MAXIMA OF LONG MEMORY STATIONARY SYMMETRIC α-stable PROCESSES, AND SELF-SIMILAR PROCESSES WITH STATIONARY MAX-INCREMENTS. 1.

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1 MAXIMA OF LONG MEMORY STATIONARY SYMMETRIC -STABLE PROCESSES, AND SELF-SIMILAR PROCESSES WITH STATIONARY MAX-INCREMENTS TAKASHI OWADA AND GENNADY SAMORODNITSKY Abstract. We derive a functional limit theorem for the partial maxima process based on a long memory stationary -stable process. The length of memory in the stable process is parameterized by a certain ergodic theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with -Fréchet marginals, and it has the property of stationary max-increments, which we introduce in this paper. The functional limit theorem is established in the space D[0, equipped with the Skorohod M 1- topology; in certain special cases the topology can be strengthened to the Skorohod J 1-topology. 1. Introduction The asymptotic behavior of the partial maxima sequence M n = max 1 k n X k, n = 1, 2,... for an i.i.d. sequence X 1, X 2,... of random variables is the subect of the classical extreme value theory, dating back to Fisher and Tippett The basic result of this theory says that only three one-dimensional distributions, the Fréchet distribution, the Weibull distribution and the Gumbel distribution, have a max-domain of attraction. If Y has one of these three distributions, then for a distribution in its domain of attraction, and a sequence of i.i.d. distribution, random variables with that M n b n 1.1 Y a n for properly chosen sequences a n, b n ; see e.g. Chapter 1 in Resnick 1987 or Section 1.2 in de Haan and Ferreira Under the same max-domain of attraction assumption, a functional version of 1.1 was established in Lamperti 1964: with the same sequences a n, b n as in 1.1, M nt b n 1.2, t 0 Y t, t 0 a n for a nondecreasing right continuous process Y t, t 0, and the convergence is weak convergence in the Skorohod J 1 topology on D[0,. The limiting process is often called the extremal process; its properties were established in Dwass 1964, 1966 and Resnick and Rubinovitch Mathematics Subect Classification. Primary 60F17, 60G70. Secondary 37A40, 60G10, 60G52. Key words and phrases. Stable process, extreme value theory, conservative flow, pointwise dual ergodicity, sample maxima. This research was partially supported by the ARO grants W911NF and W911NF , NSF grant DMS and NSA grant H at Cornell University. 1

2 2 TAKASHI OWADA AND GENNADY SAMORODNITSKY Much of the more recent research in extreme value theory concentrated on the case when the underlying sequence X 1, X 2,... is stationary, but may be dependent. In this case the extrema of the sequence may cluster, and it is natural to expect that the limiting results 1.1 and 1.2 will, in general, have to be different. The extremes of moving average processes have received special attention; see e.g. Rootzén 1978, Davis and Resnick 1985 and Fasen The extremes of the GARCH1,1 process were investigated in Mikosch and Stărică The classical work on this subect of the extremes of dependent sequences is Leadbetter et al. 1983; in some cases this clustering of the extremes can be characterized through the extremal index introduced, originally, in Leadbetter The latter is a number 0 θ 1. Suppose that a stationary sequence X 1, X 2,... has this index, and let X 1, X 2,... be an i.i.d. sequence with the same onedimensional marginal distributions as X 1, X 2,.... If 1.1 and 1.2 hold for the i.i.d. sequence, then the corresponding limits will satisfy Ỹ = d Ỹ 1, but the limit in 1.1 for the dependent sequence X 1, X 2,... will satisfy Y = d Ỹ θ. In particular, the limit will be equal to zero if the extremal index is equal to zero. This case can be viewed as that of long range dependence in the extremes, and it has been mostly neglected by the extreme value community. Long range dependence is, however, an important phenomenon in its own right, and in this paper we take a step towards understanding how long range dependence affects extremes. A random variable X is said to have a regularly varying tail with index for > 0 if P X > x = x Lx, x > 0, where L is a slowly varying at infinity function, and the distribution of any such random variable is in the max-domain of attraction of the Fréchet distribution with the same parameter ; see e.g. Resnick Recall that the Fréchet law F,σ on 0, with the tail index and scale σ > 0 satisfies 1.3 F,σ x = exp { σ x }, x > 0. Sometimes the term -Fréchet is used. In this paper we discuss the case of regularly varying tails and the resulting limits in 1.2. The limits obtained in this paper belong to the family of the so-called Fréchet processes, defined below. We would like to emphasize that, even for stationary sequences with regularly varying tails, non-fréchet limits may appear in 1.2. We are postponing a detailed discussion of this point to a future publication. A stochastic process Y t, t T on an arbitrary parameter space T is called a Fréchet process if for all n 1, a 1,..., a n > 0 and t 1,..., t n T, the weighted maximum max 1 n a Y t follows a Fréchet law as in 1.3. The best known Fréchet process is the extremal Fréchet process obtained in the scheme 1.2 starting with an i.i.d. sequence with regularly varying tails. The extremal

