LIMIT THEOREMS FOR WEIGHTED SUMS OF INFINITE VARIANCE RANDOM VARIABLES ATTRACTED TO INTEGRALS OF LINEAR FRACTIONAL STABLE MOTIONS

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1 LIMIT THEOREMS FOR WEIGHTED SUMS OF INFINITE VARIANCE RANDOM VARIABLES ATTRACTED TO INTEGRALS OF LINEAR FRACTIONAL STABLE MOTIONS MAKOTO MAEJIMA (KEIO UNIVERSITY) AND SAKURAKO SUZUKI (OCHANOMIZU UNIVERSITY) (Ruig head : Limit theorems for weighted sums) Abstract. Let {ξ j } be a sequece of radom variables which belog to the domai of attractio of a liear fractioal stable motio { H,α (t)} with ifiite variace. We study the covergece of weighted sums I (f) := A f(j/)ξ j with a suitable scalig A, to I(f) := f(u)d H,α(u) i distributio uder suitable assumptios o a class of determiistic fuctios f. We also show that if {f t, t 0} are the kerel fuctios from the movig average represetatio of a liear fractioal stable motio with aother idex H, the {I (f t )} coverges to a liear fractioal stable motio { H+H 1/α,α(t)}. Keywords: liear fractioal stable motio, itegral with respect to liear fractioal stable motio, slowly varyig fuctio, weighted sum. 1. Itroductio We call {X(t), t 0} a Lévy process if it has idepedet ad statioary icremets, X(0) = 0 a.s., it is stochastically cotiuous at ay t, ad almost all sample paths are right cotiuous ad have left limits. I this paper, we eed the Lévy process with time parameter ruig o R. Let {X (t), t 0} be a idepedet copy of {X(t), t 0} ad exted {X(t)} i such a way that X(t) = X ( t + 0) for t < 0. If, furthermore, the distributio of X(1) is symmetric α-stable, 0 < α 2, i the sese that E[exp iθx(1)] = exp{ c θ α }, θ R, for some c > 0, we call {X(t)} a symmetric α-stable Lévy process, ad deote it by {Z α (t), t R} throughout this paper. For simplicity, we always assume c = 1. Whe α = 2, it is othig but the stadard Browia motio up to a multiplicative costat. For 0 < H < 1 ad 0 < α 2 such that H 1, defie a liear fractioal stable α motio { H,α (t), t R} by { } H,α (t) = (t s) H 1/α + ( s) H 1/α + dz α (s), where x + = max(x, 0) ad 0 s = 0 eve for s 0. Whe α = 2, it is the fractioal Browia motio (fbm). I this paper, we are cocered with the case α < 2. Thus we always assume that H > 1/α, because if H < 1/α, almost all sample paths 1

2 of H,α (t) are owhere bouded ([5]), which is ot a iterestig case. Note that H > 1/α implies α > 1, sice H < 1. Let {X j } be idepedet ad idetically distributed radom variables such that (1.1) 1 1/α j=1 X j d Z α (1) as, d where deotes covergece i distributio. Sice α > 1, {X j } have the fiite first momet, ad we assume E[X j ] = 0. Defie a sequece of strogly depedet radom variables {ξ j } by ξ j = k Z c k X j k, j Z, where {c j } are suitably chose depedig o the problem. We wat to fid the class of f s ad coditios for the covergece of ( ) j I (f) := A f ξ j with a suitable scalig A to I(f) := f(u)d H,α (u) i distributio as. This problem was studied by Pipiras ad Taqqu [6] for the case α = 2. They gave sufficiet coditios for the covergece (1.2) 1 f H ( j ) ξ j R f(u)db H (u) i distributio as, where B H is a fbm with idex H (1/2, 1). They also cosidered the followig problem i [6]. If {f t, t 0} are the kerel fuctios from the movig average represetatio of fbm with aother idex H, the the limit {I(f t )} i the sese of covergece of fiite-dimesioal distributios is a fbm with idex H + H 1/2. The purpose of this paper is to study the limit theorems of Pipiras ad Taqqu [6] i the case α < 2. For it, we take liear fractioal stable motios { H,α (t)} istead of {B H (t)}. The rest of the paper is orgaized as follows. I Sectio 2, we propose a class of determiistic fuctios f for which stochastic itegrals with respect to liear fractioal stable motios are well-defied (also see [3]). I Sectio 3, we preset our first theorem of type (1.2) ad the proof is give. I Sectio 4, we also show that if {f t, t 0} are the kerel fuctios from the movig average represetatio of a liear fractioal stable motio with aother idex H, the {I (f t ), t 0} coverges to a liear fractioal stable motio C H+H 1/α,α(t) with some C > 0, i the sese of fiite-dimesioal distributios. 2

