SCALING LIMITS FOR RANDOM FIELDS WITH LONG-RANGE DEPENDENCE

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1 The Annals of Probability SCALING LIMITS FOR RANDOM FIELDS WITH LONG-RANGE DEPENDENCE By Ingemar Kaj, Lasse Leskelä, Ilkka Norros, and Volker Schmidt Uppsala University, Helsinki University of Technology, VTT Technical Research Centre, and University of Ulm This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density λ of the sets grows to infinity and the mean volume ρ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which λ and ρ are scaled. If λ grows much faster than ρ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable. 1. Introduction. Fractional Brownian motion often appears as a renormalized limit of independent superpositions of long-memory stochastic processes that are used in physics and other application areas, such as telecommunications and finance. Observing fractional Brownian motion in the limit typically requires rescaling of two model parameters, and switching the order of taking the double limit may lead to approximations with completely different statistical properties Taqqu and Levy, 1986). This was also the conclusion of Mikosch, Resnick, Rootzén and Stegeman 22), who studied data traffic models with heavy tails, and identified conditions for convergence to fractional Brownian motion and stable Lévy motion in terms of relative scaling speeds of model parameters. This type of results have been refined in studies of a special scaling regime that leads to limit processes that are not stable Gaigalas and Kaj, 23; Kaj and Martin-Löf, 24; Kaj and Taqqu, 24; Kaj, 25). Supported by a basic grant in Applied Mathematics of the Swedish Foundation for Strategic Research. Supported by the Mittag-Leffler Institute and the Finnish Graduate School in Stochastics. AMS 2 subject classifications: Primary 6F17; secondary 6G6, 6G18 Keywords and phrases: long-range dependence, self-similar random field, fractional Brownian motion, fractional Gaussian noise, stable random measure, Riesz energy 1

2 2 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT This paper extends the above trichotomy into multidimensional context by studying renormalized limits of a spatial random field generated by independently and uniformly scattered random sets in. Viewing the random field as a random linear functional indexed by suitable test functions or test measures, we find different limits for the model as the mean density λ of the random sets grows to infinity and the mean volume ρ of the sets tends to zero. If λ grows much faster than ρ shrinks, the model with heavy tails converges to a Gaussian self-similar random field with long-range dependence, which in symmetric case corresponds to fractional Gaussian noise with Hurst parameter H > 1/2. In the opposite case where ρ shrinks to zero very rapidly, the limit has infinite second moments and no spatial dependence. We also describe a special intermediate scaling regime that leads to limits that are not stable. In dimension one, these findings correspond to results obtained earlier for a stochastic process known as the infinite source Poisson model or the M/G/ model Kurtz, 1996; Konstantopoulos and Lin, 1998; Mikosch et al., 22; Kaj and Taqqu, 24). The outline of the paper is as follows. In Section 2 we construct the model and introduce a functional analytic approach suitable for asymptotic analysis of the random fields. In Section 3 we discuss the different scaling regimes and state the main limit theorems. Section 4 contains a discussion on the statistical properties of the limits, and Section 5 concludes with the proofs. 2. Random grain model. Let C be a bounded measurable set in such that C = 1 and C =, where C is the boundary of C and denotes the Lebesgue measure. The building blocks of the model are the sets x + v 1/d C, called grains, where x is a point in and v > is the volume of the grain. Our goal is to study the mass distribution generated by a family of grains X j + ρv j ) 1/d C with random locations X j and random volumes ρv j, j = 1, 2,... We assume that X j are uniformly distributed in the space according to a Poisson random measure with mean density λ >, and that V j are independent copies of a positive random variable V with E V = 1, also independent of the locations X j. Hence the scalar ρ > equals the mean grain volume. The random field J λ,ρ x) is defined as the number of grains covering x, J λ,ρ x) = # { j : x X j + ρv j ) 1/d C }, and we let J λ,ρ A) = j A Xj + ρv j ) 1/d C ) 1) be the cumulative mass induced by the grains to a measurable set A.

3 SCALING LIMITS FOR RANDOM FIELDS The random grain field as a random linear functional. The random variables J λ,ρ A) with A ranging over all measurable sets constitute a setindexed random function. More generally, we can view J λ,ρ as a random functional by replacing the measure A in 1) by an arbitrary positive measure φ, J λ,ρ φ) = φ X j + ρv j ) 1/d C ). j Denote by F v) the probability distribution of V. Then the grain volumes ρv j are distributed according to F ρ v) = F v/ρ), and J λ,ρ φ) can be conveniently described as a stochastic integral with respect to a Poisson random measure N λ,ρ dx, dv) on with intensity measure λdxf ρ dv), J λ,ρ φ) = φx + v 1/d C) N λ,ρ dx, dv). 2) To study the linear structure of J λ,ρ in a natural way, we do not want to restrict to positive measures. Let M 1 be the linear space of signed measures φ on with finite total variation φ 1 <. When φ M 1, we see by writing φa) = A φdx) and changing the order of integration that φx + v 1/d C) λdxf ρ dv) λρ φ 1 <, so the stochastic integral on the right-hand side of 2) converges in probability for all φ M 1 Kallenberg, 21). To each function φ L 1 one can uniquely associate a signed measure φ M 1 defined by φdx) = φx)dx. We will identify the space L 1 with its image in M 1 under the map φ φ, so that L 1 M 1. Accordingly, when φ L 1, we will from now on use the same symbol φ to signify both the function φx) and the measure φdx). Moreover, if A is a measurable set with A <, we identify A with the indicator function 1 A L 1 M 1. Note that then J λ,ρ 1 A ) = J λ,ρ A) agrees with 1). Denote by B r the open ball centered at the origin with radius r. Then we see that J λ,ρ has long-range dependence in the sense that lim Cov Jλ,ρ B 1 ), J λ,ρ B r \B 1 ) ) = 3) r if and only if E V 2 =. To verify this, note first that the left-hand side of 3) can be written as B 1 x + v 1/d C ) B1 c x + v 1/d C ) dx F ρ dv) 4)