3 LIMIT THEORY FOR PARTIAL MAXIMA 3 Fréchet process Y t, t 0 has finite dimensional distributions defined by 1.4 Y t1, Y t 2,..., Y t n d = X 1, max X 1 max X 1,t 1/ 1,t 1/ 1,t 1/ 1, X 2,t 2 t 1 1/,..., X n,t n t n 1 1/, X 2,t 2 t 1 1/,... for all n and 0 t 1 < t 2 :... < t n. The different random variables in the right hand side of 1.4 are independent, with X k,σ having the Fréchet law F,σ in 1.3, for any k = 1,..., n. This independence makes the extremal Fréchet processes the extremal Fréchet versions of the better known Lévy processes. The structure of general Fréchet processes has been extensively studied in the last several years. These processes were introduced in Stoev and Taqqu 2005, and their representations as a part of a much more general context were studied in Kabluchko and Stoev Stationary Fréchet processes in particular, their ergodicity and mixing were discussed in Stoev 2008, Kabluchko et al and Wang and Stoev In this paper we concentrate on the maxima of stationary -stable processes with 0 < < 2. Recall that a random vector X in R d is called -stable if for any A and B > 0 we have AX 1 + BX 2 d = A + B 1/ X + y, where X 1 and X 2 are i.i.d. copies of X, and y is a deterministic vector unless X is deterministic, necessarily, 0 < 2. A stochastic process Xt, t T is called -stable if all of its finitedimensional distributions are -stable. We refer the reader to Samorodnitsky and Taqqu 1994 for information on -stable processes. When = 2, an -stable process is Gaussian, while in the case 0 < < 2, both the left and the right tails of a nondegenerate -stable random variable X are generally regularly varying with exponent. That is, P X > x c + x, P X < x c x as x for some c +, c 0, c + + c > 0. That is, if X 1, X 2,... is an i.i.d. sequence of -stable random variables, then the i.i.d. sequence X 1, X 2,... satisfies 1.1 and 1.2 with a n = n 1/ and b n = 0, n 1. Of course, we are not planning to study the extrema of an i.i.d. -stable sequence. Instead, we will study the maxima of the absolute values of a stationary -stable process. The reason we have chosen to work with stationary -stable processes is that their structure is very rich, and is also relatively well understood. This will allow us to study the effect of that structure on the limit theorems 1.1 and 1.2. We are specifically interested in the long range dependent case, corresponding to the zero value of the extremal index. The structure of stationary symmetric -stable SS processes has been clarified in the last several years in the works of Jan Rosiński; see e.g. Rosiński 1995, The integral representation of such a process can be chosen to have a very special form. The class of stationary SS processes

4 4 TAKASHI OWADA AND GENNADY SAMORODNITSKY we will investigate requires a representation slightly more restrictive than the one generally allowed. Specifically, we will consider processes of the form 1.5 X n = f T n x dmx, n = 1, 2,..., E where M is a SS random measure on a measurable space E, E with a σ-finite infinite control measure µ. The map T : E E is a measurable map that preserves the measure µ. Further, f L µ. See Samorodnitsky and Taqqu 1994 for details on -stable random measures and integrals with respect to these measures. It is elementary to check that a process with a representation 1.5 is, automatically, stationary. Recall that any stationary SS process has a representation of the form: 1.6 X n = with 1.7 f n x = a n x E f n xdmx, n = 1, 2,..., dµ T n 1/ dµ x f T n x, x E for n = 1, 2,..., where T : E E is a one-to-one map with both T and T 1 measurable, mapping the control measure µ into an equivalent measure, and a n x = n 1 =0 u T x, x E for n = 1, 2,..., with u : E { 1, 1} a measurable function. Here M is SS and f L µ. See Rosiński The main restriction of 1.5 is, therefore, to a measure preserving map T. It has been observed that the ergodic-theoretical properties of the map T, either in 1.5 or in 1.7, have a maor impact on the memory of a stationary -stable process. See e.g. Surgailis et al. 1993, Samorodnitsky 2004, 2005, Roy 2008, Resnick and Samorodnitsky 2004, Owada and Samorodnitsky 2012, Owada The most relevant for this work is the result of Samorodnitsky 2004, who proved that, if the map T in 1.5 or in 1.7 is conservative, then using the normalization a n = n 1/ b n = 0 in 1.1, as indicated by the marginal tails, produces the zero limit, so the partial maxima grow, in this case, strictly slower than at the rate of n 1/. On the other hand, if the map T is not conservative, then the normalization a n = n 1/ in 1.1 is the correct one, and it leads to a Fréchet limit we will survey the ergodic-theoretical notions in the next section. Therefore, the extrema of SS processes corresponding to conservative flows cluster so much that the sequence of the partial maxima grows at a slower rate than that indicated by the marginal tails. This case can be thought of as indicating long range dependence. It is, clearly, inconsistent with a positive extremal index.

5 LIMIT THEORY FOR PARTIAL MAXIMA 5 The Fréchet limit obtained in 1.1 by Samorodnitsky 2004 remains valid when the map T is conservative but with the normalization of a smaller order than n 1/, as long as the map T satisfies a certain additional assumption. If one views the stationary -stable process as a natural function of the Poisson points forming the random measure M in 1.6 then, informally, this assumption guarantees that only the largest Poisson point contributes, distributionally, to the asymptotic behavior of the partial maxima of the process. In this paper we restrict ourselves to this situation as well. However, we will look at the limits obtained in the much more informative functional scheme 1.2. In this paper the assumption on the map T will be expressed in terms of the rate of growth of the so-called return sequence, which we define in the sequel. We would like to emphasize that, when this return sequence grows at a rate slower than the one assumed in this paper, new phenomena seem to arise. Multiple Poisson points may contribute to the asymptotic distribution of the partial maxima, and non-fréchet limit may appear in 1.2. We leave a detailed study of this to a subsequent work. In the next section we provide the elements of the infinite ergodic theory needed for the rest of the paper. In Section 3 we introduce a new notion, that of a process with stationary max-increments. It turns out that the possible limits in the functional maxima scheme 1.2 with b n = 0 are selfsimilar with stationary max-increments. We discuss the general properties of such processes and then specialize to the concrete limiting process we obtain in the main result of the paper, stated and proved in Section Ergodic Theoretical Notions In this section we present some basic notation and notions of, mostly infinite, ergodic theory used in the sequel. The main references are Krengel 1985, Aaronson 1997, and Zweimüller Let E, E, µ be a σ-finite, infinite measure space. We will say that A = B mod µ if A, B E and µa B = 0. For f L 1 µ we will often write µf for the integral f dµ. Let T : E E be a measurable map preserving the measure µ. The sequence T n of iterates of T is called a flow, and the ergodic-theoretical properties of the map and the flow are identified. A map T is called ergodic if any T -invariant set A i.e., a set such that T 1 A = A mod µ satisfies µa = 0 or µa c = 0. A map T is said to be conservative if 1 A T n = a.e. on A n=1 for any A E, 0 < µa < ; if T is also ergodic, then the restriction on A is not needed.

