Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \le 1$$\end{document}. In the functional setting, we complement the existing results on M1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1$$\end{document}-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document} topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).


Introduction and Announcement of Results
Let {Y j } j∈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 where the constants {c j } j∈Z are such that the above series is 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 α-stable law μ α , i.e., where Z ∼ μ α . Let us observe that by the Skorokhod theorem [25], we also have Z n (t) = 1 a n where {Z (t)} is the stable Lévy process with Z (1) ∼ μ α , and the convergence holds on the Skorokhod space 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 α ∈ (0, 2) such that where h is a function that varies slowly at x = +∞, and also lim x→∞ P(Y j > x) P(|Y j | > x) = p and lim x→∞ P(Y j < −x) P(|Y j | > x) = q, p + q = 1.
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 α ∈ (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 P-a.s. convergent if, and only if, j∈Z |c j | α h(|c j | −1 ) < +∞.
Given the above series is convergent we can define 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: In Sect. 2, we give necessary and sufficient conditions for the finite dimensional convergence where the constants a n are the same as in (2), A = j∈Z c j and {Z (t)} is an α-stable Lévy Motion such that Z (1) ∼ Z . The obtained conditions lead to tractable sufficient conditions, which in case α < 1 are new and essentially weaker than condition j∈Z |c j | β < +∞, for some 0 < β < α, 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 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 a n 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 γ > 0 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) ∼ Z .

Convergence of Finite Dimensional Distributions for Summable Coefficients
We begin with stating the main result of this section followed by its important consequence. Then Proof of Corollary 2.2 In view of Theorem 2.1, it is enough to notice that Proof of Theorem 2.1 Using Fubini's theorem, we obtain that S n (t) = 1 a n Further, we may decompose j∈Z 1 a n d [nt], j Y j = 0 j=−∞ 1 a n d [nt], j Y j Let us consider the partial sum process: First we will show

Lemma 2.3 Under the assumptions of Theorem
In particular, Proof of Lemma 2.3 Define To prove that V 0 n −→ P 0, we apply Proposition 5.5. We have to show that , j · |Y j | > a n → 0, as n → ∞.
We need a simple lemma.

Lemma 2.4
Let {a n, j ; 1 ≤ j ≤ n, n ∈ N} be an array of numbers such that max 1≤ j≤n |a n, j | → 0, as n → ∞.
Then there exists a sequence j n → ∞, j n = o(n), such that j n j=1 |a n, j | → 0.
By the above lemma and (23), we can find a sequence j n → ∞, j n = o(n), increasing so slowly that still For the remaining part we have j=1 P |δ n ||Y j | > a n ≤ [nt] δ α a α n h(a n /δ) = [nt]δ α h(a n ) a α n h(a n /δ) h(a n ) .
But δ > 0 is arbitrary, hence we have proved (22) and

Lemma 2.3 follows.
In the next step, we shall prove Proof of Lemma 2.5 By Lemma 2.3 we know that S 0 (26) implies (25) and the latter implies (24). So let us assume (24). By regular variation of a n , we have for each t ∈ (0, 1]

It follows that
Since also and E e iθ A·Z (t) = 0, θ ∈ R 1 (for Z (t) has infinitely divisible law), we conclude that Let us observe that by Proposition 5.5 (25) holds if, and only if, 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): and the finite dimensional distributions of stochastic processes A· Z n (t) are convergent to those of A · 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 ∈ (0, +∞) can be chosen as the truncation level. Hence, conditions (14) admit an equivalent reformulation in the "r -form" r · a n d n, j → 0, as n → ∞.

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 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: 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.
If the linear process {X i } is well-defined and then also functionally 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≥1 and {c − j } j≥1 ).

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 τ on D([0, 1]) which is coarser than M 1 , then should imply 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 nonmetric sequential topology, with sequentially continuous addition, which is stronger than any of mentioned above L p (μ) spaces and is functional in the sense it has the following classic property (see Theorem 3.5 of [13]).
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 can be J 1 -discontinuous at some x ∈ 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 (μ) 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 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.
and the supremum is taken over all finite partitions