3 2. Stochastic itegrals Throughout the paper we deote by f g the usual covolutio: (f g)(s) = f(s u)g(u) du for measurable fuctios f ad g. I this sectio we defie stochastic itegrals with respect to d H,α. The followig fact has bee poited out already i [3] by oe of the authors of the preset paper with other coauthors. Sice the paper [3] was ot published, we explai it here with the permissio of two coauthors. I the followig, we put β = H 1/α for otatioal coveiece, ad thus β satisfies 0 < β < 1 1/α. The H,α (t) is writte as (2.1) H,α (t) = Defie, for γ > 0, The, sice (t s) β + ( s) β + = β ρ γ (x) := (see [6]), (2.1) ca be rewritte as (2.2) H,α (t) = s {(t s) β + ( s) β +} dz α (s). { 0, x 0, γ x γ 1, x < 0. 1 [0,t) (u)(u s) β 1 du = (1 [0,t) ρ β )(s) (1 [0,t) ρ β )(s) dz α (s). O the other had, for the simple fuctio of the form f(u) = f k 1 [uk,u k+1 )(u), f k R, u k < u k+1, k = 1,...,, k=1 we should defie f(u)d H,α (u) = f k ( H,α (u k+1 ) H,α (u k )). k=1 However, by (2.2), the right-had side ca be rewritte as (f ρ β )(s) dz α (s). Thus we arrive at the followig defiitio of I(f). Defiitio 2.1. Let For f Λ α,β, we defie (2.3) I(f) = Λ α,β = {f : f ρ β L α }. f(u)d H,α (u) := (f ρ β )(s) dz α (s). 3

4 Clearly, I(f) is liear with respect to f. Here, otice that the right-had side of (2.3) is well-defied thaks to the assumptio that f ρ β L α, ad it holds E [ e iθ I(f)] = exp { θ α f ρ β α Lα}, θ R. We ext study the class Λ α,β. By the Hausdorff-Youg iequality, we immediately have Lemma 2.2. Let p 1. (i) If f L 1 L p, g L 1 ad h L p, the f (g + h) L p f L p g L 1 + f L 1 h L p. (ii) If f f i L 1 L p, g g i L 1 ad h h i L p, the f (g + h ) f (g + h) i L p. Now let α ad β be as before. Recall that 1 < α < 2 ad 0 < β < 1 1 α. Theorem 2.3. Let p > 1. (Note that the case p = α is icluded.) The, there exists 1 β C = C(β, p) < such that for every f L p L 1, (2.4) f ρ β L p f L p + C f L 1, ad, therefore, (takig p = α), we have L p L 1 Λ α,β. Proof. Let g(x) = ρ β (x)1 ( 1,0) (x) ad h(x) = ρ β (x)1 (, 1] (x). The β 1 > 1 ad p(β 1) < 1 imply g L 1 ad h L p, respectively. I fact, g L 1 = 1. Therefore, applyig the previous Lemma, we have f ρ β L p f L p + f L 1 h L p, which is (2.4). The followig corollaries follow easily from Theorem 2.3. Corollary 2.4. Let f(x), x R, be a measurable fuctio vaishig outside a compact set. If there exist costats C > 0, a R, 1 < γ < 0 such that α the f Λ α,β. f(x) C x a γ, x R, Corollary 2.5. If f(x), x R, is a bouded, itegrable fuctio, the f Λ α,β. Now we discuss aother importat fuctio belogig to Λ α,β. This fuctio will be treated agai i Sectio 4. We also metio the fact (Theorem 2.6), which is ot used i this sectio but will be used i Sectio 4. 4

