On functional weak convergence for partial sum processes

For a strictly stationary sequence of regularly varying random variables we study functional weak convergence of partial sum processes in the space D[0, 1] with the Skorohod J1 topology. Under the strong mixing condition, we identify necessary and sufficient conditions for such convergence in terms of the corresponding extremal index. We also give conditions under which the regular variation property is a necessary condition for this functional convergence in the case of weak dependence.


Introduction
Let (X n ) be a strictly stationary sequence of real-valued random variables.If the sequence (X n ) is i.i.d. then it is well known (see for example Gnedenko and Kolmogorov [8], Rvačeva [16], Feller [7]) that there exist real sequences (a n ) and (b n ) such that for some non-degenerate α-stable random variable S with α ∈ (0, 2) if and only if X 1 is regularly varying with index α ∈ (0, 2), that is, where L( • ) is a slowly varying function at ∞ and the tails are balanced: there exist p, q ≥ 0 with p + q = 1 such that P(X 1 > x) as x → ∞.As α is less than 2, the variance of X 1 is infinite.The functional generalization of (1.1) has been studied extensively in probability literature.Define the partial sum processes where the sequences (a n ) and (b n ) are chosen as Here x represents the integer part of the real number x.In functional limit theory one investigates the asymptotic behavior of the processes V n ( • ) as n → ∞.Since the sample paths of V n ( • ) are elements of the space D[0, 1] of all right-continuous real valued functions on [0, 1] with left limits, it is natural to consider the weak convergence of distributions of V n ( • ) with the one of Skorohod topologies on D[0, 1] introduced in Skorohod [18].
A functional limit theorem for processes V n ( • ) for infinite variance i.i.d.regularly varying sequences (X n ) was established by Skorohod [19].Under some weak dependence conditions, weak convergence of partial sum processes in the Skorohod J 1 topology were obtained by Leadbetter and Rootzén [12] and Tyran-Kamińska [20].They give a characterization of the J 1 convergence in terms of convergence of the corresponding point processes of jumps.Further in [20] are given sufficient conditions for such convergence when the stationary sequence is strongly mixing.One of them is a certain local dependence condition, which is implied by the local dependence condition D of Davis [5].It prevents clustering of large values of |X n |, which allows the J 1 convergence to hold, since the J 1 topology is appropriate when extreme values do not cluster.
After recalling relevant notations and background in Section 2, in Section 3 we characterize the functional J 1 convergence of the partial sum process of a strictly stationary strongly mixing sequence (X n ) of regularly varying random variables in terms of the the extremal index of the sequence (|X n |), which is a standard tool in describing clustering of large values.When clustering of large values occurs J 1 convergence fails to hold, although convergence with respect to the weaker Skorohod M 1 topology might still hold.
Recently Basrak et al. [2] gave sufficient conditions for functional limit theorem with the M 1 topology to hold for stationary, jointly regularly varying sequences for which all extremes within each cluster of high-threshold excesses have the same sign.
The regular variation property is a necessary condition for the J 1 convergence of the partial sum process in the i.i.d.case (see for example Corollary 7.1 in Resnick [15]).In Section 4 we extend this result to the weak dependent case when clustering of large values do not occur.

Preliminaries
In this section we introduce some basic tools and notions to be used throughout the paper.

Regular Variation
the Radon measure µ on E being given by where p and q are as in (1.3).Using standard regular variation arguments it can be shown that for every λ > 0 it holds that Therefore a n can be represented as where L ( • ) is a slowly varying function at ∞.