4 4 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT using the covariance formula 1) below. Because B c 1 x + v 1/d C ) v and B 1 x + v 1/d C) dx = B 1 v, expression 4) is bounded above by ρ 2 E V 2. For the other direction, B c 1 x + v1/d C) v B 1 implies that B c 1 x + v1/d C) v/2 for all v 2 B 1, so that 4) is finite only if E V 2 is finite. Observe also that in dimension one, long-range dependence as defined in 3) is equivalent to n= CovY, Y n ) =, where Y n = J λ,ρ n, n + 1]) is the discretized version of J λ,ρ Riesz energy of signed measures. To study the limiting behavior of J λ,ρ φ) as λ and ρ, we need to impose some more regularity for the measures φ M 1. The following subspaces of M 1 will turn out to be useful. For α, 1), let us define M α = { φ M 1 : φ dx) φ dy) x y 1 α)d < where φ is the total variation measure of φ, and for φ, ψ M α let φdx)ψdy) φ, ψ α = c α,d, 5) x y 1 α)d where }, c α,d = π α 1/2)d Γ 1 α)d 2 )/Γ αd 2 ). 6) A classical result in potential theory states that φ, ψ α is an inner product on the vector space M α Landkof, 1972). We denote the corresponding norm by φ α = φ, φ 1/2 α. The quantity φ 2 α is often called the Riesz energy of φ. The following proposition describes how the spaces M α can be ordered. Proposition 1. For all < α 1 < α 2 < 1, L 1 L 2 M α 1 M α 2 M 1. Remark. Let S be the space of tempered distributions on, and denote the Fourier transform by F : S S. Then M α S and φ, ψ α = Fφx)Fψx) x αd dx 7) for all φ, ψ M α Landkof, 1972, Section VI.1). Equation 7) shows that F maps M α isometrically into L 2 α, the space of square integrable functions

5 SCALING LIMITS FOR RANDOM FIELDS 5 with respect to x αd dx. It is also known that FM α ) is dense in L 2 α, so the Plancherel theorem implies that the closure of M α with respect to the norm φ α equals F 1 L 2 α), which is called the space of distributions with finite Riesz energy Landkof, 1972) Integrals with respect to centered Poisson random measures. Recall that the centered integral f dn dη) of a nonrandom function f with respect to a Poisson random measure N with intensity measure η may be defined even for functions that are not η-integrable. It is known that f dn dη) exists as a limit in probability if and only if f f 2 ) dη <, 8) in which case the distribution of fdn dη) is characterized by ) E exp i fdn dη) = exp Ψ f)dη, 9) where Ψv) = e iv 1 iv for v R Kallenberg, 21). Moreover, E f dn dη)) ) g dn dη) = fg dη, 1) when f and g are square integrable with respect to η. 3. Scaling behavior and main results Scaling behavior of the random grain model. We will next study the limiting behavior of J λ,ρ φ) as the mean grain density λ grows to infinity and the mean grain volume ρ shrinks to zero. When the grain volume distribution has finite variance, the following central limit theorem shows that the centered and renormalized version of J λ,ρ converges to white Gaussian noise. Theorem 1. Let C be a bounded set with C = 1 and C =, and assume E V 2 <. Then as λ and ρ, the following limit holds in the sense of finite-dimensional distributions of random functionals indexed by L 1 L 2 : J λ,ρ φ) E J λ,ρ φ) ρλ E V 2 ) 1/2 W φ), where W is the centered Gaussian random linear functional on L 2 with E W φ)w ψ) = φx)ψy) dx dy. 11)

6 6 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT However, our main focus will be on the model where the volume distribution is heavy-tailed with infinite variance. Hence, we will from now assume that the distribution F v) of the normalized volume V has a regularly varying tail of index γ 1, 2), that is, lim v F av)) F v) = a γ for all a >, 12) where F v) = 1 F v). This implies that E V 2 =. Let us denote fρ) gρ), if fρ)/gρ) 1 as ρ. Then 12) implies that the scaled volume distribution F ρ v) = F v/ρ) satisfies F ρ v) F ρ 1)v γ as ρ, and by Karamata s theorem see formula 21) in Section 5.1), the expected number of grains with volume larger than one that cover the origin equals {x,v): x+v 1/d C, v>1} λdxf ρ dv) = λ 1 vf ρ dv) λ F ρ 1) 1 γ 1. Consequently, we distinguish the following three scaling regimes: Large-grain scaling λ F ρ 1) Intermediate scaling λ F ρ 1) σ > Small-grain scaling λ F ρ 1) The regular variation of F v) implies that the relations λ 1/ρ) γ+ɛ and λ 1/ρ) γ ɛ for some ɛ > belong to large-grain and small-grain regimes, respectively, while in the critical intermediate scaling regime, the size of λ is roughly proportional to 1/ρ) γ. Under large-grain scaling, the number of grains that are big enough to carry statistical dependence over macroscopic distances grows to infinity. Hence, the limit of J λ,ρ in this case is expected to have long-range spatial dependence. In the opposite case of small-grain scaling, no grains survive that are big enough to represent substantial dependence over spatial distances, so the small-grain limit of J λ,ρ should have very weak dependence over space. The intermediate scaling regime is a blend of the two other, with a balanced mix of large grains providing long-range spatial dependence and small grains generating nontrivial random variations on short distances Main results. The following theorem justifies the heuristics in Section 3.1. Let γ 1, 2), and recall that the independently scattered γ-stable

