The Stable Limit Theorem

Transcription

1 The Stable Limit Theorem Thomas oy September 6, 0 Itroductio Stable distributios are a class of probability distributios that have may useful mathematical properties such as skewess ad heavy tails. They have bee used as a model for differet types of physical ad ecoomic systems. I practice it has bee difficult to use stable distributios due to the fact that the desities ad distributio fuctios have closed formulas oly i specific cases (Gaussia, Cauchy ad Lévy). However, thaks to recet reliable computer programs, it is possible to compute stable desities, distributio fuctios ad quatiles. The focus of this paper will be the Stable Limit Theorem. I the course of this paper, we will try to rewrite the proof writte by William Feller [] ad add certai details that were omitted i his book. Afterwards, we will give some examples where the Stable Limit Theorem ca be applied. Stable Distributios Defiitio. (Defiitio VI.3. of []). A distributio F is ifiitely divisible if for each it ca be represeted as the distributio of the sum S = X, X, of idepedet radom variables with commo distributio F. We deote by ϕ Y the characteristic fuctio of a radom variable Y, i.e. ϕ Y (u) = E(e iuy ), where u. The ext result gives the represetatio of the characteristic fuctio of a ifiitely divisible radom variable usig a itegral with respect to a caoical measure. (See Defiitio A., Appedix A, for the defiitio of a caoical measure.) Theorem. (Theorem XVII.. of []). The class of characteristic fuctios of ifiitely divisible distributios coicides with the class of fuctios of the form: { ϕ Y (u) = E(e iuy ) = exp iub + e iux } iu si x x M(dx) Project supported by a NSEC Udergraduate esearch Award at Uiversity of Ottawa, durig the summer of 0. (Supervisor: aluca Bala) ()

2 where M is a caoical measure ad b is a real umber. I this case, we write X ID F eller (b, M). emark.3. I the right-had side of (), the itegrad is defied to be u whe x = 0. Hece, if M = Cδ 0, where δ 0 deotes the Dirac measure at 0, the Y has ormal N(b, C) distributio. emark.4. There is a alterative way of writig the characteristic fuctio of a ifiitely divisible distributio as: ϕ Y (u) = exp {iua σ u } + (e iux iux { x } )ν(dx) where a, σ 0 ad ν is a Lévy measure o, i.e. ν satisfies: (x )ν(dx) < ad ν({0}) = 0. This is called the Lévy represetatio. Here a b deotes the miimum betwee a ad b. I this case, we write Y ID(a, σ, ν). Note that if Y ID(a, σ, ν) the Y ID F eller (b, M) where b = a + (si x x { x } )ν(dx) ad M(dx) = x ν(dx) + σ δ 0 (dx) Defiitio.5 (Defiitio VI.. of []). Let F be the law of a radom variable X, (X i ) i be i.i.d. copies of X ad S = = X i. The distributio F is stable if for each there exist costats c > 0, d such that ad F is ot cocetrated at oe poit. S d = c X + d () Theorem.6 (Theorem VI.. of []). The ormig costats i () are of the form c = /α with 0 < α. Defiitio.7. A radom variable Y with characteristic futio ϕ Y (u) = e ψ Y (u), u, where ψ Y (u) is give by ( iuδ u α γ α isg(u)β ta πα ), α ψ Y (u) = iuδ u γ( + isg(u)β π l u ), α = (3) for some α (0, ], γ > 0, β [, ] ad δ is said to have a α-stable S α (γ, β, δ) distributio. I this case, we write Y S α (γ, β, δ). Usig Example XVII.3.(g)-(h) of [], we obtai the followig lemma.

3 Lemma.8. Assume that Y ID F eller (0, M α ), i.e. its characteristic fuctio is give by () where 0 < α < ad M α (dx) = [c + x α+ (0, ) (x) + c ( x) α+ (,0) (x)]dx (4) for some c +, c 0. The Y S α (γ, β, δ) where the parameters (γ, β, δ) are give by: Γ(3 α) (c γ α + + c ) = α( α)( α) cos πα, α (c + + c ) π (5), α = β = c + c c + + c (6) si x (c + c ) dx, 0 < α < 0 xα+ δ = x si x (c + c ) 0 x dx, < α < 0, α = (7) If Y ID F eller (b, M α ) where b ad M α is give by (4), the Y S α (γ, β, δ ) where δ = b + δ ad (γ, β, δ) are the same as above. emark.9. The costat δ i (7) ca also be expressed as: si x x M α(dx), 0 < α < δ = x si x x M α (dx), < α < 0, α = (8) To see this, ote that if α (0, ) δ = (c + 0 = (c + 0 = si x x x α+ dx c si x x x α+ dx + c si x x M α(dx) The cases α (, ) ad α = are similar. 0 0 ) si x x x α+ dx ) si x x x α+ dx Lemma.0. Assume that Y ID(0, 0, ν α ), where the measure ν α is give by: ν α (dx) = [c + x α (0, ) (x) + c ( x) α (,0) (x)]dx (9) 3

