ge y \ge L. \end{aligned}$$

It follows from that fact and (48) that

$$\begin{aligned} \sup _{j\le -j_n} \Psi _{\alpha - \beta }\left( a_n, \frac{a_n}{\big |d_{n,j}\big |}\right) \rightarrow 0, \text { as}\;n\rightarrow \infty . \end{aligned}$$

Hence it is sufficient to show that

$$\begin{aligned} \sup _n\frac{1}{n} \sum _{j=-\infty }^{-j_n} \sum _{k=1-j}^{n-j}|c_k|^{\beta } < +\infty . \end{aligned}$$

In fact, more is true.

Lemma 4.2

If \(\sum _{j=0}^{\infty } |b_j| < +\infty \), then for each \(t > 0\)

$$\begin{aligned} \frac{1}{n} \sum _{j=0}^{\infty } \sum _{k=1+j}^{n+j}b_k \rightarrow 0, \text { as}\;n\rightarrow \infty . \end{aligned}$$

Proof of Lemma 4.2

We have

$$\begin{aligned} \begin{aligned} \Big |\frac{1}{n} \sum _{j=0}^{\infty } \sum _{k=1+j}^{n+j}b_k \Big |&\le \frac{1}{n} \sum _{j=0}^{\infty } \sum _{k=1+j}^{n+j}|b_k| \\&= \frac{1}{n} \sum _{k=1}^{\infty } (k\wedge n) |b_k| \\&= \left( \frac{1}{n} \sum _{k=1}^{n} k |b_k| + \sum _{k=n+1}^{\infty } |b_k|\right) . \end{aligned} \end{aligned}$$

The first sum in the last line converges to \(0\) by Kronecker’s lemma. The second is the rest of a convergent series.

Returning to the Proof of Corollary 4.1, let us notice that convergence

$$\begin{aligned} \sum _{j=n+ j_n}^{+\infty } \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \rightarrow 0,\quad \text {as}\;n\rightarrow \infty , \end{aligned}$$

can be checked the same way.

Corollary 4.3

Under the usual conditions, if \(\alpha \in (1,2)\) and \(\sum _{j\in {\mathbb {Z}}} |c_j| < +\infty \), then

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{{\mathcal {D}(S)}\ }A\cdot Z(t). \end{aligned}$$

Remark 4.4

Corollaries 4.1 and 4.3 were proved independently by Astrauskas [1] and Davis and Resnick [8]. Our approach follows direct manipulations of Astrauskas, while Davis and Resnick involved point process techniques.

Remark 4.5

For \(\alpha \le 1\) assumption (49) is unsatisfactory, for it excludes the case of strictly \(\alpha \)-stable random variables \(\{Y_j\}\) with \(\sum _j |c_j|^{\alpha } < +\infty \), but \(\sum _j |c_j|^{\beta } = +\infty \) for every \(\beta < \alpha \). With our criterion given in Theorem 2.1 we can easily prove the needed result.

Corollary 4.6

Suppose that \(\alpha \le 1\), \(\sum _{j\in {\mathbb {Z}}} |c_j|^{\alpha } < +\infty \), the usual conditions hold and \(h\) is such that

$$\begin{aligned} h(\lambda x)/h(x) \le M,\quad \lambda \ge 1, x \ge x_0, \end{aligned}$$
(50)

for some constants \(M\), \(x_0\). If the linear process \(\{X_i\}\) is well-defined, then

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{{\mathcal {D}(S)}\ }A\cdot Z(t). \end{aligned}$$

Proof of Corollary 4.6

First notice that \(\sum _j |c_j| < +\infty \) so that \(A\) is defined. Proceeding like in the Proof of Corollary 4.1, we obtain

$$\begin{aligned} \begin{aligned}&\sum _{j=-\infty }^{-j_n} \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \\&\quad = \frac{1}{n}\sum _{j=-\infty }^{-j_n} \left| \sum _{k=1-j}^{n-j}c_k\right| ^{\alpha } \frac{n h(a_n)}{a_n^{\alpha }} \frac{1}{h(a_n)} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \\&\quad \le K\cdot M \ \frac{1}{n} \sum _{j=-\infty }^{-j_n} \sum _{k=1-j}^{n-j} |c_k|^{\alpha } \rightarrow 0, \end{aligned} \end{aligned}$$

where the convergence to \(0\) holds by Lemma 4.2.