7 SCALING LIMITS FOR RANDOM FIELDS 7 random measure with unit skewness and Lebesgue control measure is the random linear functional Λ γ φ) = φx) Λ γ dx) on L γ characterized by E e iλγ φ) = exp σ γ φ 1 iβφ tan πγ )) ), 13) 2 where σ φ = φ γ and β φ = φ γ γ φ + γ γ φ γ γ), and where φ + = maxφ, ) and φ = minφ, ). For an alternative equivalent definition of Λ γ as a set-indexed random function, see Samorodnitsky and Taqqu 1994). Theorem 2. Let C be a bounded set with C = 1 and C =, and assume that V has a regularly varying tail with exponent γ 1, 2). Let α, 2 γ). Then the following three limits hold in the sense of finitedimensional distributions of random functionals as λ and ρ : i) Large-grain scaling) If λ F ρ 1), then J λ,ρ φ) E J λ,ρ φ) γλ Fρ 1) ) 1/2 W γ,c φ), φ M α, where W γ,c is the centered Gaussian random linear functional on M 2 γ with E W γ,c φ)w γ,c ψ) = φdx)k γ,c x y)ψdy), and K γ,c x) = ii) Intermediate scaling) If λ F ρ 1) σ >, then v 1/d x + C ) C v γ dv. 14) J λ,ρ φ) E J λ,ρ φ) J γ,c φ σ), φ M α, where Jγ,C is defined on M 2 γ as a centered integral with respect to the Poisson random measure N γ dx, dv) on with intensity measure dxv γ 1 dv, Jγ,Cφ) = φx + v 1/d C) N γ dx, dv) dx v γ 1 dv), 15) and where φ σ is defined by φ σ A) = φσa) with σ = γσ ) 1/γ 1)d). iii) Small-grain scaling) If λ F ρ 1), then J λ,ρ φ) E J λ,ρ φ) c γ 1/ F ρ ) γλ) Λ γ φ), φ L 1 L 2, where Λ γ is the independently scattered γ-stable random measure on with Lebesgue control measure and unit skewness, 1/ F ρ )) u) = inf{v : 1/ F ρ v) u} is the quantile function of F ρ, and Γ2 γ) c γ = γγ 1) cos πγ ) ) 1/γ. 16) 2

8 8 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT 3.3. Role of symmetry and randomly oriented grains. Fix a parameter H 1/2, 1), and let W H be the centered Gaussian random linear functional on M 2H 1 with φdx)ψdy) E W H φ)w H ψ) = c 2H 1,d x y 2 2H)d = φ, ψ 2H 1, 17) where φ, ψ 2H 1 is the Riesz inner product defined by 5) and c 2H 1,d is given by 6). When C is symmetric around the origin so that θc = C for all rotations θ of, the rotation invariance of the Lebesgue measure shows that the covariance kernel of the large-grain limit W γ,c satisfies K γ,c x) = K γ,c x e 1 ) = K γ,c e 1 ) x γ 1)d, where e 1 is an arbitrary fixed unit vector in. For symmetric grains C it hence follows that W γ,c equals cw H in the sense of finite-dimensional distributions, where H = 3 γ)/2 and c = c 1 2H 1,d K γ,ce 1 ) 1/2. We can also study the limit behavior for a slightly modified model where the grains have independent and uniform random orientations. To define this model, let dθ be the Haar measure on the compact group SOd) of rotations in, and let N λ,ρ dx, dv, dθ) be a Poisson random measure on SOd) with intensity measure λdxf ρ dv)dθ. Then J λ,ρ φ) = φx + v 1/d θc) N λ,ρ dx, dv, dθ) SOd) defines the analogue of J λ,ρ with randomly rotated grains θc compare with definition 2) in Section 2.1). Because dθ is a probability measure on the compact group SOd) that is not scaled during λ and ρ, the following result can be verified by copying the proof of Theorem 2. Note that the geometrical shape of C is not visible in the large-grain limit below. Theorem 3. Assume C is symmetric around the origin. Then under the assumptions of Theorem 2, the following three limits hold in the sense of finite-dimensional distributions of random functionals as λ and ρ : i) Large-grain scaling) If λ F ρ 1), then J λ,ρ φ) E J λ,ρ φ) γλ Fρ 1) ) 1/2 cw H φ), φ M α, where W H is the Gaussian random linear functional defined in 17) with H = 3 γ)/2, and c = c 1 2H 1,d v 1/d θe 1 + C ) C 1/2 v γ dv dθ). SOd)

9 SCALING LIMITS FOR RANDOM FIELDS 9 ii) Intermediate scaling) If λ F ρ 1) σ >, then J λ,ρ φ) E J λ,ρ φ) φ σ x + v 1/d θc) N γ dx, dv, dθ) dx v γ 1 dvdθ), SOd) φ M α, where N γ dx, dv, dθ) is a Poisson random measure on SOd) with intensity dxv γ 1 dvdθ and φ σ is as in Theorem 2. iii) Small-grain scaling) If λ F ρ 1), then J λ,ρ φ) E J λ,ρ φ) c γ 1/ F ρ ) γλ) Λ γ φ), φ L 1 L 2, where Λ γ is as in Theorem Statistical properties of the limits Properties of the large-grain limit. A change of variables shows that the covariance kernel of W γ,c given by 14) scales according to K γ,c ax) = a γ 1)d K γ,c x) for a >. Hence for H = 3 γ)/2, E W γ,c φ s )W γ,c ψ s ) = s 21 H)d E W γ,c φ)w γ,c ψ), where the dilatation φ s of the measure φ is defined for s > by φ s A) = φsa). 18) Thus, W γ,c is self-similar in the sense that W γ,c φ s ) and s 1 H)d W γ,c φ) have the same finite-dimensional distributions on M 2H 1 for all s >. Moreover, W γ,c has long-range dependence in the sense of 3). To see this, note first that lim Cov Wγ,C B 1 ), W γ,c B r \B 1 ) ) = K γ,c x y) dy dx. r By changing the order of integration, K γ,c x y) dy = B c 1 x+v 1/d C B 1 B c 1 B c 1 y v 1/d C ) dy v γ 1 dv, and because B1 c y v 1/d C ) v B 1 for all v, we see that the right-hand side above is greater than or equal to v 2 /2 for all v 2 B 1. Hence for all x, K γ,c x y) dy 1 v γ+1 dv, B1 c 2 2 B 1