6 6 TAKASHI OWADA AND GENNADY SAMORODNITSKY The conservative part of a measure-preserving T is the largest T -invariant subset C of E such that the restriction of T to C is conservative. The set D = E \ C is the dissipative part of T and the decomposition E = C D is called the Hopf decomposition of T. The dual operator T : L 1 µ L 1 µ is defined by d ν f T 2.1 T 1 f =, f L 1 µ, dµ where ν f is the signed measure ν f A = A fdµ, A E. The dual operator satisfies the duality relation 2.2 E T f g dµ = E f g T dµ for f L 1 µ, g L µ. Note that 2.1 makes sense for any nonnegative measurable function f on E, and the resulting T f is again a nonnegative measurable function. Furthermore, 2.2 holds for arbitrary nonnegative measurable functions f and g. A conservative, ergodic and measure preserving map T is said to be pointwise dual ergodic, if there exists a normalizing sequence a n such that a n k=1 n T k f µf a.e. for every f L 1 µ. The property of pointwise dual ergodicity rules out invertibility of the map T. Sometimes we require that for some functions the above convergence takes place uniformly on a certain set. A set A E with 0 < µa < is said to be a uniform set for a conservative, ergodic and measure preserving map T, if there exist a normalizing sequence a n and a nontrivial nonnegative measurable function f L 1 µ nontriviality means that f is different from zero on a set of positive measure such that a n k=1 n T k f µf uniformly, a.e. on A. If 2.4 holds for f = 1 A, the set A is called a Darling-Kac set. A conservative, ergodic and measure preserving map T is pointwise dual ergodic if and only if T admits a uniform set; see Proposition in Aaronson In particular, it is legitimate to use the same normalizing sequence a n both in 2.3 and 2.4. Let A E with 0 < µa <. T n x, x E, is naturally related to the wandering rate sequence n w n = µ T k A. The frequency of visits to the set A along the traectory k=0

7 If we define the first entrance time to A by LIMIT THEORY FOR PARTIAL MAXIMA 7 ϕ A x = min{n 1 : T n x A} notice that ϕ A < a.e. on E since T is conservative and ergodic, then w n µϕ A < n as n. Since T is also measure preserving, we have µa {ϕ A > k} = µa c {ϕ A = k} for k 1. Therefore, alternative expressions for the wandering rate sequence are n 1 w n = µa + µa c {ϕ A = k} = k=1 n 1 µa {ϕ A > k}. Suppose now that T is a pointwise dual ergodic map, and let A be a uniform set for T. turns out that, under an assumption of regular variation, there is a precise connection between the wandering rate sequence w n and the normalizing sequence a n in 2.3 and 2.4. Specifically, let RV γ represent the class of regularly varying at infinity sequences or functions, depending on the context of index γ. If either w n RV β or a n RV 1 β for some β [0, 1], then 1 n 2.6 a n as n. Γ2 βγ1 + β w n Proposition in Aaronson 1997 gives one direction of this statement, but the argument is easily reversed. k=0 It We finish this section with a statement on distributional convergence of the partial maxima for pointwise dual ergodic flows. It will be used repeatedly in the proof of the main theorem. For a measurable function f on E define M n fx = max 1 k n f T k x, x E, n 1. In the sequel we will use the convention max k K b k = 0 for a nonnegative sequence b n, if K =. Proposition 2.1. Let T be a pointwise dual ergodic map on a σ-finite, infinite, measure space E, E, µ. We assume that the normalizing sequence a n is regularly varying with exponent 1 β for some 0 < β 1. Let A E, 0 < µa <, be a uniform set for T. Define a probability measure on E by µ n = µ {ϕ A n}/µ{ϕ A n}. Let f : E R be a measurable bounded function supported by the set A. Let f = inf{m : fx M a.e. on A}. Then M nt f, 0 t 1 f 1 {Vβ t}, 0 t 1 in the M 1 topology on D[0, 1], where the law of the left hand side is computed with respect to µ n, and V β is a random variable defined on a probability space Ω, F, P with P V β x = x β, 0 < x 1.

8 8 TAKASHI OWADA AND GENNADY SAMORODNITSKY Proof. For 0 < ε < 1 let A ε = {x A : fx 1 ε f }. Note that each A ε is uniform since A is uniform. Clearly, 1 ε f 1 {ϕaε x nt} M nt fx f 1 {ϕa x nt} µ-a.e. for all n 1 and 0 t 1. Since for monotone functions weak convergence in the M 1 topology is implied by convergence in finite dimensional distributions, we can use Theorem 3.2 in Billingsley 1999 in a finite-dimensional situation. The statement of the proposition will follow once we show that, for a uniform set B which could be either A or A ε the law of ϕ B /n under µ n converges to the law of V β. Let w n B be the corresponding wandering rate sequence. Since 2.6 holds for w n B with the same normalizing constants a n, ϕb µ n n x = µϕ B nx wb nx x β µϕ A n w n for all 0 < x 1, because the wandering rate sequence is regularly varying with index β by Self-similar processes with stationary max-increments The limiting process obtained in the next section shares with any possible limits in the functional maxima scheme 1.2 with b n = 0 two very specific properties, one of which is classical, and the other is less so. Recall that a stochastic process Y t, t 0 is called self-similar with exponent H of self-similarity if for any c > 0 Y ct, t 0 d= c H Y t, t 0 in the sense of equality of all finite-dimensional distributions. The best known classes of self-similar processes arise in various versions of a functional central limit theorem for stationary processes, and they have an additional property of stationary increments. Recall that a stochastic process Y t, t 0 is said to have stationary increments if for any r Y t + r Y r, t 0 d= Y t Y 0, t 0 ; see e.g. Embrechts and Maeima 2002 and Samorodnitsky In the context of the functional limit theorem for the maxima 1.2, a different property appears. Definition 3.1. A stochastic process Y t, t 0 is said to have stationary max-increments if for every r 0, there exists, perhaps on an enlarged probability space, a stochastic process Y r t, t 0 such that Y r t, t 0 d = Y t, t 0, 3.2 Y t + r, t 0 d = Y r Y r t, t 0.