5 For γ R ad t R, we put { (t u) γ + ( u) γ f t,γ (u) = +, γ 0, 1 (0,t] (u), γ = 0, u R. Recall that this fuctio appeared i (2.1): (2.5) H,α (t) = f (u) dz α (u), β = H 1/α(> 0). If 1 α < γ < 0, the applyig Corollaries 2.4 ad 2.5 to f t,γ1 [ A,A] ad f t,γ 1 [ A,A] c, respectively, we see that f t,γ Λ α,β, ad hece its stochastic itegral is defied ad ca be calculated explicitly as follows. Theorem 2.6. If 1 α < γ < 0, the f t,γ Λ α,β ad it holds that where B(, ) is the usual beta fuctio. Proof. By a direct calculatio we see that f t,γ (u)d H,α (u) = β B(γ + 1, β) H+γ,α (t), (2.6) f t,γ ρ β = β B(γ + 1, β) f +γ, provided that β 1 < γ < 0. Therefore, it is easy to see that f t,γ ρ β L α. Also, by the defiitio of the stochastic itegral, we ca deduce that f t,γ (u)d H,α (u) = (f t,γ ρ β )(u) dz α (u) = β B(γ +1, β) f +γ (u) dz α (u). The first equality is the defiitio ad the secod follows from (2.6). However, the extreme right-had side equals β B(γ + 1, β) H+γ,α (t) by (2.5). 3. Covergece to stochastic itegrals Our first result i the preset paper is the followig. Theorem 3.1. Let 1 < α < 2 ad 1/α < H < 1. Recall that we put β = H 1, α let c j = βj β 1 L(j), j = 1, 2, 3,..., for some slowly varyig fuctio L(x), x > 0, ad defie ξ j = c k X j k, j Z, k=1 where {X j } j= is a sequece of idepedet idetically distributed radom variables itroduced i (1.1).The 1 ( ) j d f ξ H j f(u)d H,α (u) as L() for every piece-wise cotiuous fuctio f(x), x R, such that f(u) (1 + u ) (1+ɛ) for some ɛ > 0. 5

6 Here f(u) g(u) meas that there exists K > 0 satisfyig f(u) Kg(u) for all u R. Remark 3.2. Note that {c j } above satisfies that j=1 c j a < for some a < α. The the sequece of radom variables {ξ j } belogs to the domai of attractio of H,α (t), because it satisfies the coditios (a) ad (b) i the followig result, which is a kow result o the domai of attractio of the liear fractioal stable process H,α (t). See the theorem below. Therefore, if f is a simple fuctio with compact support, our assertio is already kow, although we do ot use it here, ad our purpose here is to exted it for more geeral fuctios. Theorem 3.3 ([4]). Let 1 < α 2, 1/α < H < 1 ad let {c j } j= be a sequece of real umbers such that j c j a < for some a < α. Suppose that there exists a L() which is slowly varyig at such that 1 (a) lim β L() c j = 1, j=1 (b) c j = O ( j β 1 L(j) ) as j. The [t] 1 f.d. ξ H j H,α (t) as, L() j=1 where f.d. meas covergece of all fiite dimesioal distributios. We ext proceed to the proof of Theorem 3.1. First, otice that ( ) j f ξ j = ( ) j f c k X j k = ( ) j f c j m X m. m Z j=m+1 Therefore, puttig k=1 (3.1) g (s) := we ca rewrite the sum as follows: 1 f H L() ( j j>[s] f ) ξ j = 1 ( ) j cj [s] β L(), ( j g 1/α ) X j. Thus the weighted sum of ξ j is represeted by that of X j. The covergece of the latter is already discussed ad our mai tool here is the followig theorem, which is essetially due to [4] although the coditios are modified a little. A similar result is also foud i Avram ad Taqqu [1]. Theorem 3.4 ([4]). Let {X j } be radom variables i (1.1) ad suppose that fuctios g, g 1, g 2,... defied o R satisfy the followig coditios: (A) g g a.e. as, where g (u) g(u) at u = u 0 meas that g (u ) g(u 0 ) wheever u u 0. 6