10 1 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT which is infinite for γ 1, 2). The symmetric large-grain limit W H can be represented in terms of the white Gaussian noise W defined in Theorem 1, when W is viewed as an independently scattered Gaussian random measure with Lebesgue control measure as in Samorodnitsky and Taqqu 1994). Proposition 2. For H 1/2, 1), the random linear functional W H on M 2H 1 equals W H φ) = c H 1/2,d φdy) W dx) x y 3/2 H)d 19) in the sense of finite-dimensional distributions, where c H 1/2,d is given by 6). Specializing to dimension one, we see that for φ, ψ L 1 M 2H 1, E W H φ)w H ψ) = c 2H 1,1 φs)ψt) t s 2H 2 ds dt, from which we recognize that W H φ) equals a constant multiple of the stochastic integral of φ with respect to fractional Brownian motion with Hurst parameter H > 1/2 Gripenberg and Norros, 1996). The functional W H may thus be viewed as a natural extension of fractional Gaussian noise Mandelbrot and van Ness, 1968) into multidimensional parameter spaces. Moreover, choosing φ = 1 [,t] in 19) yields the well-balanced representation of fractional Brownian motion Samorodnitsky and Taqqu, 1994, Section 7.2.1) Properties of the intermediate limit. Defining the dilatation φ s as in 18), a change of variables shows that Ψφ s x + v 1/d C)) dx v γ 1 dv = s γ 1)d Ψφx + v 1/d C)) dx v γ 1 dv. Comparing this with the characteristic functional of Jγ,C given by 9) shows that Jγ,C is not self-similar nor stable. However, the sum of n independent copies of Jγ,C has the same finite-dimensional distributions as φ J γ,c φ s) where s = n 1/γ 1)d). This property is called aggregate-similarity in Kaj 25).

11 SCALING LIMITS FOR RANDOM FIELDS 11 Using 1) and changing the order of integration, E Jγ,C φ)j γ,c ψ) = = φx + v 1/d C)ψx + v 1/d C) dx v γ 1 dv R + φdx)k γ,c x y)φdy), which shows that Jγ,C and W γ,c share the same second order statistical structure. Especially, this implies that Jγ,C has long-range dependence in the sense of 3) Properties of the small-grain limit. Note that when φ is a function in L 1 L 2 M 1, the dilatation φ s defined in 18) becomes φ s A) = φx) dx = s d φsx) dx, sa so for functions, φ s x) = s d φsx). Inspection of the characteristic functional 13) of Λ γ shows that Λ γ φ s ) and s 1 H)d Λ γ φ) have the same finitedimensional distributions, where H = 1/γ. Thus, Λ γ is self-similar with Hurst parameter 1/γ. The random measure Λ γ can also be represented by Λ γ φ) = c 1 γ v φx) N γ dx, dv) dxv γ 1 dv), where N γ dx, dv) is the Poisson random measure appearing in 15) and c γ is given by 16), see Samorodnitsky and Taqqu 1994). Specializing to dimension one, we remark that the stochastic process t Λ γ 1 [,t] ) = t A Λ γ ds) is the centered γ-stable Lévy motion with unit skewness. 5. Proofs. By definition, J λ,ρ together with the four limit fields defined in Theorem 1 and Theorem 2 are linear in the sense that for all test measures φ 1,..., φ n and all scalars a 1,..., a n, J λ,ρ a 1 φ a n φ n ) = a 1 J λ,ρ φ 1 ) + + a n J λ,ρ φ n ) almost surely. Hence, convergence of the finite-dimensional distributions of the centered and renormalized version of J λ,ρ is equivalent to the convergence

12 12 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT of the one-dimensional distributions. Recall from 9) that for b >, the characteristic functional of J λ,ρ φ) E J λ,ρ φ))/b is given by E exp i J ) λ,ρφ) E J λ,ρ φ) φx + v 1/d ) C) = exp Ψ λf ρ dv)dx, b b 2) where Ψv) = e iv 1 iv. Recall that Ψ satisfies the following bounds, which can be verified by differentiation: Ψv) 2 v v 2 /2) and Ψv) v 2 /2) v 2 v 3 /6) for all v. Moreover, Ψ is Hölder continuous with Ψv) Ψu) 2 v u. Before going to the proofs of the main theorems, we first introduce some preliminary results on regular variation Section 5.1) and on maximal functions Section 5.2). Proposition 1 is proved in Section 5.3, while in Section 5.4 we develop the key results on the regularity of the characteristic functional 2). Sections contain the proofs of the main theorems, and Section 5.9 concludes with the proof Proposition Regular variation. Let F be a probability distribution on with a regularly varying tail of exponent γ >, and let < p < γ < q. Then using integration by parts and Karamata s theorem Bingham et al., 1987, Theorem ) it follows that as a, v p F dv) γ γ p F a) a p, 21) a, ) [,a] v q F dv) γ q γ F a) a q, 22) where F a) = 1 F a). The next two lemmas summarize the theory on regular variation that is later used to analyze the distribution of the normalized volume V. Lemma 1. Let F be a probability distribution on with a regularly varying tail of exponent γ >, and define the scaled distribution F ρ for ρ > by F ρ v) = F v/ρ). Assume that fv) is a continuous function on such that for some < p < γ < q, Then lim sup v v p fv) < and lim sup v q fv) <. v fv)f ρ dv) F ρ 1) fv) γv γ 1 dv as ρ.