Definition 3.4 (S-convergence and the S topology)
We shall say that x n S-converges to x 0 (in short x n → S x 0 ) if for every ε > 0 one can find elements v n,ε ∈ V([0, 1]), n = 0, 1, 2, . . . which are ε-uniformly close to x n 's and weakly- * convergent: 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.
such that Then from any sequence {x n } ⊂ K one can extract a subsequence {x n k } and find is relatively compact with respect to −→ S , then it satisfies both (32) and (33).
Then x n → x 0 in S. Remark 3.8 The S topology is sequential, i.e., it is generated by the convergence −→ S . By the Kantorovich-Kisyński recipe [17] x n → 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 −→ 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 −→ M 1 x 0 implies x n −→ 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 ∈ D([0, 1]) and δ > 0 where H (a, b, c) is the distance between b and the interval with endpoints a and c:   H (c, a, d) + H (c, b, d). H (c, a, d) + H (c, b, d).
The other cases can be reduced to the considered above. .
Assume first that x(t 2 ) − x(t 1 ) > η. We claim that To see this, suppose that x(t 3 ) < x(t 2 ) − β. Then the distance between x(t 2 ) and the interval with endpoints x(t 1 ) and x(t 3 ) is greater than β, which is a contradiction.
On the other hand, if we assume that x(t 4 ) − x(t 3 ) < −η, we obtain that which means that the distance between x(t 3 ) and the interval with endpoints x(t 1 ) and x(t 4 ) is greater than β, again a contradiction.
Repeating this argument, we infer that: Taking the sum of these inequalities, we conclude that: On the other hand, by Corollary 3.11, we have: Combining (37) and (38), we obtain that which is the desired upper bound.
Assuming that x(t 2 ) − x(t 1 ) < −η, we come in a similar way to the inequality 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 ⊂ D([0, 1]) which is relatively M 1compact is also relatively S-compact.
Proof Let x n −→ M 1 x 0 . By Proposition 3.9 on the dense set of points of continuity of x 0 and for t = 1. Suppose, we know that 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 → x 0 in S. Thus, it remains to check conditions (39) follows.
The Proof of (40) will be complete once we estimate the number N by a constant independent of n.
The η-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, the number of oscillations in Group 1 is at most M.
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., We now use Lemma 3.12 in each of intervals [t j , t j+1 ], j = 0, 1, . . . , m. Note that β n, j := sup Since there are M intervals of the form [t j , t j+1 ], we conclude that the number of oscillations in Group 2 is at most Summing (44) and (46), we obtain that 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≥1 in D[0, 1] which is S-convergent, but does not converge in the M 1 topology. Example 3.14 Let x = 0 and To see this, we take v n,ε = x n . Then v n,ε ⇒ v ε = 0 since for any The fact that (x n ) n≥1 cannot converge in M 1 follows by Proposition 3.9 since if

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.
for each subset Q 0 = {0 ≤ q 1 < q 2 < · · · < q k } of a dense set Q ⊂ [0, 1], 1 ∈ Q. If {X n } and {Y n } are uniformly S-tight, then 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 x + y ∞ ≤ x ∞ + y ∞ and valid for arbitrary functions x, y ∈ D[0, 1] and η > 0.

Remark 3.17
In linear topological spaces, the algebraic sum K 1 +K 2 = {x 1 +x 2 ; x 1 ∈ K 1 , x 2 ∈ K 2 } of compact sets K 1 and 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 D([0, 1]) equipped with S, we are, however, able to prove that the addition is only sequentially continuous, i.e., if x n −→ S x 0 and y n −→ S y 0 , then x n + y n −→ 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 K 1 + K 2 of relatively S-compact K 1 and K 2 is relatively S-compact. For the uniform tightness purposes, we also need that the S-closure of K 1 + K 2 is again relatively S-compact. This is guaranteed by the lower-semicontinuity in S of · ∞ and N η (see [13], Corollary 2.10).

The Main Result
Theorem 3.18 Let {Y j } be an i.i.d. sequence satisfying the usual conditions and j |c j | < +∞. Let S n (t) be defined by (11) and T n (t) by (16). Then where A + = i∈Z c + i and A − = i∈Z c − i . It follows from Proposition 3.1 that Now a direct application of Proposition 3.16 completes the proof of the theorem.

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 → ∞, We will write −→ D(S) when convergence in distribution with respect to the S topology takes place.

Corollary 4.1 Under the assumptions of Theorem
Proof We have to check (47). By simple manipulations and taking into account that due to (6) K = sup n na −α n h(a n ) < +∞ we obtain where lim x→∞ c(x) = c ∈ (0, ∞) and lim x→∞ (x) = 0, be the Karamata representation of the slowly varying function h(x) (see e.g., Theorem 1.3.1 in [5]). Take 0 < γ < min{α − β, c} and let L > a be such that for Then, we have for It follows from that fact and (48) that sup j≤− j n α−β a n , a n d n, j → 0, as n → ∞.
Hence it is sufficient to show that In fact, more is true.
Proof of Lemma 4. 2 We have 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  A · Z (t).

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 α ≤ 1 assumption (49) is unsatisfactory, for it excludes the case of strictly α-stable random variables {Y j } with j |c j | α < +∞, but j |c j | β = +∞ for every β < α. With our criterion given in Theorem 2.1 we can easily prove the needed result.

Corollary 4.6
Suppose that α ≤ 1, j∈Z |c j | α < +∞, the usual conditions hold and h is such that for some constants M, x 0 . If the linear process {X i } is well-defined, then S n (t) −→
h a n d n, j where the convergence to 0 holds by Lemma 4.2.

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).
− j n j=−∞ d n, j α a α n h a n d n, j where γ (x, y) was defined in the Proof of Corollary 4.1 and sup j≤− j n γ a n , a n d n, j → 0, as n → ∞.
Thus it is enough to prove We have This is again more than needed. The proof of +∞ j=n+ j n d n, j α a α n h a n d n, j → 0, as n → ∞.
goes the same way.