9 LIMIT THEORY FOR PARTIAL MAXIMA 9 Notice the analogy between the definition 3.1 of stationary increments when Y 0 = 0 and Definition 3.1. Since the operations of taking the maximum is not invertible unlike summation, the latter definition, by necessity, is stated in terms of existence of the max-increment process Y r t, t 0. Theorem 3.2. Let X 1, X 2,... be a stationary sequence. Assume that for some sequence a n, and a stochastic process Y t, t 0 such that, for every t > 0, Y t is not a constant, 1 M a nt, t 0 Y t, t 0 n in terms of convergence of finite-dimensional distributions. Then Y t, t 0 is self-similar with exponent H > 0 of self-similarity, and has stationary max-increments. Furthermore, the sequence a n is regularly varying with index H. Proof. The facts that the limiting process Y t, t 0 is self-similar with exponent H > 0 of selfsimilarity, and that the sequence a n is regularly varying with index H, follow from the Lamperti theorem; see Lamperti 1962, or Theorem in Embrechts and Maeima Note that the distinction between convergence along a discrete sequence n, or along a continuous sequence λ, disappears for maxima of stationary processes. We check now the stationarity of the max-increments of the limiting process. Let r > 0, and t i > 0, i = 1,..., k, some k 1. Write M a nti +r = 1 1 M n a nr max n a X, i = 1,..., k. n nr< nt i +r By the assumption of the theorem and stationarity of the process X 1, X 2,..., 1 1 M a nr Y r, max X, i = 1,..., k Y t 1,..., Y t k n nr< nt i +r a n as n. Using the standard tightness argument, we conclude that for every sequence n m there is a subsequence n ml such that 1 1 M a nml r, max nml a X, i = 1,..., k Y r, Y r t 1,..., Y r t k nml n ml r< n ml t i +r as l, where Y r t 1,..., Y r t k d = Y t1,..., Y t k. Let now τ i, i = 1, 2,... be an enumeration of the rational numbers in [0,. A diagonalization argument shows that there is a sequence n m and a stochastic process Y r τ i, i = 1, 2,... with Y r τ i, i = 1, 2,... d = Y τi, i = 1, 2,... such that M a nmr, max X, i = 1, 2,... Y r, Y r τ i, i = 1, 2,... nm n mr< n mτ i +r a nm

10 10 TAKASHI OWADA AND GENNADY SAMORODNITSKY in finite-dimensional distributions, as m. We extend the process Y r to the entire positive half-line by setting Y r t = 1 lim 2 Y r τ + lim Y r τ, t 0. τ t, rational τ t, rational Then 3.4 and monotonicity imply that as m, M a nmr, max X, t 0 Y r, Y r t, t 0 nm n mr< n mt+r a nm in finite-dimensional distributions. Now the stationarity of max-increments follows from 3.3, 3.5 and continuous mapping theorem. Remark 3.3. Self-similar processes with stationary max-increments arising in a functional maxima scheme 1.2 are close in spirit to the stationary self-similar extremal processes of O Brien et al. 1990, while extremal processes themselves are defined as random sup measures. A random sup measure is, as its name implies, indexed by sets. They also arise in a limiting maxima scheme similar to 1.2, but with a stronger notion of convergence. Every stationary self-similar extremal processes trivially produces a self-similar process with stationary max-increments via restriction to sets of the type [0, t] for t 0, but the connection between the two obects remains unclear. Our limiting process in Theorem 4.1 below can be extended to a stationary self-similar extremal processes, but the extension is highly nontrivial, and will not be pursued here. It is not our goal in this paper to study in details the properties of self-similar processes with stationary max-increments, so we restrict ourselves to the following basic result, Proposition 3.4. Let Y t, t 0 be a nonnegative self-similar process with stationary maxincrements, and exponent H of self-similarity. Suppose Y t, t 0 is not identically zero. Then H 0, and the following statements hold. a If H = 0, then Y t = Y 1 a.s. for every t > 0. b If 0 < EY 1 p < for some p > 0, then H 1/p. c If H > 0, Y t, t 0 is continuous in probability. Proof. By the stationarity of max-increments, Y t is stochastically increasing with t. This implies that H 0. If H = 0, then Y n d = Y 1 for each n = 1, We use 3.2 with r = 1. Using t = 1 we see that, in the right hand side of 3.2, Y 1 = Y 1 1 a.s. Since Y 1 n Y 1 1 a.s., we conclude, using t = n in the right hand side of 3.2, that Y 1 = Y 1 n a.s for each n = 1, 2,.... By monotonicity, we conclude that the process Y 1 t, t 0, hence also the process Y t, t 0, is a.s. constant on [1, and then, by self-similarity, also on 0,.