13 SCALING LIMITS FOR RANDOM FIELDS 13 Proof. Fix a constant a, 1). Then 21) implies that for all v >, v p F ρ dv) F ρ 1)γ v γ 1+p dv, v, ) which shows that the finite measures F ρ 1) 1 v p F ρ dv) restricted to a, ) converge weakly to the finite measure γv γ 1+p dv on a, ) as ρ. Because the function v p fv) is continuous and bounded on a, ), this implies F ρ 1) 1 fv) F ρ dv) fv) γv γ 1 dv. 23) a, ) v a, ) Moreover, the second assumption on f implies that fv) cv q for all v [, 1], so 22) implies that a F ρ 1) 1 fv) F ρ dv) fv) γv γ 1 dv [,a] a c F ρ 1) 1 v q F ρ dv) + c v q γv γ 1 γ dv 2c q γ aq γ. [,a] The claim now follows because the right-hand side above can be made arbitrary small by choosing a small enough, and because 23) holds for all a, 1). Lemma 2. Let F and F ρ be defined as in Lemma 1, and let f ρ v) be a family of measurable functions on. Assume that for some < p < γ < q, either or Then lim sup ρ v>a lim sup ρ a v v p Fρ 1) f ρ v) = for all a >, and Fρ 1) f ρ v) cv q for all ρ, v, v q Fρ 1) f ρ v) = for all a >, and Fρ 1) f ρ v) cv p for all ρ, v, lim f ρ v) F ρ dv) =. ρ 24) 25) Proof. Assume that the functions f ρ v) satisfy the conditions 24) and fix a >. Denote c a ρ) = sup v>a v p Fρ 1) f ρ v). Then by 21), f ρ v) F ρ dv) c a ρ) F ρ 1) 1 v p γ F ρ dv) c a ρ) γ p ap γ, a, ) a, )

14 14 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT and by 22),,a] f ρ v) F ρ dv) c F ρ 1) 1 [,a] v q γ F ρ dv) c q γ aq γ. Because c a ρ) as ρ, the two above bounds imply that γ lim sup f ρ v) F ρ dv) c ρ q γ aq γ. Since this is true for all a >, the claim follows by letting a. The proof under assumption 25) is analogous Maximal functions. Let C be a bounded measurable set in with C = 1. If φ is a locally integrable function, define the averages m φ x, v) by m φ x, v) = v 1 φy) dy, 26) x+v 1/d C and let φ be the maximal function of φ given by φ x) = sup v 1 φy) dy. 27) v> x+v 1/d C The following lemma summarizes the known facts about maximal functions that are used to find integrable upper bounds for the characteristic functional of J λ,ρ in the proofs of Theorem 1 and Theorem 2 part iii). Lemma 3. Let C be a bounded measurable set in with C = 1. i) If φ L 1, then lim v m φ x, v) = φx) for almost all x. ii) If φ L 1, then φ x) < for almost all x. iii) If φ L p for some p > 1, then φ L p. Proof. Let B be the open ball centered at the origin with unit volume, and fix a > such that C a 1/d B. Assume first that φ L 1. Then v 1 φy) φx) dy aav) 1 φy) φx) dy x+v 1/d C x+av) 1/d B for all x and v >. The right-hand side tends to zero as v, because almost all points x are Lebesgue points of φ Rudin, 1987, Theorem 7.7). Thus, the first claim is valid. Next, let φ x) = sup v 1 φy) dy. v> x+v 1/d B

15 SCALING LIMITS FOR RANDOM FIELDS 15 be the Hardy Littlewood maximal function of φ. Then C a 1/d B implies that φ x) aφ x) for all x. The second claim now follows, because for φ L 1, φ x) < almost everywhere Rudin, 1987, Theorem 7.4). The third claim follows directly from the Hardy Littlewood maximal theorem Rudin, 1987, Theorem 8.18), which states that if φ L p for some p > 1, then φ L p Proof of Proposition 1. Assume < α 1 < α 2 < 1. When φ L 1 L 2, then denoting D = {x, y) : x y 1} we see that φx) φy) x y 1 α 1)d dxdy φ ) D c Moreover, writing φx) φy) x y 1 α 1)d dxdy = D D D D φx) φy) x y 1 α 1)d/2 x y 1 α 1)d/2 dxdy and applying Hölder s inequality we see that φx) φy) x y 1 α 1)d dxdy φx) 2 x y 1 α 1)d dxdy = φ 2 2 {x: x 1} dx x 1 α 1)d. This bound together with 28) shows that φ M α 1, so the first inclusion has been shown. To verify the second inclusion, note that x y 1 α2)d x y 1 α1)d on D, and x y 1 α2)d 1 on D c. Thus, φ dx) φ dy) x y 1 α 2)d which is finite for φ M α 1. D φ dx) φ dy) x y 1 α 1)d dxdy + D c φ dx) φ dy), 5.4. Regularity properties of the characteristic functional. In this section we prove the key continuity and boundedness properties of the characteristic functional of J λ,ρ that are required for the asymptotical analysis of the model. We start with a continuity property of the Lebesgue measure. Denote the symmetric difference of sets A and B by A B = A \ B) B \ A). Lemma 4. Let C be a bounded measurable set in such that C < and C =. Then lim C rc =. r 1