4 for some c +, c 0 ad α (0, ). The Y S α (γ, β, δ ) where (γ, β) are give by (5) ad (6), ad x { x } ν α (dx) 0 < α < δ = x { x } ν α (dx), < α < (0) (si x x { x } )ν α (dx), α = Proof. By emark.4, Y ID F eller (b, M α ), where M = M α (M α is give by (4)) ad b = (si x x { x } )ν α (dx) By Lemma.8, Y S α (γ, β, δ ), where (γ, β) are give by (5) ad (6), δ = b+δ ad δ is give by (8) (see emark.9). Hece, whe α < δ = (si x x { x } )ν α (dx) si x ν α (dx) = x { x } ν α (dx) The desired expressio for δ i the case α (, ) or α = follows similarly. 3 Domai of attractio Defiitio 3.. Let (X i ) i be idepedet idetically distributed (i.i.d.) radom variables with law F ad S = i= X i. Let Y be a radom variable with law G. We say that X belogs to the domai of attractio of Y if there exist > 0 ad b such that: emark 3.. Note that () is equivalet to S b d Y () (ϕ (u) ) ψ Y (u) for ay u () where ϕ is the characteristic fuctio of F, the probability distributio of X b, i.e. ϕ (u) = ϕ X a b(u) = ϕ X ( u ) e iub ad ϕ Y (u) = e ψ Y (u). To see this, ote that by the cotiuity of characteristic fuctios, () is equivalet to: ϕ S a b (u) ϕ Y (u) for ay u. (3) 4

5 Note that the characteristic fuctio of S b is give by: { [ ( )]} S E exp iu b ( ) = E exp Xj iu b a j= )]} (iu Xa b = { E [ exp = [ϕ (u)]. The (3) is equivalet to: ( [ϕ (u)] = + (ϕ ) (u) ) ϕ Y (u) = e ψ Y (u) for ay u which is i tur equivalet to (). We cosider the fuctio: U(x) = E ( X ( X x) ), x > 0. (4) Theorem 3.3. Let X, (X i ) i be i.i.d. radom variables with commo distributio F ad let S = i= X i. Suppose that there exist > 0 ad b such that () holds. The Y has a stable distributio with characteristic fuctio () where [ ( )] X b = lim β, β = E si b M = Cδ 0 or M = M α ad M α is give by (4). Proof. We deote We cosider separately two cases: M (dx) = x F (dx) (5) Case : Assume that F is symmetric, (i.e. P (X < x) = P (X > x) for ay x > 0) ad () holds with b = 0. I this case, F is the probability distributio of X. Note that, if F is symmetric, we cosider (X i ) i, a idepedet copy of (X i ) i ad we let X i = X i X i. The ( X i ) i is a sequece of i.i.d. radom variables with a symmetric distributio ad S = X i = S S i= 5

6 where S = i= X i. By Theorem.8 of [] ( S S ) b, b d (Y, Y ) where Y is a idepedet copy of Y. By applyig the Cotiuous Mappig Theorem of [] to the map of h :, h(x, y) = x y, we get: ( ) ( ) S S S = b b d Y Y = Ỹ. Hece, the symmetrized sequece ( X i ) i satisfies () with b = 0 ad the same as the origial sequece (X i ) i. Usig Corollary A.4, with C = ad b = 0, we get, by () that there exist a caoical measure M ad b [ such ( that )] M M properly ad X β b, where β = si xf (dx) = E si. I this case ψ Y (u) = iub + + e iux iu si x x M(dx). We ow prove various properties of the sequece ( ), which are cosequeces of the fact that S d Y. Sice M M properly, we have (accordig to Defiitio A..(i)) M ([ x, x]) M([ x, x]) for ay cotiuity poit x of M. Usig the defiitio (5) of M ad the fact that i this case F is the distributio of X, we have: M ([ x, x]) = [ x,x] (y)y F (dy) = ( ) X E { X = a a E ( X { X ax}). a We obtai that ( ) must satisfy: U( x) M([ x, x]) for ay cotiuity poit x of M. (6) a From Defiitio A..(ii), we get: M + (x) = x y M (dy) M + (x) = 6 x y (dy)

