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 $\alpha$-stable L\'evy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $\alpha \leq 1$. In the functional setting we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $S$ topology, introduced by Jakubowski (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, 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 6.1 below) that the series (1) defining the linear process X i is P-a.s. convergent if, and only if, 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 Section 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 Section 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 non-zero) 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 is applicable. This can be seen by analysis of the following simple example ( [2], p. 488). Set But we see that sup 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 welldefined, 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 S n (t) = 1 a n if, and only if, where Corollary 2.2 Under the assumptions of Theorem 2.1, define where c + = c ∨ 0, c − = (−c) ∨ 0, c ∈ R 1 , and set T n (t) = 1 a n 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 Further, we may decompose Let us consider the partial sum process: First we will show Lemma 2.3 Under the assumptions of Theorem 2.1 we have for each t > 0 In particular, Proof of Lemma 2.3 Define To prove that V 0 n −→ P 0 we apply Proposition 6.2. We have to show that ,j · |Y j | > a n → 0, as n → ∞.
Since a n → ∞ and 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 jn j=1 |a n,j | → 0.
Proof of Lemma 2.4 For each m ∈ N there exists N m > max{N m−1 , m 2 } such that for n ≥ N m m j=1 |a n,j | < 1 m .
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 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 (i) 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 E e iθS 0 n (t) → E e iθA·Z(t) , θ ∈ R 1 , and E e iθA·Z(t) = 0, θ ∈ R 1 (for Z(t) has infinitely divisible law), we conclude that E e iθ S − Thus S − n (t) + S + n (t) −→ P 0 and by Lemma 2.3 also S 0 Let us observe that by Proposition 6.2 (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 6.2 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" 3 Functional convergence 3.1 Convergence in the M 1 topology As outlined in Introduction (see also Section 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 non-negative coefficients. Let us recall the notation introduced in Corollary 2.2: Notice, that in general X ± i (ω) is not equal to X i (ω) ± and that we have 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 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 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 (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.
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).
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 1topology 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: if, and only if, for some dense subset Q ⊂ [0, 1] containing 0 and 1, and lim δ→0 lim sup In particular, if x n −→ M 1 x 0 , then for t = 1 and at every point of continuity of x 0 .  H(c, b, d).
The other cases can be reduced to the considered above. .
If η > 2β then 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. Hence 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 1 -compact is also relatively S-compact.
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 . (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.
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 f ∈ C[0, 1], The fact that (x n ) n≥1 cannot converge in M 1 follows by Proposition 3.9 since if t 1 < 1 2 − 1 n < t 2 < 1 2 + 1 n < t 3 , then H x n (t 1 ), x n (t 2 ), x n (t 3 ) = 1.

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.

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).
on the Skorokhod space D([0, 1]) equipped with the S topology.
Proof. By Corollary 2.2 where A + = i∈Z c + i and A − = i∈Z c − i . It follows from Proposition 3.1 that T + n −→ D A + · Z on 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 A · Z(t).
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 → ∞, j n = o(n), such that We will write −→ D(S) when convergence in distribution with respect to the S topology takes place.
It follows from that fact and (48) that sup j≤−jn Ψ 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 +∞ j=n+jn d n,j α a α n h a n d n,j → 0, as n → ∞, can be checked the same way.  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 Proof of Corollary 4.6 First notice that j |c j | < +∞ so that A is defined. Proceeding like in the proof of Corollary 4.1 we obtain −jn where the convergence to 0 holds by Lemma 4.2. Remark 4.8 Notice that if α < 1, then j |c j | α h(|c j | −1 ) < +∞, with h slowly varying, automatically implies j |c j | < +∞.

Corollary 4.9
Under the usual conditions, if α < 1, then if j∈Z |c j | α < +∞, and the coefficients c j are regular in a very weak sense: there exists a constant 0 < γ < α such that (with the convention that 0/0 ≡ 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). Ψ γ a n , a n d n,j , where Ψ γ (x, y) was defined in the proof of Corollary 4.1 and sup j≤−jn Ψ 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+jn d n,j α a α n h a n d n,j → 0, as n → ∞.
goes the same way. A · Z(t).

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

S-continuous functionals
A phenomenon of self-cancelling 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 Section 3.4. Here is another positive example of an S-continuous functional. Let µ be an atomless measure on [0, 1] and let h : R 1 → R 1 be a continuous function. Consider a smoothing operation s µ,h on D([0, 1]) given by the formula 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 = ℓ (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.
Proof of Corollary 5.1 The expression to be analyzed has the form where H β (x) = |x| β and by (26)

0.
We have checked in the course of the proof of Theorem 3.18, that {S n } is uniformly S-tight. By (3) {A · 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 · Z n } is uniformly S-tight. Now an application of Proposition 3.3 gives on the Skorokhod space D([0, 1]) equipped with the S topology.

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. (3) we obtain for t ∈ [k/n, (k + 1)/n) S n (t) = 1 a n k j=1 X j = 1 a n ζY k − ξY 0 + ζ − ξ)Z n ((k − 1)/n).
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 .
Notice that on G c n there are no two consecutive values of Y j exceeding ε n a n .
It follows that {S n (t)} are uniformly M 1 -tight if, and only if, { S n (t)} are. Let w M 1 (x, δ) be given by (34). Since P G c n → 1 we have for any δ > 0 and η > 0 lim sup And on G c n , by the property (54) and if 2/n < δ we have ω( S n (·), δ) ≥ 1 a n ζ − ξ max j Y n,j .

Linear space of convergent linear processes
We can explore the machinery of Section 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 A · Z(t), Proof of Proposition 5.
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 A · Z(t),

Appendix
We provide two results of a technical character. The first one is well-known ( [1]) and is stated here for completeness. Proposition 6.2 might be of independent interest.  (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.
where h(x) is slowly varying at x = ∞.
To get (62) we proceed similarly. First, by Karamata's Theorem Hence for some x 0 we have Since α > 1, we have E |Y | < +∞ and (62) follows. Proof of Proposition 6.1 We begin with specifying the conditions of the Kolmogorov Three Series Theorem in terms of our linear sequences. We have Applying (60) we obtain Similarly, if α ∈ (0, 1), then by (61) j∈Z E c j Y j I |c j Y j | ≤ 1 ≤ j∈Z |c j |E |Y j |I |Y j | ≤ 1/|c j | ≤ C 1 j∈Z |c j |(1/|c j |) 1−α h(|c j | −1 ) If α = 1, then by the symmetry we have E Y j I |Y j | ≤ a = 0, a > 0, and the series of truncated expectations trivially vanishes j∈Z E c j Y j I |c j Y j | ≤ 1 = 0.
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 6.1 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 6.2.
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,jȲj , 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,jn (Y jn −Ȳ jn ) + j∈Z,j =jn 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,jn | < +∞.
This is in contradiction with V n −V n −→ P 0. Hence c = 0, c n,jn → 0 and since j n was chosen arbitrary, (68) follows. Now let us choose k n such that both |j|>kn c n,j Y j −→ P 0, as n → ∞, and |j|>kn P |c n,j Y j | > 1 → 0, as n → ∞.
This completes the proof of Proposition 6.2.