Remark 4.7

As mentioned before, the above corollary covers the important case when \(h(x) \rightarrow C > 0\), as \(x\rightarrow \infty \), i.e., when the law of \(Y_i\) is in the domain of strict (or normal) attraction. Many other examples can be produced using Karamata’s representation of slowly varying functions. Assumption (50) is much in the spirit of Lemma A.4 in [21]. Our final result goes in different direction.

Remark 4.8

Notice that if \(\alpha <1\), then \(\sum _j |c_j|^{\alpha } h(|c_j|^{-1}) < +\infty \), with \(h\) slowly varying, automatically implies \(\sum _j |c_j| < +\infty \).

Corollary 4.9

Under the usual conditions, if \(\alpha < 1\), then

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{{\mathcal {D}(S)}\ }A\cdot Z(t), \end{aligned}$$

if

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}} |c_j|^{\alpha } < +\infty , \end{aligned}$$

and the coefficients \(c_j\) are regular in a very weak sense: there exists a constant \(0 < \gamma <\alpha \) such that

$$\begin{aligned} \frac{\max _{j+ 1 \le k \le j+n} |c_k|^{\frac{(1-\alpha )(\alpha -\gamma )}{(1 - \alpha + \gamma )}}}{\sum _{k = j+1}^{j+n} |c_k|^{\alpha }}&\le K_+< +\infty , \quad j \ge 0.\end{aligned}$$
(51)
$$\begin{aligned} \frac{\max _{j-n \le k\le j-1} |c_k|^{\frac{(1-\alpha )(\alpha -\gamma )}{(1 - \alpha + \gamma )}}}{\sum _{k= j -n}^{j-1} |c_k|^{\alpha }}&\le K_- < +\infty , \quad j \le 0. \end{aligned}$$
(52)

(with the convention that \(0/0 \equiv 1\).)

Remark 4.10

Notice that we always assume that the linear process is well- defined. This may require more than demanded in Corollary 4.9.

Proof of Corollary 4.9

As before, we have to check (47).

$$\begin{aligned} \begin{aligned}&\sum _{j=-\infty }^{-j_n}\frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \\&\quad = \frac{1}{n} \sum _{j=-\infty }^{-j_n} \Big |\sum _{k=1-j}^{n-j}c_k\Big |^{\alpha - \gamma } \frac{n h(a_n)}{a_n^{\alpha }} \big |d_{n,j}\big |^{\gamma } \frac{1}{h(a_n)} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \\&\quad \le K \frac{1}{n} \sum _{j=-\infty }^{-j_n} \Big |\sum _{k=1-j}^{n-j}c_k\Big |^{\alpha - \gamma } \Psi _{\gamma }\left( a_n, \frac{a_n}{\big |d_{n,j}\big |}\right) , \end{aligned} \end{aligned}$$

where \(\Psi _{\gamma }(x,y)\) was defined in the Proof of Corollary 4.1 and

$$\begin{aligned} \sup _{j\le -j_n} \Psi _{\gamma }\left( a_n, \frac{a_n}{\big | d_{n,j}\big |}\right) \rightarrow 0, \text { as}\;n\rightarrow \infty . \end{aligned}$$

Thus it is enough to prove

$$\begin{aligned} \sup _n \frac{1}{n} \sum _{j=-\infty }^{-j_n} \Big |\sum _{k= 1-j}^{n-j}c_k\Big |^{\alpha - \gamma } < +\infty . \end{aligned}$$