Skorohod J 1 and M 1 topologies
Since the stochastic processes that we consider in this paper have discontinuities, for the function space of sample paths of these stochastic processes we take the space D[0, 1] of all right-continuous real valued functions on [0, 1] with left limits.Usually the space D[0, 1] is endowed with the Skorohod J 1 topology, which is appropriate when clustering of large values do not occur.
The metric d J1 that generates the J 1 topology on D[0, 1] is defined in the following way.Let ∆ be the set of strictly increasing continuous functions λ : [0, 1] → [0, 1] such that λ(0) = 0 and λ(1) = 1, and let e ∈ ∆ be the identity map on [0, 1], i.e. e(t) = t for all t ∈ [0, 1].For x, y ∈ D[0, 1] define When stochastic processes exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump, the J 1 topology become inappropriate since the J 1 convergence fails to hold.This difficulty can be overcame by using a weaker topology in which the functional convergence may still hold.i.e. the Skorohod M 1 topology.
The M 1 metric d M1 that generates the M 1 topology is defined using the completed graphs.For x ∈ D[0, 1] the completed graph of x is the set where x(t−) is the left limit of x at t. Thus the completed graph of x besides the points of the graph {(t, x(t)) : t ∈ [0, 1]} contains also the vertical line segments joining (t, x(t)) and (t, x(t−)) for all discontinuity points t of x.We define an order on the graph Γ x by saying that (t This definition introduces d M1 as a metric on D[0, 1].The induced topology is called the Skorohod M 1 topology.
The J 1 and M 1 metrics are related by the following inequality (see for instance Theorem 6.3.2 in Whitt [21]).
Nothing that we obtain lim inf n→∞ T (x n ) ≥ T (x).Therefore lim n→∞ T (x n ) = T (x), and we conclude that T is continuous at x.
Precisely, the functionals T + and T − are continuous on D[0, 1] when D[0, 1] is endowed with the Skorohod J 1 topology.This can be proven using a slight modification of the proof of Lemma 4.1 in Pang and Whitt [14] and the procedure used in the proof of Lemma 2.1 (we omit the details here).
Functional weak convergence

Weak dependence
A strictly stationary sequence (ξ n ) has extremal index θ if for every τ > 0 there exists a sequence of real numbers (u n ) such that (2.4) It holds that θ ∈ [0, 1].In particular, if the ξ n are i.i.d. then (2.4) can hold only for θ = 1.Dependent random variables can also have extremal index equal to 1.For this it suffices that they satisfy the extreme mixing conditions D(u n ) and D (u n ) introduced by Leadbetter [10], [11].The extremal index can be interpreted as the reciprocal mean cluster size of large exceedances (cf.Hsing et al. [9]).When θ < 1 clustering of extreme values occurs.If the sequence (ξ n ) is strongly mixing and the ξ n 's are regularly varying then for θ to be the extremal index of (ξ n ) it suffices that (2.4) holds for some τ > 0 (cf. Leadbetter and Rootzén [12], page 439).
In order to restrict the dependence in the sequence (X n ) we will use the strong mixing condition.Let (Ω, F, P) be a probability space.For any σ-field A ⊂ F, let L 2 (A) denote the space of square-integrable, A-measurable, real-valued random variables.For any two σ-fields Let (u n ) be a sequence of real numbers and (q n ) any sequence of positive integers with q n → ∞ as n → ∞ and q n = o(n).O'Brien [13] showed that if the sequence (X n ) is strongly mixing and there exists a sequence (p n ) of positive integers such that p n = o(n), nα qn = o(p n ), q n = o(p n ), and either lim inf[P(X where M i,j = max{X k : k = i, . . ., j}.
3 Limit theorem with J 1 convergence Let (X n ) be a strongly mixing and strictly stationary sequence of regularly varying random variables with index α ∈ (0, 2).Let (a n ) be a sequence of positive real numbers such that Tyran-Kamińska [20] showed that under a certain "vanishing small values" condition when α ∈ [1, 2) (see Condition 3.2 below), the partial sum process , t ∈ [0, 1], satisfies a nonstandard functional limit theorem in the space D[0, 1] equipped with the Skorohod J 1 topology, with a Lévy α-stable process as a limit if and only if the following local dependence condition holds: Condition 3.1.For any x > 0 there exist sequences of integers p n = p n (x), q n = q n (x) → ∞ such that and Here (α n ) is the sequence of α-mixing coefficients of (X n ).
We will show a similar result, but with a certain condition involving the extremal E[e izW (1) that is, µ is a Lévy measure.For a textbook treatment of Lévy processes we refer to Bertoin [3] and Sato [17].
In this section we identify some necessary and sufficient conditions for the J 1 convergence of partial sum processes V n ( • ) to a Lévy stable process.In case α ∈ [1, 2), we will need to assume that the contribution of the smaller increments of the partial sum process is close to its expectation.Condition 3.2.For all δ > 0, Theorem 3.3.Let (X n ) be a strictly stationary sequence of regularly varying random variables with index α ∈ (0, 2).Assume the sequence (X n ) is strongly mixing, and if 1 ≤ α < 2, also suppose that Condition 3.2 holds.Then in D[0, 1] endowed with the J 1 topology, where V ( • ) is an α-stable Lévy process with characteristic triplet (0, µ, 0) and µ as in (2.1), if and only if the sequence (|X n |) has extremal index θ = 1.
Proof.First assume the sequence (|X n |) has extremal index θ = 1.Let (q n ) be any sequence of positive integers such that q n → ∞ and q n = o(n).Fix an arbitrary x > 0 and put p n = max{ n √ α qn , √ nq n + 1}, where (α n ) is the sequence of α-mixing coefficients of (X n ).Then it can easily be seen that p n = o(n), nα qn = o(p n ) and q n = o(p n ).Since by a standard regular variation argument i.e. lim inf ) we obtain that, as n → ∞, where Then by Theorem 2.2.1 in Leadbetter and Rootzén [12] P max where