7 for ay cotiuity poit x of M. Note that: x y M (dy) = x y y F (dy) = (x, ) (y)f (dy) ( )) ( ) X = E ( (x, ) = P > x Xa We obtai: Similarly, = P (X > x) = ( F ( x)) [ F ( x)] M + (x) for ay cotiuity poit x of M. (7) F ( x) M ( x) for ay cotiuity poit x of M. Due to (), we have ϕ (u) = (ϕ (u) ) 0 i.e. ϕ (u) = ϕ X ( u ). This proves that ϕ X ( u ) = E (e i u X) a. Hece, Feller cocludes i [] (he details of Feller s reasoig are still ot clear to us), that u 0 i.e.. It remais to show the details of Feller s reasoig. We ow prove that To see this, ote that [ ( ϕ S (u) = E exp iu S )] = a Similarly, ϕ S + (u) = [ ϕ X ( u + j= + [ ( E exp iu X )] [ ( )] j u = ϕ X = [ϕ (u)] ϕ Y (u). )] = [ϕ +(u)] = [ϕ +(u)] + ϕ + (u) ϕ Y (u). The we get S d S d + Y ad Y. So, Feller cocludes that. + It looks ituitively right, but it remais to show the details of to prove this statemet. Usig (6) ad Lemma A.7 (Appedix A) with λ = a, χ(x) = M([ x, x]) ad D the set of cotiuity poits of M, we obtai that U is regularly varyig of idex ρ ad M([ x, x]) = Cx ρ, for some C > 0. Note that ρ 0. (If ρ < 0 7

8 the M() = lim x M([ x, x]) = lim x Cx ρ = 0, which is impossible). We deote ρ = α for some α. Hece for a slowly varyig fuctio l, ad U(x) = x α l(x) (8) M([ x, x]) = Cx α (9) If α =, the M([ x, x]) = C for ay x > 0 i.e. M = Cδ 0. If α <, the M is give by: ( M(dx) = C( α) x α+ (0, ) (x) + ) ( x) α+ (,0) (x) dx (0) because x x C( α) ( y α+ (0, ) (y) + ) ( y) α+ (,0) (y) dy = Cx α I this case, M has o atom at the origi, i.e. M({0}) = 0. Sice M is a caoical measure, M + (x) = Usig (0), we have M + (x) = We obtai that: x y C( α) y α+ dy = M + (x) = C y = = M([ x, x]). M(dy) <, for ay x > 0. y C( α) y α dy x C( α) x α. α α α x α, for ay x > 0. () Case : Assume ow that F is ot symmetric. Usig () ad Corollary A.4 with C = ad b = 0, we coclude that there exist a caoical measure M ad b such that M M properly ad β b, where I this case β = [ ( )] X si(x)f (dx) = E si b. + ψ Y (u) = iub + e iux iu si x x M(dx). 8

9 Sice M M properly, we have:. M ([ x, x]) M([ x, x]). M + (x) = x y M (dy) x y M(dy) = M + (x) for ay cotiuity poit x of M. We ote that, similarly to Case, x y M (dy) = [ F ( (x + b ))]. x 3. M x ( x) = y M (dy) y M(dy) = M ( x) for ay cotiuity poit x of M. Similarly, x y M (dy) = [F ( ( x + b ))]. Usig the defiitio (5) of M ad the fact that, i this case, F is the distributio of X b, we see that M ([ x, x]) = [ x,x] (y)y F (dy) = E [ ( ) ] X b { X b x} a Therefore from. we obtai that [ ( ) ] X E b { X b x} M([ x, x]) a which is the aalogue of (7) i the o-symmetric case. O the other had: M + (x) = x y y F (dy) = (x, ) (y)f (dy) ( )] ( ) X = E [ (x, ) b = P b > x Xa = P (X > (x + b )) = [ F ( (x + b )]. Similarly, M ( x) = F [ ( x + b )]. Usig. ad 3. we get: [ F ( (x + b ))] M + (x) ad F ( ( x + b )) M ( x) () for ay x > 0 such that x ad x are cotiuity poits of M. Note that b 0 To see this, recall that ϕ (u) (due to ()). But ϕ (u) = ϕ X ( u )e iub ad ϕ X ( u ) sice. (Note that we proved that i the case whe F is symmetric ad b = 0. I the geeral case, the symmetrized sequece ( X i ) i satisfies () with the same as (X i ) i. Hece by Case,. ) Hece e iub, i.e. b 0. 9

