Sums of extreme values of subordinated long-range dependent sequences: moving averages with finite variance

In this paper we characterize the limiting behavior of sums of extreme values of long range dependent sequences defined as functionals of linear processes with finite variance. The extremal sums behave completely different by compared to the i.i.d case. In particular, though we still have asymptotic normality, the scaling factor is relatively bigger than in the i.i.d case, meaning that the maximal terms have relatively smaller contribution to the whole sum. Also, the scaling need not depend on the tail index of the underlying marginal distribution, as it is well-known to be so in the i.i.d. situation. Furthermore, subordination may completely change the asymptotic properties of sums of extremes.


Introduction
Let {ǫ i , i ≥ 1} be a centered sequence of i.i.d. random variables. Consider the class of stationary linear processes We assume that the sequence c k , k ≥ 0, is regularly varying with index −β, β ∈ (1/2, 1). This means that c k ∼ k −β L 0 (k) as k → ∞, where L 0 is slowly varying at infinity. We shall refer to all such models as long range dependent (LRD) linear processes. In particular, if the variance exists (which is assumed throughout the whole paper), then the covariances ρ k := EX 0 X k decay at the hyperbolic rate, ρ k = k −(2β−1) L(k), where lim k→∞ L(k)/L 2 0 (k) = B(2β − 1, 1 − β) and B(·, ·) is the beta-function. Consequently, the covariances are not summable (cf. [11]).
Assume that X 1 has a continuous distribution function F . For y ∈ (0, 1) define Q(y) = inf{x : F (x) ≥ y} = inf{x : F (x) = y}, the corresponding (continuous) quantile function. Given the ordered sample X 1:n ≤ · · · ≤ X n:n of X 1 , . . . , X n , let F n (x) = n −1 n i=1 1 {X i ≤x} be the empirical distribution function and Q n (·) be the corresponding left-continuous sample quantile function, i.e. Q n (y) = X k:n for k−1 n < y ≤ k n . Define U i = F (X i ) and E n (x) = n −1 n i=1 1 {U i ≤x} , the associated uniform empirical distribution. Denote by U n (·) the corresponding uniform sample quantile function.
The aim of this paper is to study the asymptotic behavior of trimmed sums based on the ordered sample X 1:n ≤ · · · ≤ X n:n coming from the long range dependent sequence defined by (1). Let T n (m, k) = n−k i=m+1 X i:n and note that (see below for a convention concerning integrals) Ho and Hsing observed in [14] that, under appropriate conditions on F , as n → ∞, where 0 < y 0 < y 1 < 1. Equation (4) means that, in principle, the quantile process can be approximated by partial sums, independently of y. This observation, together with (3), yields the asymptotic normality of the trimmed sums in case of heavy trimming m = m n = [δ 1 n], k = k n = [δ 2 n], where 0 < δ 1 < δ 2 < 1 and [·] is the integer part (see [14,Corollary 5.2]). This agrees with the i.i.d. situation (see [22]). However, the representation (3) requires some additional assumptions on F . In order to avoid them, we may study asymptotics for the trimmed sums via the integrals of the form α n (y)dQ(y). This approach was initiated in two beautiful papers by M. Csörgő, S. Csörgő, Horváth and Mason, [2], [3]. Then, S. Csörgő, Haeusler, Horváth and Mason took this route to provide the full description of the weak asymptotic behavior of the trimmed sums in the i.i.d. case. The list of the papers written by these authors on this particular topic is just about as long as this introduction. Therefore we refer to [7] for an extensive up-to-date discussion and a survey of results.
In the LRD case, instead of using the Brownian bridge approximation, we can use the reduction principle for the general empirical processes as studied in [11], [14], [16] or [24] (see Lemma 9 below). We can then use an approach that is similar to that the above mentioned authors to establish asymptotic normality in case of light, moderate and heavy trimming with the scaling factor σ −1 n,1 , which is the same as for the whole partial sum. So, in this context the situation is similar to the i.i.d. case and for details we refer the reader to the technical report [17].
The most interesting phenomena, however, occur when one deals with the k n -extreme sums, n i=n−kn+1 X i . If F (0) = 0 and 1 − F (x) = x −α , α > 2, then in the i.i.d situation we have a n n i=n−kn+1 where the scaling factor is a n = nk −1 n 1/2−1/α n −1/2 , c n is a centering sequence and Z is a standard normal random variable (see [9]). In the LRD case we still obtain asymptotic normality. However, although the Ho and Hsing result (4) does not say anything about the behavior of the quantile process in the neighborhood of 0 and 1, the somewhat imprecise statement that the quantile process can be approximated by partial sums, independently of y suggests that • a required scaling factor would not depend on the tail index α.
Indeed, we will show in Theorem 1 that the appropriate scaling in case Removing the scaling for the whole sums (n −1/2 and σ −1 n,1 in the i.i.d. and LRD cases, respectively), we also see that • the scaling in the LRD situation is greater, meaning that the k nextreme sums contribute relatively less to the whole sum compared to the i.i.d situation. This also is quite intuitive. Since the dependence is very strong, it is very unlikely that we have few big observations, which is a typical case in the i.i.d. situation. Rather, if we have one big value, we have a lot of them.
One may ask, whether such phenomena are typical for all LRD sequences. Not likely. Define Y i = G(X i ), i ≥ 1, with some real-valued measurable function G. In particular, taking G = F −1 Y F we may obtain a LRD sequence with the arbitrary marginal distribution function F Y . Assume for a while that F , the distribution of X 1 , is standard normal and that q n (·) is the quantile process associated with the sequence {Y i , i ≥ 1}. Following [6] we observed in [4, Section 2.2] and [5] that q n (·) is, up to a constant, approxi- Here, f Y is the density of F Y and φ, Φ are the standard normal density and distribution, respectively. In the non-subordinated case, Y i = X i , and the factor φ(Φ −1 (y))/f Y (F −1 Y (y)) disappears. Nevertheless, from this discussion it should be clear that the limiting behavior of the extreme sums in the subordinated case Y i = G(X i ) is different, namely (see Theorem 1) • the scaling depends on the marginal distributions of both X i and Y i .
In particular, if the distribution F of X 1 belongs to the maximal domain of attraction of the Fréchet distribution Φ α , then though the distribution F Y of Y 1 belongs to the maximal domain of attraction of the Gumbel distribution, the scaling factor depends on α. This cannot happen in the i.i.d. situation and, intuitively, it means that in the subordinated case the long range dependent sequence {X i , i ≥ 1} also contributes information to the asymptotic behavior of extreme sums.
Moreover, we may have two LRD sequences (1), with the same covariance, with the same marginals, but completely different behavior of extremal terms.
Of course, it would be desirable to obtain some information about limiting behaviour not only of extreme sums, but for sample maxima as well. It should be pointed out that our method is not appropriate. This is still an open problem to derive limiting behaviour of maxima in the model (1). In a different setting, the case of stationary stable processes generated by conservative flow, the problem is treated in [20].
We will use the following convention concerning integrals. If −∞ < a < b < ∞ and h, g are left-continuous and right-continuous functions, respectively, then hdg, whenever these integrals make sense as Lebesgue-Stjeltjes integrals. The integration by parts formula yields We shall write g ∈ RV α (g ∈ SV ) if g is regularly varying at infinity with index α (slowly varying at infinity).
In what follows C will denote a generic constant which may be different at each of its appearances. Also, for any sequences a n and b n , we write a n ∼ b n if lim n→∞ a n /b n = 1. Further, let ℓ(n) be a slowly varying function, possibly different at each place it appears. On the other hand, L(·), L 0 (·), L 1 (·), L * 1 (·), etc., are slowly varying functions, fixed form the time they appear. Moreover, g (k) denotes the kth order derivative of a function g and Z is a standard normal random variable. For any stationary sequence {V i , i ≥ 1}, we will denote by V the random variable with the same distribution as V 1 .