16 16 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT Proof. If y belongs to the interior of C, then 1 rc y) = 1 C y/r) converges to 1 as r 1. Moreover, because C =, it follows that lim r 1 1 rc y) = 1 for almost all y C. Hence, the dominated convergence theorem implies lim C rc = lim 1 rc y) dy = C, r 1 r 1 and by writing we see that the claim is valid. C rc = C C rc + rc C rc = 1 + r d ) C 2 C rc, Lemma 5. Assume that φ M 1 and let C be as in Lemma 4. Then the functions v φx + v 1/d C) 2 dx and v Ψφx + v 1/d C)) dx are continuous on. Proof. Define for u, v, du, v) = C φx + v 1/d C) φx + u 1/d C) dx. Then, using v 2 u 2 = u + v u v we see that φx + v 1/d C) 2 φx + u 1/d C) 2 dx 2 φ 1 du, v), 29) while Ψv) Ψu) 2 v u implies Ψφx + v 1/d C)) Ψφx + u 1/d C)) dx 2du, v). 3) Next, observe that ) φx + v 1/d C) φx + u 1/d C) φ x + u 1/d C) x + v 1/d C) ) = φ x + u 1/d C v 1/d C). Because φ x + A) dx = φ 1 A for all measurable sets A, the above inequality implies that du, v) φ 1 u 1/d C v 1/d C. Moreover, because u 1/d C v 1/d C = u C v/u) 1/d C, it follows using Lemma 4 that lim u v du, v) = for all v. This fact together with the bounds 29) and 3) completes the proof.

17 SCALING LIMITS FOR RANDOM FIELDS 17 Lemma 6. For each φ M α with α, 1) there is a constant c such that for all v, φx + v 1/d C) 2 dx cv v 2 α ). Proof. Let φ be a measure in M α with α, 1). Because the total variation measure φ also belongs to M α, we may assume without loss of generality that φ is positive. Then by changing the order of integration we see that φx + v 1/d C) 2 dx φ 1 φx + v 1/d C) dx = φ 2 1v. 31) Next, let a be large enough such that C ab, where B is the open unit ball. Then φx + v 1/d C) 2 φx + av) 1/d B) 2, so again by changing the order of integration we see that φx+v 1/d C) 2 dx x av) 1/d B) y av) 1/d B) φdx)φdy). 32) Let e 1 be an arbitrary fixed unit vector in. Because the Lebesgue measure is rotation and translation invariant, we see that for all x and y, x av) 1/d B) y av) 1/d B) = x y e 1 + av) 1/d B) av) 1/d B av 1 x y < 2av) 1/d ) 21 α)d av) 2 α x y 1 α)d. where the first inequality holds because the set x y e 1 + av) 1/d B) av) 1/d B is empty for av) 1/d x y /2. Combining the above bound with 32), we get φx + v 1/d C) 2 dx 2 1 α)d a 2 α φ 2 α v2 α. 33) Combining inequalities 31) and 33) together, we conclude that the claim holds by taking c = max φ 2 1, 21 α)d a 2 α φ 2 α ) Proof of Theorem 1. Let φ L 1 L 2 and define b = ρλ E V 2 ) 1/2. Without loss of generality, choose ρ as the basic model parameter and consider λ and b as functions of ρ. Because φ L 1 L 2, φx + v 1/d C) = φy) dy = vm φ x, v). x+v 1/d C

18 18 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT where m φ x, v) is the average of φ defined in 26). Using 2) and the definition of b we thus see that E exp i J ) ) λ,ρφ) E J λ,ρ φ) vmφ x, ρv) = exp λψ b λ E V 2 ) 1/2 F dv)dx. By Lemma 3, lim ρ m φ x, ρv) = φx) for all v and almost all x. Because Ψv) = 1 2 v2 + ɛv) where ɛv) v 3 /6, it follows that ) lim λψ vmφ x, ρv) ρ λ E V 2 ) 1/2 = v2 φx) 2 2 E V 2. 34) Moreover, letting φ be the maximal function of φ defined in Lemma 3, the bound Ψv) v 2 /2 implies that ) vmφ x, ρv) λψ λ E V 2 ) 1/2 v2 φ x) 2 2 E V 2. Because by Lemma 3 the right-hand side is integrable with respect to dxf dv), the dominated convergence theorem combined with 34) shows that lim E exp i J ) λ,ρφ) E J λ,ρ φ) = exp 1 ) φx) 2 dx, ρ b 2 where the right-hand side is the characteristic functional of the white Gaussian noise W on L 2. Hence, the proof of Theorem 1 is complete Proof of Theorem 2, large-grain scaling. Fix γ 1, 2), let φ M 2 γ, and assume first that φ is positive. Then by changing the order of integration it follows that φx + v 1/d ) 2 dx = x v 1/d C ) y v 1/d C ) φdx)φdy), so by the translation invariance of the Lebesgue measure, φx + v 1/d ) 2 v γ 1 dv dx = φdx)k γ,c x y)φdy), 35) where K γ,c is the covariance kernel defined in 14). Choose a > large enough so that C ab, where B is the open unit ball. Then letting e 1 be an arbitrary unit vector in we see that K γ,c x) K γ,ab x) = K γ,ab e 1 ) x 1 γ)d,