We have

$$\begin{aligned} \Big |\sum _{k=1-j}^{n-j}c_k\Big | \le \sum _{k=1-j}^{n-j}|c_k| \le \left( \sum _{k=1-j}^{n-j}|c_k|^{\alpha } \right) \cdot \max _{1 - j \le k \le n-j}|c_k|^{1-\alpha }, \end{aligned}$$
(53)

hence

$$\begin{aligned} \begin{aligned} \frac{1}{n} \sum _{j=-\infty }^{-j_n} \Big |\sum _{k=1-j}^{n-j} c_k\Big |^{\alpha - \gamma }&=\frac{1}{n} \sum _{j=-\infty }^{-j_n} \left( \sum _{k=1-j}^{n-j}|c_k|^{\alpha }\right) \frac{\Big |\sum _{k=1- j}^{n-j}c_k\Big |^{\alpha - \gamma }}{\left( \sum _{k=1-j}^{n-j}| c_k|^{\alpha }\right) }\\&\le \frac{1}{n} \sum _{j=-\infty }^{-j_n} \left( \sum _{k=1-j}^{n -j}|c_k|^{\alpha }\right) \frac{ \max _{ 1 - j \le k \le n-j} |c_k|^{(1-\alpha )(\alpha - \gamma )}}{\left( \sum _{k=1-j}^{n-j}| c_k|^{\alpha }\right) ^{1 -\alpha + \gamma }} \\&\le \left( K_+\right) ^{1-\alpha + \gamma } \frac{1}{n} \sum _{j= -\infty }^{-j_n} \left( \sum _{k=1-j}^{n-j}|c_k|^{\alpha }\right) \rightarrow 0. \end{aligned} \end{aligned}$$

This is again more than needed. The proof of

$$\begin{aligned} \sum _{j=n+ j_n}^{+\infty } \frac{\big |d_{n,j}\big |^{\alpha }}{a_n^{\alpha }} h\left( \frac{a_n}{\big |d_{n,j}\big |}\right) \rightarrow 0,\quad \text {as}\;n\rightarrow \infty . \end{aligned}$$

goes the same way.

Example 4.11

If \(\alpha < 1\),

$$\begin{aligned} |c_j| = \frac{1}{|j|^{1/\alpha } \log ^{(1+\varepsilon )/\alpha } |j|},\quad |j| \ge 3, \end{aligned}$$

and \(\{X_i\}\) is well-defined, then under the usual conditions

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{{\mathcal {D}(S)}\ }A\cdot Z(t). \end{aligned}$$

Remark 4.12

In our considerations, we search for conditions giving functional convergence of \(\{S_n(t)\}\) with the same normalization as \(\{Z_n(t)\}\) (by \(\{a_n\}\)). It is possible to provide examples of linear processes, which are convergent in the sense of finite dimensional distribution with different normalization. Moreover, it is likely that also in the heavy-tailed case one can obtain a complete description of the convergence of linear processes, as it is done by Peligrad and Sang [23] in the case of innovations belonging to the domain of attraction of a normal distribution. We conjecture that whenever the limit is a stable Lévy motion our functional approach can be adapted to the more general setting.

5 Some Complements

5.1 \(S\)-Continuous Functionals

A phenomenon of self-canceling oscillations, typical for the \(S\) topology, was described in Example 3.14. This example shows that supremum cannot be continuous in the \(S\) topology. In fact, supremum is lower semi-continuous with respect to \(S\), as many other popular functionals—see [13], Corollary 2.10. On the other hand addition is sequentially continuous and this property was crucial in consideration given in Sect. 3.4.

Here is another positive example of an \(S\)-continuous functional.