7 (B) g g as i L α+ɛ L α ɛ for some ɛ > 0, where g (u) = g ([u]/), u R, = 1, 2, 3,.... The we have 1 ( ) j d g 1/α X j g(u)dz α (u) as. We shall see that {g } i (3.1) satisfy the coditios (A) ad (B) with g = f ρ β. Let φ(x) = βx β 1 L(x), (x 0) so that c j = φ(j) (j N). Defie { c[ x] ρ () (x) = β 1 L(), x < 0, 0, x 0. The we have for x 0, ρ () (x) ρ β (x) as, where ρ β (x) is the same as before. Sice φ is regularly varyig, for ay ɛ > 0, there exists a C > 0 such that ( ) { β 1 (x ) ɛ ( ) } ɛ φ(x) x x φ(y) C + y y y for all sufficietly large x, y > 0. Sice it is harmless to chage the values of fiitely may c j s, we may, without loss of geerality, modify the value of φ(x) i a eighborhoods of the origi. Therefore, we may assume that ρ () (x) C ( ρ β+ɛ (x) + ρ β ɛ (x) ) holds for all x R ad all sufficietly large. Now we prepare the followig two lemmas to check that the above g satisfies the coditios (A) ad (B) i Theorem 3.4. Lemma 3.5. Let p > 1 1 β. (Note that the case p = α is icluded.) If ϕ ϕ i L p L 1, the (3.2) ϕ ρ () ϕ ρ β i L p. Proof. Let ρ () 1 = ρ () 1 ( 1,0). The ρ () 1 ρ β 1 ( 1,0), where the covergece is i L 1 because ρ () 1 (x) C x β 1 ɛ ad β > 0. Similarly, let ρ () 2 = ρ () 1 (, 1]. The ρ () 2 ρ β 1 (, 1], where the covergece is i L p because ρ () 2 (x) C x β 1+ɛ ad p(β 1 + ɛ) < 1. Therefore (3.2) follows from Lemma 2.2. Lemma 3.6. Let ϕ ad ϕ be measurable fuctios satisfyig (i) ϕ ϕ at all cotiuity poits of ϕ ad (ii) there exits a costat C > 0 ad a F L 1 such that ϕ (x) C ad ϕ (x) F (x). 7

8 The (3.3) ϕ ρ () ϕ ρ β a.e. Proof. Let ρ () 1 ad ρ () 2 be as i the proof of the previous lemma. If x x, the ϕ (x u) ρ () (u) ϕ(x u) ρ β (u), a.e. Therefore, because (ϕ ρ () 1 )(x ) = 0 1 ϕ (x u) ρ () (u) du 0 ϕ (x u) ρ () (u) C u β 1 ɛ, 1 < u < 0. (Choose ɛ > 0 so that β ɛ > 0.) Similarly, sice (ϕ ρ () 2 )(x ) = ϕ (u) ρ () 2 (x u) du 1 ϕ (u) ρ () 2 (x u) cost. F (u). (Notice that ρ () 2 is bouded). Thus we have which is (3.3). ϕ(x u) ρ β (u) du ϕ(u) (ρ β 1 (, 1] )(x u) du, (ϕ ρ () )(x ) = (ϕ ρ () 1 )(x ) + (ϕ ρ () 2 )(x ) (ϕ ρ β )(x), Now we are ready to prove Theorem 3.1. Sice for s R, ( ) j + [s] ( cj [s] (3.4) g (s) = g (s) = f β L() = f we ca rewrite as j=1 (3.5) g (s) = (f ρ () )([s]/) where f (x) = f([x]/). [x] ) ρ () (x) dx, It follows that f i Theorem 3.1 ad the above f satisfy the coditios of ϕ ad ϕ i Lemmas 3.5 ad 3.6. Furthermore, ϕ ρ () ad ϕ ρ β i the previous two lemmas ca be replaced with g = f ρ () ad g = f ρ β i Theorem 3.4, respectively. Therefore, by otig the followig propositio derived easily from Lemmas 3.5 ad 3.6, the proof of Theorem 3.1 is completed by Theorem 3.4. Propositio 3.7. Let g be the fuctios defied by (3.1). The, uder the coditios of Theorem 3.1, g f ρ β a.e. Furthermore, the covergece also holds i L α+ɛ L α ɛ for some ɛ > 0. 8