19 SCALING LIMITS FOR RANDOM FIELDS 19 so the right-hand side of 35) is bounded by K γ,ab e 1 ) φ 2 2 γ and hence finite. Thus by Fubini s theorem, equation 35) holds also for nonpositive φ M 2 γ. Because the left-hand side of 35) is nonnegative, φdx)k γ,c x y)ψdy) is a positive definite bilinear form in M 2 γ M 2 γ, and hence defines the distribution of a centered Gaussian random linear functional on M 2 γ, which we call W γ,c. Assume next that φ M α M 2 γ for some α, 2 γ) and let b = γλ F ρ 1)) 1/2. As before, we choose ρ as the basic model parameter and consider λ and b as functions of ρ. Define the functions f ρ and f on by f ρ v) = Ψ φx + v 1/d C) b ) dx, fv) = 1 φx + v 1/d C) 2 dx. 2 By Lemma 5, the function fv) is continuous, and fv) cv v 2 α )/2 by Lemma 6. Application of Lemma 1 with p = 1 and q = 2 α hence yields fv) F ρ dv) F ρ 1) fv) γv γ 1 dv. so that by the definition of b, lim ρ fv) λb 2 F ρ dv) = fv) γv γ 1 dv. 36) Next, define g ρ v) = λf ρ v) λb 2 fv), and observe that φx + v 1/d ) C) g ρ v) = λ Ψ + 1 φx + v 1/d ) ) C) 2 dx. b 2 b Because Ψv) v 2 /2) v 3 /6 and φx + v 1/d C) 3 dx φ 2 1 φ x + v 1/d C) dx = φ 3 1 v, we see that g ρ v) λb 3 φ 3 1v/6. Using the definition of b we thus see that F ρ 1)v 1 g ρ v) φ 3 1 6γb 37) for all v. Moreover, using Ψv) v 2 /2 and Lemma 6 it follows that g ρ v) cλb 2 v 2 α. Hence, F ρ 1)v 2 α) g ρ v) c/γ

20 2 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT for all v. The large-grain assumption λ F ρ 1) implies that b, so that the right-hand side of 37) tends to zero as ρ. Thus, using Lemma 2 with p = 1 and q = 2 α, we conclude that lim ρ Combining 36) and 38) we get lim ρ g ρ dv) F ρ dv) =. 38) f ρ v) λf ρ dv) = In light of 2) and 35), this is equivalent to lim E exp i J ) λ,ρφ) E J λ,ρ φ) = exp 1 ρ b 2 which completes the proof of Theorem 2 part i). fv) v γ 1 dv. ) φdx)k γ,c x)φdy), 5.7. Proof of Theorem 2, intermediate scaling. Let φ be a measure in M 2 γ. In the beginning of the proof of Theorem 2, part i), we saw that φx + v 1/d ) 2 dx v γ 1 dv <. Thus by 8), the stochastic integral φx + v 1/d C) N γ dx, dv) dxv γ 1 dv) converges in probability, so the right-hand side of 15) is well-defined on M 2 γ. Assume next that φ M α for some α, 2 γ), and choose again ρ as the basic model parameter and consider λ as a function of ρ. Define fv) = Ψφx + v 1/d C)) dx. Note that fv) is continuous by Lemma 5. Moreover, Ψv) v 2 /2 together with Lemma 6 shows that fv) cv v 2 α )/2. Lemma 1 with p = 1 and q = 2 α thus shows that fv) λf ρ dv) λ F ρ 1) fv) γv γ 1 dv.

21 SCALING LIMITS FOR RANDOM FIELDS 21 The characteristic functional formula 2) together with the intermediate scaling assumption λ F ρ 1) σ now implies that ) lim E exp i J λ,ρφ) E J λ,ρ φ))) = exp γσ fv) v γ 1 dv. ρ Denoting σ = γσ ) 1/γd) and defining φ σ A) = σ d φσ 1 A), a change of variables shows that the right-hand side above equals exp Ψφ σ x + v 1/d C)) dx v γ 1 dv. By 9), this is the characteristic functional of 15), so the proof of Theorem 2 part ii) is complete Proof of Theorem 2, small-grain scaling. Define b = 1/ F ρ ) γλ) for notational convenience, we de not include the constant c γ into b). As before, λ and b are considered to be functions of ρ. Because λ, it follows from 1/ F ρ ) γλ) = ρ1/ F ) γλ) that b/ρ, and by Theorem in Bingham et al. 1987), λ F b/ρ) γ 1 as ρ. 39) The small-grain scaling assumption λ F ρ 1) implies that λ F ρ ɛ) for all ɛ >. Observe that for all ρ such that b ɛ we have γ 1 λ F ρ b) λ F ρ ɛ). Because the right-hand side above converges to zero, b must eventually become less than ɛ as ρ. In other words, b. Let φ L 1 L 2. We start by showing that for almost all x, lim ρ Ψ vm φ x, bv)) λf ρ/b dv) = Ψvφx)) v γ 1 dv. 4) Observe first that because Ψv) is continuous and Ψv) 2 v v 2 /2), we can apply Lemma 1 with p = 1 and q = 2 and with ρ/b in place of ρ) to the function v Ψvφx)) and conclude that Ψvφx)) λf ρ/b dv) λ F ρ/b 1) Ψvφx)) γv γ 1 dv.

22 22 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT Using 39), this shows that for all x, Next, let lim ρ Ψvφx)) λf ρ/b dv) = g ρ v) = Ψvm φ x, bv)) Ψvφx)). Ψvφx)) v γ 1 dv. 41) Using Ψv) v 2 /2 and v u) 2 = v + u)v u) we see that for all a >, sup v 2 F ρ/b 1) g ρ v) φ x) + φx) ) sup m φ x, bv) φx) /2. <v a <v a By Lemma 3, the right-hand side tends to zero for almost all x, as ρ recall that b tends to zero together with ρ). Moreover, Ψv) 2 v shows that v 1 Fρ/b 1) g ρ v) 2φ x) + φx) ) for all ρ and v. Thus, we can now use Lemma 2 again with p = 1 and q = 2 and ρ/b in place of ρ) to conclude that lim g ρ v) F ρ/b dv) =. ρ Combining this with 41) now shows the validity of 4). Property 4) implies that lim Ψ vm φ x, bv)) λf ρ/b dv) dx ρ = Ψ vφx)) v γ 1 dv dx, 42) provided we can take the limit in 42) inside the dx-integral. To justify this interchange of the limit and the integral, choose a small enough ɛ > such that γ 1 + ɛ, 2 ɛ). Then Ψv) 2 min v, v 2 ) 2 min v γ ɛ, v γ+ɛ ) and m φ x, bv) φ x) imply that Ψvm φ x, bv)) 2φ x) γ ɛ + φ x) γ+ɛ )v γ ɛ v γ+ɛ ). Moreover, by Lemma 1 with ρ/b in place of ρ), v γ ɛ v γ+ɛ ) λf ρ/b dv) λ F ρ/b 1) v γ ɛ v γ+ɛ ) γv γ 1 dv,