11 LIMIT THEORY FOR PARTIAL MAXIMA 11 Next, let p > 0 be such that 0 < EY 1 p <. It follows from 3.2 with r = 1 that Therefore, This means that ph 1. 2 H Y 1 d = Y 2 d = max Y 1, Y ph EY 1 p = EY 2 p = E [ Y 1 p Y 1 1 p] 2EY 1 p Finally, we take arbitrary 0 < s < t. We use 3.2 with r = s. For every η > 0, P Y t Y s > η = P Y s Y s t s Y s > η P Y s t s > η = P t s H Y 1 > η. Hence continuity in probability. We now define the limiting process obtained in the main limit theorem of Section 4, and place it in the general framework introduced earlier in this section. Let > 0, and consider the extremal Fréchet process Z t, t 0, defined in 1.4, with the scale σ = 1. For 0 < β < 1 we define a new stochastic process by Z,β t = Z t β, t 0. Since the extremal Fréchet process is self-similar with H = 1/, it is immediately seen that the process Z,β is self-similar with H = β/. We claim that the process Z,β has stationary max-increments as well. To show this, we start with a useful representation of the extremal Fréchet process Z t, t 0 in terms of the points of a Poisson random measure. Let k, s k be the points of a Poisson random measure on R 2 + with mean measure ρ λ, where ρ x, = x, x > 0 and λ is the Lebesgue measure on R +. Then an elementary calculation shows that Z t, t 0 d = sup { k : s k t }, t 0. Therefore, Z,β t, t 0 d = U,β t, t 0, where 3.6 U,β t = sup { k : s k t β }, t 0. Given r > 0, we define U r,β t = sup { k : t + r β t β s k t + r β }. Since t1 + r β t β 1, t 1 + r β t 2 + r β t β 2, t 2 + r β for 0 t 1 < t 2, it follows that U r,β t, t 0 d = U,β t, t 0.

12 12 TAKASHI OWADA AND GENNADY SAMORODNITSKY Furthermore, since t + r β t β r β, we see that U,β t + r = U,β r U r,β t for all t 0. This means that the process U,β has stationary max-increments and, hence, so does the process Z,β. It is interesting to note that, by part b of Proposition 3.4, any H-self-similar process with stationary max-increments and -Fréchet marginals, must satisfy H 1/. The exponent H = β/ with 0 < β 1 of the process Z,β with β = 1 corresponding to the extremal Fréchet process Z covers the entire interval 0, 1/]. Therefore, the upper bound of part b of Proposition 3.4 is, in general, the best possible. As a general phenomenon, the stationary increments of a self-similar stochastic process arising in a functional central limit theorem for partial sums exhibit dependence reflecting, at least partially the dependence in the original stationary process. This can be seen, for example, in Owada and Samorodnitsky 2012 that starts with a stationary process similar in its nature to the one considered in this paper. Do the stationary max-increments of a process arising in the functional maxima scheme 1.2 reflect similarly the dependence structure in the original process? The fact the operation of taking partial maxima tends to reduce the memory in the original process has been noticed before; for example, for stationary Gaussian processes partial maxima converge, with the same normalization, to the extremal Gumbel process, as long as the correlations decay more rapidly than log n 1 ; see Berman Within that range of correlation decay, on the other hand, one would observe that the partial sums stop converging to the Brownian motion, and start converging to the Fractional Brownian motion, and the normalization will change as well. A related example with power tails is given by the infinite moving average models of the form X n = =0 c Z n, where Z n are i.i.d. random variables, belonging to the max-domain of attraction of an -Fréchet law, and c is a deterministic sequence satisfying certain conditions for the process to be well-defined. In this case it was shown by Davis and Resnick 1985 that the normalized partial maxima of X n converge in the functional sense to an extremal -Fréchet process; this paper assumes absolute summability of c, but it is only needed for the tail estimate 2.7, which is valid without the absolute summability, see Mikosch and Samorodnitsky In contrast, partial sums of the infinite moving average processes can converge, if 0 < < 2, in certain cases some of which prevent lack of summability of the coefficients, to the linear fractional -stable motions, and not to the -stable Lévy motions; see e.g. Section 4.7 in Whitt With the discussion in view, we investigate the dependence properties of the stationary maxincrements of the process Z,β defined above. This process will appear, in the case 0 < < 2 and 1/2 < β < 1, in the main limit theorem, Theorem 4.1 below. Equivalently, we will look at the

13 max-increments of the process U,β in 3.6. Let LIMIT THEORY FOR PARTIAL MAXIMA V,β n = sup { k : n β 1 < s k n β}, n = 1, 2,.... An important observation is that this sequence is independent only when β = 1, that is, when Z,β and U,β coincide with the extremal -Fréchet process. In general, we have the following statement. Proposition 3.5. The stationary process V n,β, n 1 is mixing and is generated by a dissipative flow. Proof. Notice that V,β n, n 1 d = M n β 1, n β ], n 1, where M is an independently scattered -Fréchet sup-measure on R + with the Lebesgue control measure. See Stoev and Taqqu It follows by Theorem 3.4 in Stoev 2008 that V n,β, n 1 is mixing. Further, n=1 1 n β 1,n β ]x < a.e., and Theorem 5.2 of Wang and Stoev 2010 shows that V n,β is generated by a dissipative flow. The main result in Theorem 4.1 below deals with the partial maxima of stationary SS processes for which the underlying flows are conservative; these are associated with long memory in the process. The resulting limiting process is not an extremal process, hence its max-increments are dependent. The memory in the original SS is long enough to guarantee that. On the other hand, the max-increments of the latter are generated by a dissipative flow, which is associated with shorter memory. We observe, therefore, another example of memory reduction by the operation of taking partial maxima. We finish this section by mentioning that an immediate conclusion from 3.6 is the following representation of the process Z,β on the interval [0, 1]: 3.8 Z,β t, 0 t 1 = d Γ 1/ 1 {V t}, 0 t 1, =1 where Γ, = 1, 2,..., are arrival times of a unit rate Poisson process on 0,, and V are i.i.d. random variables with P V 1 x = x β, 0 < x 1, independent of Γ. 4. A Functional Limit Theorem for Partial Maxima In this section we state and prove our main result, a functional limit theorem for the partial maxima of the process X = X 1, X 2,... given in 1.5. Recall that T is a conservative, ergodic and measure preserving map on a σ-finite, infinite, measure space E, E, µ. We will assume that T is a pointwise dual ergodic map with normalizing sequence a n that is regularly varying with exponent 1 β; equivalently, the wandering sequence w n in 2.5 is assumed to be regularly