9 4. The case that the limit is aother liear fractioal stable motio I this sectio, we exted the followig kow story for fbm ([7]) to the case of the liear fractioal stable motio. It is show i [7] that if {f t, t 0} are the kerel fuctios from the movig average represetatio of a fbm with idex H, the the limit is a fbm with idex H + H 1. We ca ow ask what happes if we take 2 { H,α (t)} istead of {B H (t)}. The followig theorem gives us a aswer. Theorem 4.1. Let 1 < α < 2, 1/α < H < 1, 0 < H < 1/α, β = H 1/α ad f (u) = (t u) β + ( u) β +, u R. Suppose that the sequece {ξ j } satisfies the coditios of Theorem 3.1. The, the processes 1 ( ) j f H ξ j, t R L() coverge (up to costat) to the liear fractioal stable motio {βb(β +1, β) H+H 1/α,α(t), t R} i the sese of fiite-dimesioal distributios. Proof. Put ad Y (t) = Y (t) = 1 H L() f ( ) j ξ j, t R f (s)d H,α (s), t R. Sice 1 α < β < 0, it follows from Theorem 2.6 that f Λ α,β ad f (s)d H,α (s) = (f ρ β )(s)dz α (s) = βb(β + 1, β) H+H 1/α,α(t). We are goig to show that as, Y (t) coverges to Y (t) i the sese of fiitedimesioal distributios. By the Cramer-Wold method (a radom vector coverges i distributio if ad oly if all liear combiatios of its elemets do), it is eough to show that for each t R, Y (t) coverges to Y (t) i distributio. To prove it, otice that we ca rewrite the sum as follows: (4.1) where 1 H L() j=1 f ( j g,t (s) := j>[s] ) ξ j = 1 f ( j g 1/α,t ( ) j cj [s] β L(). ) X j, Moreover we ca rewrite as ( ) j + [s] ( cj [s] g,t (s) = f β L() = f [x] ) (4.2) ρ () (x) dx (4.3) = (f () ρ () )([s]/), 9

10 where f () (x) = f ([x]/). We shall show that g,t = f () ρ () satisfies the coditios (A) ad (B) i Theorem 3.4 with g = f ρ β i a similar way as before. However, this time, we caot apply Lemma 3.6, because f (u) is ot bouded. Thus we eed the followig lemma, which is a modificatio of Lemma 3.6. Lemma 4.2. Suppose ϕ ad ϕ are measurable fuctios such that (i) ϕ ϕ at all cotiuity poits of ϕ, ad (ii) there exits a costat C > 0 ad γ ( 1 < γ < 1/α) satisfyig { C x (4.4) ϕ (x) γ, t < x < 0, 0, otherwise. The ϕ ρ () ϕ ρ β a.e. Proof. Let ρ () 1, ρ () 2 be as i the proof of Lemma 3.5. By (i), if x x, the ϕ (x u)ρ () (u) ϕ(x u)ρ β (u), a.e. Therefore (ϕ ρ () 2 )(x ) = ϕ (u)ρ () 2 (x u)du because ϕ (u)ρ () 2 (x u) cost. u γ (γ > 1). (Notice that ρ () 2 is bouded.) Next we shall show (4.5) (ϕ ρ () 1 )(x ) = 0 1 ϕ (x u)ρ () (u)du ϕ(u)ρ β 1 (, 1] (x u)du 0 1 ϕ(x u)ρ β (u)du by dividig the itegral ito three cases: (1) 0 < x, (2) 1 > x ad (3) 1 < x < 0. Case (1) is easily verified, because ϕ (x u)ρ () (u) = 0. Case (2) is the same case as Lemma 3.6, sice ϕ is bouded o 1 < u < 0 ad it belogs to L 1 too. As to Case (3), let u = y be a itersectio poit of x u γ ad ρ () 1 (u), ad divide the itegral ito two parts as follows: 0 1 ϕ (x u)ρ () (u)du = Whe x u y, it follows that ( y 0 + x y ) ϕ (x u)ρ () (u)du = I 1 + I 2. ϕ (x u)ρ () (u) cost. x u γ (γ > 1) sice ρ () 1 is bouded. Whe y u 0, it follows that ϕ (x u)ρ () (u) cost. u β 1 ɛ (Note that β 1 ɛ > 1 ). Therefore Case (3) is verified. Thus we have (ϕ ρ () )(x ) = (ϕ ρ () 1 )(x ) + (ϕ ρ () 2 )(x ) (ϕ ρ β )(x). 10