Let \(\mu \) be an atomless measure on \([0,1]\) and let \(h : {\mathbb {R}}^1 \rightarrow {\mathbb {R}}^1\) be a continuous function. Consider a smoothing operation \(s_{\mu ,h}\) on \({\mathbb {D}}([0,1])\) given by the formula

$$\begin{aligned} s_{\mu ,h}(x)(t) = \int _0^t h(x(s))\,\mathrm{d}\mu (s). \end{aligned}$$

Then, \(s_{\mu ,h}(x)(\cdot )\) is a continuous function on \([0,1]\) and a slight modification of the Proof of Proposition 2.15 in [13] shows that the mapping

$$\begin{aligned} \left( {\mathbb {D}}([0,1]), S\right) \ni x \mapsto s_{\mu ,h}(x) \in \left( {\mathbb {C}}([0,1]), \Vert \cdot \Vert _{\infty }\right) \end{aligned}$$

is continuous. In particular, if we set \(\mu = \ell \) (the Lebesgue measure), \(h(0) = 0\), \(h(x) \ge 0\), and suppose that \(x_n \ \mathop {\longrightarrow }_{S}\ 0\), then

$$\begin{aligned} \int _0^1 h(x_n(s))\,\mathrm{d}s \rightarrow 0. \end{aligned}$$

In the case of linear processes, such functionals lead to the following result.

Corollary 5.1

Under the conditions of Corollaries 4.1, 4.3, 4.6 or 4.9 we have for any \(\beta > 0\)

$$\begin{aligned} \frac{1}{n a_n^{\beta }} \sum _{k=1}^n \Big | \sum _{i=1}^k \left( \sum _j c_{i-j} Y_j\right) - A Y_i\Big |^{\beta } \ \mathop {\longrightarrow }_{{\mathcal P}}\ 0. \end{aligned}$$

Proof of Corollary 5.1

The expression to be analyzed has the form

$$\begin{aligned} \int _0^1 H_{\beta }\left( S_n(t) - A\cdot Z_n(t)\right) \,\mathrm{d}t, \end{aligned}$$

where \(H_{\beta }(x) = |x|^{\beta }\) and by (26)

$$\begin{aligned} S_n(t) - A\cdot Z_n(t) \ \mathop {\longrightarrow }_{f.d.d.}\ 0. \end{aligned}$$

We have checked in the course of the Proof of Theorem 3.18, that \(\{S_n\}\) is uniformly \(S\)-tight. By (3) \(\{A\cdot Z_n\}\) is uniformly \(J_1\)-tight, hence also \(S\)-tight. Similarly as in the Proof of Proposition 3.16 we deduce that \(\{S_n - A\cdot Z_n\}\) is uniformly \(S\)-tight. Now an application of Proposition 3.3 gives

$$\begin{aligned} S_n - A\cdot Z_n \ \mathop {\longrightarrow }_{{\mathcal D}}\ 0, \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the \(S\) topology.

5.2 An Example Related to Convergence in the \(M_1\) Topology

In Introduction, we provided an example of a linear process (\(c_0=1, c_1 = -1\)) for which no Skorokhod’s convergence is possible. In this example \(A=0\) and the limit is degenerate, what might suggest that another, more appropriate norming is applicable, under which the phenomenon disappears. Here, we give an example with a non-degenerate limit showing that in the general case \(M_1\)-convergence need not hold.

Example 5.2

Let \(c_0 =\zeta > -c_1 = \xi >0\). Then \(X_j = \zeta Y_j - \xi Y_{j-1}\) and defining \(Z_n(t)\) by (3) we obtain for \(t\in [k/n, (k+1)/n)\)

$$\begin{aligned} S_n(t) = \frac{1}{a_n} \sum _{j=1}^{k} X_j = \frac{1}{a_n} \left( \zeta Y_{k} - \xi Y_0\right) + (\zeta - \xi ) Z_n((k-1)/n). \end{aligned}$$

Clearly, the f.d.d. limit \(\{(\zeta - \xi ) Z(t)\}\) is non-degenerate. We will show that the sequence \(\{S_n(t)\}\) is not uniformly \(M_1\)-tight and so cannot converge to \(\{ (\zeta - \xi ) Z(t)\}\) in the \(M_1\) topology.