14 14 TAKASHI OWADA AND GENNADY SAMORODNITSKY varying with exponent β. Crucially, we will assume that 1/2 < β < 1. See Remark 4.2 after the statement of Theorem 4.1 below. Define 4.1 b n = max f T n x 1/ µdx, n = 1, 2,.... E 1 k n The sequence b n is known to play an important role in the rate of growth of partial maxima of an -stable process of the type 1.5. It also turns out to be a proper normalizing sequence for our functional limit theorem. In Samorodnitsky 2004 it was shown that, for a canonical kernel 1.7, if the map T is conservative, then the sequence b n grows at a rate strictly slower than n 1/. The extra assumptions imposed in this paper will guarantee a more precise statement: b n RV β/. Specifically, b n 4.2 lim = f n w n where w n is the wandering sequence. This fact has an interesting message, because it explicitly shows that the rate of growth of the partial maxima is determined both by the heaviness of the marginal tails through and by the length of memory through β. In contrast, if the map T has a non-trivial dissipative component, then the sequence b n grows at the rate n 1/, and so do the partial maxima of a stationary SS process; see Samorodnitsky This is the limiting case of the setup in the present paper, as β gets closer to 1. Intuitively, the smaller is β, the longer is the memory in the process. The functional limit theorem in Theorem 4.1 below involves weak convergence in the space D[0, equipped with two different topologies, the Skorohod J 1 -topology and the Skorohod M 1 - topology, introduced in Skorohod The details could be found, for instance, in Billingsley 1999 for the J 1 -topology, and in Whitt 2002 for the M 1 -topology. See Remark 4.3. We recall the tail constant of an -stable random variable given by { 1 C = x 1 / sin x dx Γ2 cosπ/2 if 1, = 2/π if = 1; see Samorodnitsky and Taqqu Theorem 4.1. Let T be a conservative, ergodic and measure preserving map on a σ-finite infinite measure space E, E, µ. Assume that T is a pointwise dual ergodic map with normalizing sequence a n RV 1 β, 0 β 1. Let f L µ L µ, and assume that f is supported by a uniform set A for T. Let > 0. Then the sequence b n in 4.1 satisfies 4.2.

15 LIMIT THEORY FOR PARTIAL MAXIMA 15 Assume now that 0 < < 2 and 1/2 < β < 1. If M is a SS random measure on E, E with control measure µ, then the stationary SS process X given in 1.5 satisfies max X k, t 0 C 1/ Z,β t, t 0 in D[0, 1 k nt b n in the Skorohod M 1 -topology. Skorohod J 1 -topology as well. Moreover, if f = 1 A, then the above convergence occurs in the Remark 4.2. Convergence to a Fréchet limit in 4.3 no longer holds in the range 0 < β < 1/2. We will discuss elsewhere what non-fréchet limiting processes may appear in that case. In the boundary case β = 1/2, however, the statement 4.3 may still hold. This will be the case when the one ump property 4.6 below is satisfied. This is the case, for example, for the Markov shift operator T presented at the end of the paper. See also Samorodnitsky Remark 4.3. It is not difficult to see why the weak convergence in 4.3 holds in the J 1 -topology for indicator functions, but only in the M 1 -topology in general. Indeed, for a general function f the values of the process M n f in Proposition 2.1 may have multiple umps on the time scale on before reaching the limiting value f. Since the limiting process has a single ump, one does not expect the J 1 -convergence. On the other hand, if f = 1 A, then the value f = 1 is reached in a single ump, matching the single ump in the limiting process. Proof of Theorem 4.1. We start with verifying 4.2. Obviously, b n f µϕ A n, and the definition of the wandering rate sequence gives us the upper bound b n lim sup f. n w n On the other hand, take an arbitrary ɛ 0, f. The set B ɛ = { x A : fx f ɛ }. is a uniform set for T. Let w n ε be the corresponding wandering rate sequence. Then a lower bound for b n is obtained by so Letting ε 0, we obtain 4.2. b n f ɛ n µ lim inf n b n = lim inf w n n =1 b n w ε n T B ɛ, f ɛ. Suppose now that 0 < < 2 and 1/2 < β < 1. We continue with proving convergence in the finite dimensional distributions in 4.3. Since for random elements in D[0, with nondecreasing

16 16 TAKASHI OWADA AND GENNADY SAMORODNITSKY sample paths, weak convergence in the M 1 -topology is implied by the finite-dimensional weak convergence, this will also establish 4.3 in the sense of weak convergence in the M 1 -topology. Fix 0 = t 0 < t 1 < < t d, d 1. We may and will assume that t d 1. We use a series representation of the random vector X 1,..., X n : with f k = f T k, k = 1, 2,..., 4.4 X k, k = 1,..., n = d b n C 1/ ɛ Γ 1/ f k U n =1 max 1 i n f i U n, k = 1,..., n. Here ɛ are i.i.d. Rademacher random variables symmetric random variables with values ±1, Γ are the arrival times of a unit rate Poisson process on 0,, and U n are i.i.d. E-valued random variables with the common law η n defined by 4.5 dη n dµ x = 1 b max f kx, x E. n 1 k n The sequences ɛ, Γ, and U n are taken to be independent. We refer to Section 3.10 of Samorodnitsky and Taqqu 1994 for series representations of -stable random vectors. The representation 4.4 was also used in Samorodnitsky 2004, and the argument below is structured similarly to the corresponding argument ibid. The crucial consequence of the assumption 1/2 < β < 1 is that, in the series representation 4.4, only the largest Poisson ump will play an important role. It is shown in Samorodnitsky 2004 that, under the assumptions of Theorem 4.1, for every η > 0, 4.6 ϕ n η P as n. n { k=1 Γ 1/ f k U n max 1 i n f i U n > η for at least 2 different = 1, 2,... We will proceed in two steps. First, we will prove that 4.7 Γ 1/ max 1 k nti f k U n =1 max 1 k n f k U n, i = 1,..., d Z,β t i, i = 1,..., d in R d +. Next, we will prove that, for fixed λ 1,..., λ d > 0, for every 0 < δ < 1, 4.8 P and that C 1/ =1 P b 1 n max X k > λ i, i = 1,..., d 1 k nt i Γ 1/ max 1 k nti f k U n max 1 k n f k U n P b 1 n max X k > λ i, i = 1,..., d 1 k nt i } > λ i 1 δ, i = 1,..., d + o1 0