Example 4.11
If α < 1, and {X i } is well-defined, then under the usual conditions

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.

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 Then, s μ,h (x)(·) is a continuous function on [0, 1] and a slight modification of the Proof of Proposition 2.15 in [13] shows that the mapping is continuous. In particular, if we set μ = (the Lebesgue measure), h(0) = 0, h(x) ≥ 0, and suppose that x n −→ S 0, then 1 0 h(x n (s)) ds → 0.
In the case of linear processes, such functionals lead to the following result.

0.
We have checked in the course of the Proof of Theorem 3.18, that {S n } is uniformly S-tight. By

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.
Clearly, the f.d.d. limit {(ζ −ξ)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 {(ζ − ξ)Z (t)} in the M 1 topology.
For the sake of simplicity, let us assume that Y j 's are non-negative and with α < 1. Then, we can choose a n = n 1/α . Consider sets Y j > ε n a n , Y j+1 > ε n a n .
where ε n = n −1/(3α) . Then, Notice that on G c n there are no two consecutive values of Y j exceedingε n a n .
Let us define Y n, j = Y j 1 I{Y j > ε n a n } and set for t ∈ [k/n, (k + 1)/n) We have by (61) E sup 1] S n (t) − S n (t) ≤ ζ a n n j=0 E Y j 1 I{Y j ≤ ε n a n } ≤ C 1 ζ (n + 1)(ε n a n ) 1−α a n → 0.

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 C Y the set of sequences {c i } i∈Z such that if Now both terms tend to 0 by Remark 2.7. Identical reasoning can be used in the proof of the "dual" condition in (55).

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 and that on the Skorokhod space 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].  (7) and (8). Consider an array {c n, j ; n ∈ N, j ∈ Z} of numbers such that for each n ∈ N j∈Z |c n, j | α h(|c n, j | −1 ) < +∞.
Set V n = j∈Z c n, j Y j , n ∈ N. Then if, and only if, j∈Z |c n, j | α h(|c n, j | −1 ) → 0, as n → ∞.
In the proofs, we shall need some estimates which seem to be a part of the probabilistic folklore.

Lemma 5.6 Assume that
where h(x) is slowly varying at x = ∞.
For α ∈ (1, 2) we have E X j = 0 and by (62) By (63)-(67) we obtain that j∈Z |c j | α h(|c j | −1 ) < +∞ if, and only if, all the assumptions of the Three Series Theorem are satisfied. Hence j∈Z c j Y j is a.s. convergent if, and only if, (56) holds.
Proof of Proposition 6.2 By Proposition 5.4, all random variables V n = j∈Z c n, j Y j are well-defined. Let us consider a decomposition of each V n into a sum of another three (convergent!) series: V n = j∈Z c n, j Y j I (|c n, j Y j | ≤ 1) − E c n, j Y j I (|c n, j Y j | ≤ 1) + j∈Z E c n, j Y j I (|c n, j Y j | ≤ 1) + j∈Z c n, j Y j I (|c n, j Y j | > 1) = V n,1 + V n,2 + V n, 3 .
We have proved the sufficiency part of Proposition 5.5.
To prove the "only if" part, we show first that V n −→ P 0 implies uniform infinitesimality of the coefficients, that is sup j∈Z |c n, j | → 0, as n → ∞.
Let {Ȳ j } be an independent copy of {Y j }. IfV n = j∈Z c n, jȲ j , then also V n − V n −→ P 0 and these are series of symmetric random variables. For each n select some arbitrary j n ∈ Z and consider decomposition into independent symmetric random variables V n −V n = c n, j n (Y j n −Ȳ j n ) + j∈Z, j = j n c n, j (Y j −Ȳ j ) = W n + W n .
Since {V n −V n } n∈N is uniformly tight, so is {W n } n∈N (it follows from the Lévy-Ottaviani inequality, see e.g., Proposition 1.1.1 in [18]). Since the law of Y j −Ȳ j is non-degenerate, we obtain sup n |c n, j n | < +∞.
If along some subsequence n , we would have c n , j n → c = 0, then for some θ ∈ R 1 lim n →∞ E e iθ W n = |E e iθcY | 2 < 1.
It follows that also lim n →∞ E e iθ(V n −V n ) = lim n →∞ E e iθ W n E e iθ W n < 1. This is in contradiction with V n −V n −→ P 0. Hence c = 0, c n, j n → 0 and since j n was chosen arbitrary, (68) follows. Now let us choose k n such that both | j|>k n c n, j Y j −→ P 0, as n → ∞, and | j|>k n P |c n, j Y j | > 1 → 0, as n → ∞.
Then {X n, j = c n, j Y j ; | j| ≤ k n , n ∈ N} is an infinitesimal array of row-wise independent random variables, with row sums convergent in probability to zero. Applying the general central limit theorem (see e.g., Theorem 5.15 in [15]), we obtain | j|≤k n P |X n, j | > 1 = | j|≤k n P |c n, j Y j | > 1 = | j|≤k n |c n, j | α h(|c n, j | −1 ) → 0.
This completes the Proof of Proposition 5.5.