23 SCALING LIMITS FOR RANDOM FIELDS 23 so by 39) we see that the integral on the left-hand side converges to 2ɛ 1, and hence becomes eventually less than 1 + 2ɛ 1 as ρ. Thus, for all ρ small enough, Ψvm φ x, bv)) λf ρ/b dv) ɛ 1 )φ x) γ ɛ + φ x) γ+ɛ ). By Lemma 3, the right-hand side above is dx-integrable. Thus, the dominated convergence theorem shows the validity of 42). Further, using 2) lim E exp i J ) λ,ρφ) E J λ,ρ φ) ) = exp Ψ vφx)) v γ 1 dv dx. ρ b By splitting the integration over into {x : φx) } and {x : φx) < } and performing a change of variables, one can verify that the right-hand side above equals exp d γ φ + γ γ + d γ φ γ γ), where d γ is the complex conjugate of d γ = Ψv) v γ 1 dv. Moreover, d γ = Γ2 γ) γγ 1) cos πγ ) 1 i tan πγ ) ), 2 2 see Exercise 3.24 in Samorodnitsky and Taqqu 1994). Comparing the definition of c γ given in 16) with the characteristic functional of Λ γ in 13), we conclude that lim E exp ρ i J ) λ,ρφ) E J λ,ρ φ) = E e icγλγφ), b which completes the proof of Theorem 2 part iii) Proof of Proposition 2. Assume that φ is a positive measure in M 2H 1), let α = 2H 1, and define φdy) fx) = c α/2,d x y 1 α/2)d. Then the composition rule for Riesz kernels Landkof, 1972, Section 1.1) shows that c α/2,d x y 1 α/2)d x y 1 α/2)d dx = c 1 α,d y y 1 α)d.

24 24 I. KAJ, L. LESKELÄ, I. NORROS, AND V. SCHMIDT Hence, by changing the order of integration we see that fx) 2 φdy) φdy ) dx = c α,d y y 1 α)d = φ, φ α, where φ, ψ α is the Riesz inner product defined by 5). Comparing this with 11) shows that the Gaussian random variables on the left and the righthand side of 19) have the same variance, and thus equal in distribution. Equality of the finite-dimensional distributions follows by linearity. References. Bingham, N. H., Goldie, C. M. and Teugels, J. L. 1987). Regular Variation. Cambridge University Press. Gaigalas, R. and Kaj, I. 23). Convergence of scaled renewal processes and a packet arrival model. Bernoulli Gripenberg, G. and Norros, I. 1996). On the prediction of fractional Brownian motion. J. Appl. Prob Kaj, I. 25). Limiting fractal random processes in heavy-tailed systems. In Proc. Fractals in Engineering V, Tours, France J. Lévy-Véhel and E. Lutton, eds.). Springer. Kaj, I. and Martin-Löf, A. 24). Scaling limit results for the sum of many inverse Lévy subordinators. Preprint 23, 24/25 Fall. Institut Mittag Leffler, Stockholm. Kaj, I. and Taqqu, M. S. 24). Convergence to fractional Brownian motion and to the Telecom process: The integral representation approach. Preprint 24:16. Dept. of Math., Uppsala University. Kallenberg, O. 21). Foundations of Modern Probability. Springer. Konstantopoulos, T. and Lin, S.-J. 1998). Macroscopic models for long-range dependent network traffic. Queueing Systems Kurtz, T. G. 1996). Limit theorems for workload input models. In Stochastic networks: Theory and applications F. P. Kelly, S. Zachary and I. Ziedins, eds.). Oxford University Press, Landkof, N. S. 1972). Foundations of Modern Potential Theory. Springer.

25 SCALING LIMITS FOR RANDOM FIELDS 25 Mandelbrot, B. B. and van Ness, J. W. 1968). Fractional Brownian motions, fractional noises and applications. SIAM Review Mikosch, T., Resnick, S. I., Rootzén, H. and Stegeman, A. 22). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab Rudin, W. 1987). Real and Complex Analysis. 3rd ed. McGraw Hill. Samorodnitsky, G. and Taqqu, M. S. 1994). Stable Non-Gaussian Random Processes: Stochastics Models with Infinite Variance. Chapman & Hall. Taqqu, M. S. and Levy, J. B. 1986). Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics E. Eberlein and M. S. Taqqu, eds.). Birkhäuser, Address of the First author Ingemar Kaj Department of Mathematics Uppsala University P.O. Box 48, S-751 6, Uppsala, Sweden ikaj@math.uu.se Address of the Third author Ilkka Norros VTT Technical Research Centre P.O. Box 122, FI-244 VTT, Finland ilkka.norros@vtt.fi Address of the Second author Lasse Leskelä Institute of Mathematics Helsinki University of Technology P.O. Box 11, FI-215 TKK, Finland lasse.leskela@iki.fi Address of the Fourth author Volker Schmidt Department of Stochastics University of Ulm D-8969 Ulm, Germany volker.schmidt@mathematik.uni-ulm.de