17 4.9 P C 1/ =1 LIMIT THEORY FOR PARTIAL MAXIMA 17 Γ 1/ max 1 k nti f k U n max 1 k n f k U n > λ i 1 + δ, i = 1,..., d + o1. Since the Fréchet distribution is continuous, the weak convergence b 1 n max X k, i = 1,..., d Z,β t i, i = 1,..., d in R d + 1 k nt i will follow by taking δ arbitrarily small. We start with proving 4.7. For n = 1, 2,..., N n = =1 δ Γ,U n is a Poisson random measure on 0, n k=1 T k A with mean measure λ η n. Define a map S n : R + n k=1 T k A R d + by S n r, x = r 1/ M n fx 1 M nt1 fx,..., M ntd fx, r > 0, x n k=1 T k A. Then, for λ 1,..., λ d > 0, P =1 Γ 1/ max 1 k nti f k U n max 1 k n f k U n λ i, i = 1,..., d c ] = P [N n Sn 1 0, λ 1 ] 0, λ d ] = 0 { c } = exp λ η n Sn 1 0, λ 1 ] 0, λ d ] d = exp λ η n r, x : λ M nt fx Mn fx > r = exp b n =1 d λ E =1 M nt f dµ. We use 4.2 and the weak convergence in Proposition 2.1 to obtain Therefore, b n E i=1 d =1 λ d Ω =1 M nt f dµ f 1 E λ 1 {Vβ t }dp = d =1 d d t β i tβ i 1 i=1 λ M nt f dµ n P Γ 1/ max 1 k nti f k U n =1 max 1 k n f k U n λ i, i = 1,..., d d d exp t β i tβ i 1 λ = P Z,β t i λ i, i = 1,..., d. =i The claim 4.7 has, consequently, been proved. =i λ.

18 18 TAKASHI OWADA AND GENNADY SAMORODNITSKY We continue with the statements 4.8 and 4.9. Since the arguments are very similar, we only prove 4.8. Let K N and 0 < ɛ < 1 be constants so that K + 1 > 4 and δ ɛk > 0. Then P C 1/ C 1/ P b 1 n max X k > λ i, i = 1,..., d 1 k nt i Γ 1/ max 1 k nti f k U n =1 max 1 k n f k U n > λ i 1 δ, i = 1,..., d + ϕ n C 1/ ɛ min λ i 1 i d d + P C 1/ max 1 k nt i ɛ Γ 1/ f k U n i=1 =1 max 1 l n f l U n > λ i, Γ 1/ max 1 k nti f k U n =1 max 1 k n f k U n λ i 1 δ, and for each m = 1,..., n, C 1/ Γ 1/ f m U n max 1 l n f l U n > ɛ min By 4.6, it is enough to show that for all λ > 0 and 0 t 1, 4.10 P C 1/ =1 C 1/ C 1/ max 1 k nt =1 1 l d λ l for at most one = 1, 2,... ɛ Γ 1/ f k U n max 1 i n f i U n > λ, Γ 1/ max 1 k nt f k U n max 1 k n f k U n λ1 δ, and for each m = 1,..., n, Γ 1/ f m U n max 1 i n f i U n > ɛλ for at most one = 1, 2, For every k = 1, 2,..., n, the Poisson random measure represented by the points ɛ Γ 1/ f k U n max f iu n 1 i n 1, = 1, 2,... has the same mean measure as that represented by the points ɛ Γ 1/ f b 1 n, = 1, 2,.... Hence, these two Poisson random measures coincide distributionally. We conclude that the probability in 4.10 is bounded by nt P C 1/ ɛ Γ 1/ f k U n k=1 =1 max 1 i n f i U n > λ, C1/ =1 C 1/ Γ 1/ f k U n max 1 i n f i U n Γ 1/ f k U n > ɛλ for at most one = 1, 2,... max 1 i n f i U n λ1 δ,.

19 = nt P C 1/ =1 LIMIT THEORY FOR PARTIAL MAXIMA 19 ɛ Γ 1/ > λ f 1 b n, C 1/ =1 Γ 1/ C 1/ Γ 1/ > ɛλ f 1 b n for at most one = 1, 2,... np Due to the choice K + 1 > 4/, C 1/ =K+1 n f 4 C 4/ δ ɛk 4 λ 4 b 4 n E =K+1 ɛ Γ 1/ E ɛ Γ 1/ > δ ɛkλ f 1 =K+1 4 ɛ Γ 1/ < ; 4. λ1 δ f 1 b n, see Samorodnitsky 2004 for a detailed proof. Since n/b 4 n 0 as n, 4.10 follows. Suppose now that f = 1 A. In that case the probability measure η n defined in 4.5 coincides with the probability measure µ n of Proposition 2.1. In order to prove weak convergence in the J 1 -topology, we will use a truncation argument. First of all, we may and will restrict ourselves to the space D[0, 1]. Let K = 1, 2,.... First of all, we show, in the notation of 3.8, the convergence 4.11 C 1/ max 1 k nt K ɛ Γ 1/ 1 A T k U n =1, 0 t 1 C 1/ b n K =1 1 {V t}, 0 t 1 Γ 1/ in the J 1 -topology on D[0, 1]. Indeed, by 4.6, outside of an event of asymptotically vanishing probability, the process in the left hand side of 4.11 is K 4.12 C 1/ Γ 1/ max 1 A T k U n, 0 t 1. 1 k nt =1 By Proposition 2.1, we can put all the random variables involved on the same probability space so that the time of the single step in the th term in 4.12 converges a.s. for each = 1,..., K to V. Then, trivially, the process in 4.12 converges a.s. in the J 1 -topology on D[0, 1] to the process in the right hand side of Therefore, the weak convergence in 4.11 follows. Next, we note that in the J 1 -topology on the space D[0, 1], K C 1/ Γ 1/ 1 {V t}, 0 t 1 C 1/ Γ 1/ 1 {V t i } 0 t 1 as K a.s. =1 =1