Statement of results
Let F ǫ be the marginal distribution function of the centered i.i.d. sequence {ǫ i , i ≥ 1}. Also, for a given integer p, the derivatives F of F ǫ are assumed to be bounded and integrable. Note that these properties are inherited by the distribution function F of X 1 as well (cf. [14] or [24]). Furthermore, assume that Eǫ 4 1 < ∞. These conditions are needed to establish the reduction principle for the empirical process and will be assumed throughout the paper.
To study sums of k n largest observations, we shall consider the following forms of F . For the statements below concerning regular variation and domain of attractions we refer to [12], [10,Chapter 3] or [15].
The first assumption is that the distribution F satisfies the following Von-Mises condition: Using notation from [10], the condition (5) will be referred as X ∈ M DA(Φ α ), since (5) implies that X belongs to the maximal domain of attraction of the Fréchet distribution with index α. Then and the density-quantile function f Q(y) = f (Q(y)) satisfies where The second type of assumption is that F belongs to the maximal domain of attraction of the double exponential Gumbel distribution, written as X ∈ M DA(Λ). Then the corresponding Von-Mises condition implies and L 3 is slowly varying at infinity.
To study the effect of subordination, we will consider the corresponding assumptions on F Y , referred to later as Y ∈ M DA(Φ α 0 ) and Y ∈ M DA(Λ), respectively: The main result of this paper is the following theorem.
Assume that EY < ∞. Let p be the smallest positive integer such that (p + 1)(2β − 1) > 1 and assume that for r = 1, . . . , p, Let where L 21 , L 22 , L 23 , L 24 are slowly varying functions to be specified later on. Then The corresponding cases concerning assumptions on X and Y will be referred as Case 1, Case 2, Case 3 and Case 4.