For the sake of simplicity, let us assume that \(Y_j\)’s are non-negative and

$$\begin{aligned} {\mathbb {P}}\left( Y_1 > x\right) = x^{-\alpha }, \quad x \ge 1, \end{aligned}$$

with \(\alpha < 1\). Then, we can choose \(a_n = n^{1/\alpha }\). Consider sets

$$\begin{aligned} G_n = \bigcup _{j=0}^{n-1} \left\{ Y_j > \varepsilon _n a_n, Y_{j+1} > \varepsilon _n a_n\right\} . \end{aligned}$$

where \(\varepsilon _n = n^{-1/(3\alpha )}\). Then,

$$\begin{aligned} {\mathbb {P}}\left( G_n\right) \le (n+1) {\mathbb {P}}\left( Y_i > \varepsilon _n a_n\right) ^2 = (n+1) \varepsilon _n^{-2\alpha } \left( n^{1/\alpha }\right) ^{-2\alpha } \longrightarrow 0. \end{aligned}$$

Notice that

$$\begin{aligned} \text {on}\;G_n^c\;\hbox {there are no two consecutive values of}\; Y_j\hbox { exceeding} \varepsilon _n a_n. \end{aligned}$$
(54)

Let us define \(Y_{n,j} = Y_j { 1{\mathrm I}}\{Y_j > \varepsilon _n a_n\}\) and set for \(t\in [k/n, (k+1)/n)\)

$$\begin{aligned} \widetilde{S}_n(t) = \frac{1}{a_n} \left( \zeta Y_{n,k} - \xi Y_{n,0}\right) + \frac{\zeta - \xi }{a_n} \sum _{j=1}^{k-1} Y_{n,j}. \end{aligned}$$

We have by (61)

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t\in [0,1]} \big | S_n(t) -\widetilde{S}_n(t)\big |\right]&\le \frac{\zeta }{a_n}\sum _{j=0}^n{\mathbb {E}}\left[ Y_j { 1{\mathrm I}}\{ Y_j \le \varepsilon _n a_n\}\right] \\&\le C_1 \zeta \frac{(n+1) (\varepsilon _n a_n)^{1 - \alpha }}{a_n} \rightarrow 0. \end{aligned}$$

It follows that \(\{S_n(t)\}\) are uniformly \(M_1\)-tight if, and only if, \(\{\widetilde{S}_n(t)\}\) are. Let \(w^{M_1}(x,\delta )\) be given by (34). Since \({\mathbb {P}}\left( G_n^c\right) \rightarrow 1\) we have for any \(\delta > 0\) and \(\eta > 0\)

$$\begin{aligned} \limsup _n {\mathbb {P}}\left( w^{M_1}(\widetilde{S}_n(\cdot ),\delta ) > \eta \right) = \limsup _n {\mathbb {P}}\left( \{w^{M_1}(\widetilde{S}_n(\cdot ),\delta ) > \eta \}\cap G_n^c \right) . \end{aligned}$$

And on \(G_n^c\), by the property (54) and if \(2/n <\delta \) we have

$$\begin{aligned} \omega (\widetilde{S}_n(\cdot ),\delta ) \ge \frac{1}{a_n} \left( \zeta - \xi \right) \max _j Y_{n,j}. \end{aligned}$$

If \(\eta / (\zeta - \xi ) > \varepsilon _n\), then

$$\begin{aligned}&{\mathbb {P}}\left( (1/a_n) \max _j Y_{n,j} > \eta / \left( \zeta - \xi \right) \right) \\&\quad ={\mathbb {P}}\left( (1/a_n) \max _j Y_j > \eta / \left( \zeta - \xi \right) \right) \\&\quad \longrightarrow 1 -\exp \left( - \left( (\zeta - \xi )/\eta \right) ^{\alpha } \right) = \theta > 0. \end{aligned}$$

Hence for each \(\delta > 0\)

$$\begin{aligned} \liminf _n {\mathbb {P}}\left( w^{M_1}(\widetilde{S}_n(\cdot ),\delta ) > \eta \right) \ge \theta > 0, \end{aligned}$$

and the sequence \(\{\widetilde{S}_n(t)\}\) cannot be uniformly \(M_1\)-tight.