20 20 TAKASHI OWADA AND GENNADY SAMORODNITSKY This is so because, as K, sup 0 t 1 =1 Γ 1/ 1 {V t} K =1 Γ 1/ 1 {V t} Γ 1/ K+1 0 a.s.. According to Theorem 3.2 in Billingsley 1999, the J 1 -convergence in 4.3 will follow once we show that lim K for every ɛ > 0. Write ɛ/2 e x 0 xk 1 lim sup P n K 1! dx+ P max max 1 k n 1 k n ɛ/2 e x =K+1 =K+1 ɛ Γ 1/ 1 A T k U n ɛ Γ 1/ 1 A T k U n xk 1 K 1! P max 1 k n =1 > ɛ = 0 > ɛ ɛ Γ + x 1/ 1A T k U n > ɛ dx. Clearly, the first term vanishes when K. Therefore, it is sufficient to show that for every x ɛ/2, 4.13 P max as n. 1 k n ɛ Γ + x 1/ 1A T k U n > ɛ 0 =1 To this end, choose L N and 0 < ξ < 1/2 so that 4.14 L + 1 > 4 and 1 2 ξl > 0. By 4.6 we can write 4.15 P max 1 k n ɛ Γ + x 1/ 1A T k U n =1 > ɛ P max 1 k n ɛ Γ + x 1/ 1A T k U n > ɛ, and for each m = 1,..., n, =1 Γ + x 1/ 1A T m U n > ξɛ for at most one = 1, 2,... + o1. Notice that for every k = 1,..., n, the Poisson random measure represented by the points ɛ Γ + x 1/ 1A T k U n, = 1, 2,... is distributionally equal to the Poisson random measure represented by the points ɛ b n µa 1 Γ + x 1/, = 1, 2,....

21 LIMIT THEORY FOR PARTIAL MAXIMA 21 Therefore, the first term on the right hand side of 4.15 can be bounded by n P ɛ Γ + x 1/ 1A T k U n > ɛ, k=1 =1 Γ + x 1/ 1A T k U n > ξɛ for at most one = 1, 2,... = np ɛ b n µa 1 Γ + x 1/ =1 > ɛ, b n µa 1 Γ + x 1/ > ξɛ for at most one = 1, 2,... np =L+1 ɛ b n µa 1 Γ + x 1/ > 1 2 ξl ɛ. In the last step we used the fact that, for x ɛ/2, the magnitude of each term in the infinite sum does not exceed ɛ/2. By the contraction inequality for Rademacher series see e.g. Proposition of Kwapień and Woyczyński 1992, np ɛ b n µa 1 Γ + x 1/ 1 =L+1 > 2 ξl ɛ 2nP ɛ Γ 1/ 1 > 2 ξl ɛµa 1/ b n. =L+1 As before, by Markov s inequality and using the constraints of the constants L N and 0 < ξ < 1/2 given in 4.14, 2nP =L+1 ɛ Γ 1/ > 1 2 ξl as n and, hence, 4.13 follows. ɛµa 1/ b n 2nµA 4/ 2 1 ξl 4 ɛ 4 b 4 E n =L+1 ɛ Γ 1/ Remark 4.4. There is not doubt that the convergence result in Theorem 4.1 can be extended to more general infinitely divisible random measures M in 1.5, under appropriate assumptions of regular variation of the Lévy measure of M and integrability of the function f. In particular, processes Z,β with any > 0 are likely to appear in the limit in 4.3 and not only with 0 < < 2 allowed by the assumption of stability. Furthermore, the symmetry of the process X has very little to do with the limiting distribution of the partial maxima. For example, a straightforward symmetrization argument allows one to extend 4.3 to skewed -stable processes, at least in the sense of convergence of finite-dimensional distributions. 4 0 The reason we decided to restrict the

22 22 TAKASHI OWADA AND GENNADY SAMORODNITSKY presentation to the symmetric stable case had to do with a particularly simple form of the series representation 4.4 available in this case. This has allowed us to avoid certain technicalities that might have otherwise blurred the main message, which is the effect of memory on the functional limit theorem for the partial maxima. We conclude by mentioning that the result of Theorem 4.1 applies, among others, to the two examples in Owada and Samorodnitsky 2012, that of a flow generated by a null recurrent Markov chain, and that of an AFN-system. We only remind the reader of the setup of the former example, since it also appears in Samorodnitsky Consider an irreducible null recurrent Markov chain x n, n 0 defined on an infinite countable state space S with the transition matrix p i. Let π i, i S be its unique up to constant multiplication invariant measure with π 0 = 1. Note that π i is necessarily an infinite measure. Define a σ-finite and infinite measure on E, E = S N, BS N by µb = π i P i B, B S N, i S where P i denotes the probability law of x n starting in state i S. Let T x 0, x 1,... = x 1, x 2,... be the usual left shift operator on S N. Then T preserves µ. Since the Markov chain is irreducible and null recurrent, T is conservative and ergodic see Harris and Robbins Let A = {x S N : x 0 = i 0 } for a fixed state i 0 S, and ϕ A x = min{n 1 : x n A}, x S N be the first entrance time. Assume that n P 0 ϕ A k RV β k=1 for some β 1/2, 1. Then all of the assumptions of Theorem 4.1 are satisfied for any f L µ L µ, supported by A; see Aaronson References Aaronson, J. 1997: An Introduction to Infinite Ergodic Theory, vol. 50 of Mathematical Surveys and Monographs, Providence: American Mathematical Society. Berman, S. 1964: Limit theorems for the maximum term in stationary sequences, Annals of Mathematical Statistics, 35, Billingsley, P. 1999: Convergence of Probability Measures, New York: Wiley, second ed.

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