P max
1≤i≤n Therefore, from (3.4) and (3.6) we obtain, as n → ∞, and taking into account relation (3.3) it follows that P max Therefore (3.2) holds and an application of Theorem 1.1 in Tyran-Kamińska [20] yields with the J 1 topology.Since by Lemma 2.1 the functional T is continuous, by the continuous mapping theorem we obtain Following the Lévy-Ito representation of Lévy processes, V ( • ) can be represented as x µ(dx) , with N = k δ (t k ,j k ) being a Poisson process with mean measure λ × µ, where λ is the Lebesgue measure (see Resnick [15], page 150).Since T (V ) = sup{|j k | : t k ≤ 1}, we have for every x > 0, By Karamata's theorem, as n → ∞, where p and q are as in (2.
Hence (3.8) yields and an application of Slutsky's theorem (cf.Theorem 3.4 in Resnick [15]) leads to in D[0, 1] endowed with the J 1 topology.The characteristic triplet of the limiting process is therefore (0, µ, b).
Remark 3.5.The J 1 convergence in Theorem 3.3 fails to hold when the extremal index θ < 1.For example, the extremal index of the moving average process where (Y n ) is an i.i.d.sequence of regularly varying random variables, is equal to 1/2 (cf.Leadbetter and Rootzén [12] or Embrechts et al. [6], page 415).By Theorem 1 of Avram and Taqqu [1], the J 1 convergence does not hold for this process.
in D[0, 1] with the J 1 topology.Therefore, the functional J 1 convergence holds for the sequence (Y i ), where Y i = X i + Z.
Let show now that random variables Y i are not regularly varying.For large x > 0 it holds that P(Y 1 > x) ≥ P(Z > x).If we assume Y 1 is regularly varying, then it would hold P(Y 1 > x) = x −β L(x) for some β > 0 and some slowly varying function L( • ).Hence x −β L(x) ≥ (log x) −1 , i.e.
Letting x → ∞, we obtain 0 on the left hand side of this inequality (by Proposition 1.3.6 in Bingham et al. [4]) and ∞ on the right hand side, which is a contradiction.Therefore, Y 1 is not regularly varying.
Theorem 4.2.Let (X n ) be a strictly stationary sequence of random variables.Suppose that (|X n |) has extremal index θ = 1 and that the sequences (X ] endowed with the J 1 topology, where V ( • ) is an α-stable Lévy process with characteristic triplet (0, µ, 0), i.e.X 1 is regularly varying with index α.
Proof.Let x > 0 be arbitrary.As in the second part of the proof of Theorem 3.3, applying the functional T : In the same way, applying the functional T + to the convergence Here we used the fact that P(T + (V ) ≤ x) = P(N ([0, 1] × (x, ∞]) = 0) = e −px −α , with N being a Poisson process with mean measure λ × µ (as described in the proof of Theorem 3.3).Using Theorem 2.2.1 in Leadbetter and Rootzén [12], from (4.3) we obtain , where θ 1 is the extremal index of (X + n ) and ( X n ) is the associated independent sequence of (X n ).Now by Lemma 1.2.2 in Leadbetter and Rootzén [12] (cf.also Proposition 7.1 in Resnick [15]) it holds that nP(X + 1 > xa n ) → p θ 1 x −α as n → ∞.Repeating the same procedure for the functional T − we obtain where θ 2 is the extremal index of (X − n ).In the same manner from (4.2) we get On the other hand, and hence we conclude that p θ 1 + q θ 2 = 1. (4.6) Since θ 1 , θ 2 ∈ (0, 1] and p + q = 1, from relation (4.6) we obtain θ 1 = θ 2 = 1.Now, from (4.4) and (4.5) it follows (cf.Lemma 6.1 in Resnick [15]) as n → ∞.
Remark 4.3.If the random variables X n that appear in Theorem 4.2 are positive, then the conditions on extremal indexes of (|X n |), (X + n ) and (X − n ) reduce to a single condition, i.e. that (X n ) has extremal index equal to 1.The same holds if the X n are negative.
Remark 4.4.From the functional M 1 convergence of the partial sum process, using arguments as in the proof of Theorem 4.2, we can not obtain the regular variation property for random variables X n .This is due to the fact that the maximum jump functional T is not continuous on D[0, 1] with respect to the Skorohod M 1 topology (see the comment after Lemma 4.1 in Pang and Whitt [14]).
with r being the time component and u being the spatial component.Denote by Π(x) the set of parametric representations of the graph Γ