11 The followig lemma plays the same role as Propositio 3.7 i the proof of Theorem 4.1. Therefore, oce we could prove Lemma 4.3, we ca complete the proof of Theorem 4.1. Lemma 4.3. Let g,t be the fuctios defied by (4.2). The, uder the coditios of Theorem 4.1, g,t f ρ β a.e. Furthermore, the covergece also holds i L α+ɛ L α ɛ for some ɛ > 0. Proof. We check that g,t = f () ρ () g = f ρ β i L α+ɛ L α ɛ ad i the sese of. To prove f () ρ () f ρ β i L α+ɛ L α ɛ, it is eough to show that f () f i L p L 1 by Lemma 3.5. Sice f () ( ) are icreasig fuctios, it follows that (4.6) f () (u) f (u). Furthermore, sice it follows that f L p L 1 ad we ca verify (4.7) lim f () (u) = f (u) a.e. ad we have that f () f L α+ɛ L α ɛ, by Lemma 3.5 as metioed. i L p L 1, which implies that f () ρ () f ρ β i Next, we prove the covergece i the sese of. For this purpose, it is eough to check that f () ad f satisfy coditios (i) ad (ii) i Lemma 3.6 or Lemma 4.2 with the replacemet of ϕ ad ϕ by f () ad f, respectively. As to coditio (i), it is trivial because f (u) is piece-wise cotiuous. As to coditio (ii), observe that f explodes aroud u = t ad u = 0. Suppose that t < 0. (I the case that t 0, the proof is similar, so that we omit the detail.) We ca divide f ito four parts: f (u) 1 (,2t) (u) + f (u) 1 [2t,t) (u) + f (u) 1 [t,0) (u) + f (u) 1 [0, ) (u) =: f 1 (u) + f 2 (u) + f 3 (u) + 0. It follows that f 2 (u) ad f 3 (u) satisfy the coditio (ii) i Lemma 4.2, sice β > γ. As to f 1 (u), ote that f (u) = O( u β 1 ) as u. Therefore f (u) is bouded ad f (u) F (u), where F L 1, because β < 0. Applyig Lemma 3.6, we have that g,t f ρ β a.e. ACKNOWLEDGEMENT The authors would like to thak Yuji Kasahara for his may valuable commets durig this work. They are also grateful to a associate editor for his/her helpful commets o the fial versio of the paper. 11

12 Refereces [1] Avram, F. ad Taqqu, M.S.: Weak covergece of movig averages with ifiite variace. I: Eberlei, E. ad Taqqu, M.S. (Eds.), Depedece i Probability ad Statistics. Birkauser, Bosto, (1986). [2] Bigham, N.H., Goldie, C.M. ad Teugels, J.L.: Regular Variatio. New York: Cambridge Uiv. Press [3] Irisawa, M., Maejima, M. ad Shimomura, S.: A limit theorem for weighted sums of ifiite variace radom variables with log-rage depedece, KSTS/RR-03/005, Dept. of Math., Keio Uiv [4] Kasahara, Y. ad Maejima, M.: Weighted sums of i.i.d. radom variables attracted to itegrals of stable processes. Probab. Th. Rel. Fields 78, (1988). [5] Maejima, M.: A self-similar process with owhere bouded sample paths. Z. Wahrscheilichkeitstheorie verw. Gebiete 65, (1983). [6] Pipiras, V. ad Taqqu, M.S.: Covergece of weighted sums of radom variables with log-rage depedece. Stoch. Proc. Appl. 90, (2000). [7] Samoroditsky, G. ad Taqqu, M.S.: Stable No-Gaussia Processes. Lodo: Chapma ad Hall (1994). Preset Addresses: MAKOTO MAEJIMA DEPARTMENT OF MATHEMATICS, KEIO UNIVERSITY, HIYOSHI, YOKOHAMA, JAPAN. maejima@math.keio.ac.jp SAKURAKO SUZUKI GRADUATE SCHOOL OF SCIENCE AND HUMANITIES, OCHANOMIZU UNIVERSITY, TOKYO, JAPAN. suzukisakurako@aol.com 12