5.3 Linear Space of Convergent Linear Processes

We can explore the machinery of Sect. 4 to obtain a natural

Proposition 5.3

We work under the assumptions of Theorem 2.1. Denote by \(\mathcal{C}_Y\) the set of sequences \(\{c_i\}_{i\in {\mathbb {Z}}}\) such that if

$$\begin{aligned} X_i=\sum _{j \in {\mathbb {Z}}}c_j Y_{i-j}, \quad i \in {\mathbb {Z}}, \end{aligned}$$

then

$$\begin{aligned} S_n(t)=\frac{1}{a_n} \sum _{i=1}^{[nt]}X_i \ \mathop {\longrightarrow }_{f.d.d.}\ A\cdot Z(t), \end{aligned}$$

with \(A = \sum _{i\in {\mathbb {Z}}} c_i\).

Then \(\mathcal{C}_Y\) is a linear subspace of \({\mathbb {R}}^{{\mathbb {Z}}}\).

Proof of Proposition 5.3

Closeness of \(\mathcal{C}_Y\) under multiplication by a number is obvious. So let us assume that \(\{c_i'\}\) and \(\{c_i''\}\) are elements of \(\mathcal{C}_Y\). By Theorem 2.1, we have to prove that

$$\begin{aligned} \begin{aligned} \sum _{j=-\infty }^0 {\mathbb {P}}\left( \big |\sum _{k=1-j}^{n-j} (c_k' + c_k'') \big | |Y_j| > a_n\right)&\rightarrow 0,\quad \text {as}\;n\rightarrow \infty .\\ \sum _{j=n+ 1}^{\infty } {\mathbb {P}}\left( \big |\sum _{k=1-j}^{n-j}(c_k' + c_k'') \big | |Y_j| > a_n\right)&\rightarrow 0,\quad \text {as}\;n\rightarrow \infty . \end{aligned} \end{aligned}$$
(55)

But

$$\begin{aligned}&\sum _{j=-\infty }^0 {\mathbb {P}}\left( \big |\sum _{k=1-j}^{n-j} (c_k' + c_k'') \big | |Y_j| > a_n\right) \\&\quad \le \sum _{j=-\infty }^0 {\mathbb {P}}\left( \big |\sum _{k=1-j}^{n-j} c_k'\big | |Y_j| + \big |\sum _{k=1-j}^{n-j} c_k'' \big | |Y_j| > a_n\right) \\&\quad \le \sum _{j=-\infty }^0 {\mathbb {P}}\left( \big |\sum _{k=1-j}^{n-j} c_k'\big | |Y_j| > a_n/2\right) + \sum _{j=-\infty }^0{\mathbb {P}}\left( \big |\sum _{k=1-j}^{n -j} c_k'' \big | |Y_j| > a_n/2\right) . \end{aligned}$$

Now both terms tend to \(0\) by Remark 2.7. Identical reasoning can be used in the proof of the “dual” condition in (55).

5.4 Dependent Innovations

In the main results of the paper, we studied only independent innovations \(\{Y_j\}\). It is however clear that the functional \(S\)-convergence can be obtained under much weaker assumptions. In order to apply crucial Proposition 3.16 we need only that

$$\begin{aligned} S_n(t) \ \mathop {\longrightarrow }_{f.d.d.}\ A\cdot Z(t), \end{aligned}$$

and that

$$\begin{aligned} T_n^+ \ \mathop {\longrightarrow }_{{\mathcal D}}\ A_+\cdot Z, \quad \text {and }\ T_n^- \ \mathop {\longrightarrow }_{{\mathcal D}}\ A_-\cdot Z, \end{aligned}$$

on the Skorokhod space \({\mathbb {D}}([0,1])\) equipped with the \(M_1\) topology. For the latter relations, Theorem 1 of [20] seems to be an ideal tool for associated sequences (see our Proposition 3.1). A variety of potential other possible examples is given in [27].