Remark 2 . 2 .
Similar results as in Lemma 2.1 hold for the maximum positive and negative jump functionals T + , T − : D[0, 1] → R defined by index of the sequence (|X n |) instead of Condition 3.1.Roughly speaking Condition 3.1 prevents clustering of large values of |X n |.In terms of the extremal index θ of the sequence (|X n |), the non-clustering of large values occurs when θ = 1.Hence it is expected that the functional J 1 convergence of the partial sum process holds if and only if the sequence (|X n |) has extremal index equal to 1, which we formally prove in the theorem below.Recall that the distribution of a Lévy process W ( • ) is characterized by its characteristic triplet, i.e. the characteristic triplet of the infinitely divisible distribution of W (1).The characteristic function of W (1) and the characteristic triplet (a, µ, b) are related in the following way:

Remark 3 . 4 .
i | ≤ x → e −x −α .Taking x = 1, we obtainlim n→∞ P n i=1 |X i | ≤ a n → e −1 ,and from this, taking into account relation (3.1) we conclude that (|X n |) has extremal index equal to 1.If random variables X n are regularly varying with index α ∈ (0, 1), the centering function in the definition of the process V n ( • ) can be removed and this removing affects the characteristic triplet of the limiting process in the way we describe here.
1).Put b = (p − q)α/(1 − α) and define the function x 0 : [0, 1] → R by x 0 (t) = bt.The function x 0 is continuous, and hence it belongs to D[0, 1].Further, by standard arguments one can show that the function h :D[0, 1] → D[0, 1] defined by h(x) = x + x 0 is continuous (with respect to the J 1 topology on D[0, 1]).Hence by the continuous mapping theorem from V n d − → V we obtain h(V n ) d − → h(V ).Since the J 1 metric on D[0, 1] is bounded above by the uniform metric on D[0, 1], for every δ > 0 it holds The space E is equipped with the topology by which the Borel σ-algebras B(E) and B(R) coincide on R \ {0}.A set B ⊆ E is relatively compact if it is bounded away from origin, that is, if there exists u > 0 such that B ⊆ E \ [−u, u].Let M + (E)be the class of all Radon measures on E, i.e. all nonnegative measures that are finite on relatively compact subsets of E. A useful topology for M + (E) is the vague topology which renders M + (E) a complete separable metric space.If µ n ∈ M + (E), n ≥ 0, then µ n converges vaguely to µ 0 (written µ n