Corollary 2 Under the conditions of Theorem
In the subordinated case we have chosen to work with G = Q Y F to illustrate phenomena rather then deal with technicalities. One could work with general functions G, but then one would need to assume that G has the power rank 1 (see [14] for the definition). Otherwise the scaling σ −1 n,1 is not correct. To see that G(·) = Q Y F (·) has the power rank 1, note that for Substituting x = 0 and changing variables y = F (t) we obtain Furthermore, we must assume that the distribution of Y = G(X) belongs to the appropriate domain of attraction. For example, if X ∈ M DA(Φ α ) and Y i = X ρ i , ρ > 0, then Y ∈ M DA(Φ α/ρ ), provided that the map x → x ρ is increasing on IR. Otherwise, if for example ρ = 2, one needs to impose conditions not only on the right tail of X, but on the left one as well.

Remarks
Remark 4 From the beginning we assumed that Eǫ 4 1 < ∞, thus, in Cases 1 and 2 we have the requirement α ≥ 4 and this is the only constrain on this parameter. Condition EY < ∞ requires α 0 > 1 in case of Y ∈ M DA(Φ α 0 ). In view of (*), to be able to choose ξ < 1 we need to have α 0 > (1−β) −1 > 2. The same restriction comes in Case 3.

Remark 5
The conditions (*)-(****) on ξ are somehow restrictive. They come form the quality of the rates in the reduction principle for the empirical processes.

Remark 6
Appropriate results concerning the law of the iterated logarithm for the extreme sums can be also stated, at least in the case of Y ∈ M DA(Φ α 0 ), by replacing σ −1 n,1 in Theorems 1 with σ −1 n,1 (log log n) −1/2 . In view of [13], the most interesting phenomena occur if k n is small (k n = o(log log n) in the i.i.d. case). This, in view of the previous remark, cannot be treated in our situation at all.

Remark 7
The conditions D r := 1 1/2 F (r) (Q(y))/f Y Q Y (y)dy < ∞ are not restrictive at all, since they are fulfilled for most distributions with a regularly varying density-quantile function f Q(1 − y), for those we refer to [19]. Consider for example Case 1, and assume that the density f is nonincreasing on some interval [x 0 , ∞). Then F (r) is regularly varying at infinity with index r + α. Thus, for some x 1 > x 0 (CsR1) f exists on (a, b), where a = sup{x : then in view of (CsR2) and the assumed boundness of derivatives F (r) (·), the integral D r is finite.

Remark 8
In the proof of Theorem 1 we have to work with both Q(·) and f Q(·). Therefore, we assumed the Von-Mises condition (5) since it implies both (6) and (7). If one assumes only (6), then (5) and, consequently, (7) hold, provided a monotonicity of f is assumed. Moreover, the von-Mises condition is natural, since the existence of the density f is explicitly assumed.

Consequences of the reduction principle
Let p be a positive integer and let where F (r) is the rth order derivative of F . Setting U i = F (X i ) and x = Q(y) in the definition of S n (·), we arrive at its uniform version, We shall need the following lemma, referred to as the reduction principle.
We have by Karamata's Theorem: In Case 1, In Case 2, In Case 3, In Case 4, Thus, in either case, A n K n ∼ 1. Thus, the result follows by noting that U n (1 − k n /n) = U n−kn:n . ⊙ An easy consequence of (11) is the following result.

Proof of Theorem 1
To obtain the limiting behavior of sums of extremes, we shall use the following decomposition: Since E n (·) has no jumps after U n:n and Y j = We will show that I 1 yields the asymptotic normality. Further, we will show that the latter two integrals are asymptotically negligible. Each term will be treated in a separate section. Let p be the smallest integer such that (p + 1)(2β − 1) > 1, so that d n,p = n −(1−β) ℓ(n).

Second term
We have Since EJ 1 = J 2 , it suffices to show that J 2 = o(1).

Case 1: We have by Karamata's Theorem
which converges to 0 using the assumption (*).