10 We claim that, due to the fact that b 0, we have: [ F ( (x + b )) + F ( x)] 0 ad [F ( ( x + b )) F ( x)] 0. It remais to show this claim is true. Hece, [ F ( x)] M + (x) ad [F ( x)] M ( x) (3) for ay x > 0 such that x ad x are cotiuity poits of M. This coclusio also eeds to be prove. We ow use Lemma A.7 three times:. Sice ( F )( x) M + (x) for ay x > 0 cotiuity poit of M, we have two situatios: either M + (x) = 0 for ay x or F is regularly varyig of idex α + ad M + (x) = A + x α+ for some A + > 0. Note that, because 0 = lim x M + (x) = A + lim x x α+, it is ecessary that α + > 0.. Sice ( F )( x) M ( x) for ay x > 0 such that x is a cotiuous poit of M, we have two situatios: either M ( x) = 0 for ay x or F is regularly varyig of idex α ad M ( x) = A x α for some A > 0. Note that, similarly, α > P ( X > x) M + (x) + M ( x) (4) for ay x > 0 such that x ad x are cotiuity poits of M. We deote V (x) = P ( X > x) (V is called the tail sum of X). We have two situatios: either M + (x) + M ( x) = 0 for ay x or V is regularly varyig of idex α ad M + (x) + M ( x) = Ax α for some A > 0. Note that, similarly, α > 0. If M + ad M do ot vaish idetically (i.e. some x), we get: M + (x) = M ( x) > 0 for A + x α+ + A x α = Ax α for ay x Hece α + = α = α (i.e. we have the same expoet α for both tails) ad A + + A = A. To summarize, we have two cases: either M + (x) = M ( x) = 0 for ay x (i.e. M = Cδ 0 ) or, M = M α where M α is give by (0) for some α > 0 ad c +, c > 0. Note that if M = M α, the α <. We get this by usig the fact that x M([0, x]) = c + y α+ dy < for ay x >

11 emark 3.4. Whe α (0, ), M α ([0, x]) = c + α x α ad M([ x, 0]) = c α. This gives us: Mα([ y, x]) = Cpx α + Cqy α (5) where C = c + + c α, p = c + c ad q =. (6) c + + c c + + c Whe α (0, ), we also have: M α + (x) = c + α x α = Cp α α x α ad Mα ( x) = c α x α = Cq α α x α. (7) elatios (5) ad (7) also hold whe α =, replacig M α by M = Cδ 0. Whe M = Cδ 0, the characteristic fuctio of Y is E[e iuy ] = exp{iub u C} i.e. Y has ormal distributio with mea b ad variace C. Whe M = M α, the log-characteristic fuctio of Y is give by: + e iuα iu si x ψ Y (u) = iub + x M α (dx), i.e. Y ID F eller (b, M α ) ( ) where b = lim β ad β = E si( X b ). I this case, accordig to Lemma.8, Y has S α (γ, β, δ ) distributio, with δ = δ + b. Theorem 3.5 (Theorem XVII.5. of []). (0) A distributio G posesses a domai of attractio if ad oly if it is stable. (i) The class of stable distributios coicides with the class of ifiitely divisible distributios with caoical measures give by M = Mδ 0 or M = M α, where M α is give by (4) for some c +, c 0, 0 < α <. (ii) The log-characteristic fuctio of a stable distributio is give by (3) for some (α, γ, β, δ) with α (0, ], γ > 0, β [, ], δ. (iii) If X has a stable distributio of idex α, the { x α P (X > x) Cp α α as x x α P (X < x) Cq α α as x where M([ y, x]) = Cpx α + Cqy α for some C > 0, p, q 0, p + q =. Proof. (0) By Theorem.6, if a radom variable Y possesses a domai of attractio, the Y has a stable distributio. Coversely, suppose that Y has a stable distributio. We show that Y belogs to its ow domai of attractio. Let (Y i ) i be i.i.d. (as Y ). Let S = Y Y.

12 By the defiitio of the stable distributio, S d = /α Y + d for a costat d. Hece S d /α d = Y d Y. (i) Suppose Y has a stable distributio. By (0), there exist i.i.d. radom variables (X i ) i such that S d b Y. By Theorem.6 Y IDF eller (b, M), with M = M α or M = Cδ 0. Coversely, suppose that Y ID F eller (b, M). By Lemma.8, Y has a logcharacteristic fuctio (3). (ii) We show that Y has a stable distributio if ad oly if Y has a logcharacteristic fuctio (3). Suppose Y has a stable distributio. The by (i), Y ID F eller (b, M) with M = M α or M = Cδ 0. By Lemma.8, Y has a log-characteristic fuctio (3). Suppose that Y has a log-characteristic fuctio (3). Let Y,...Y be i.i.d. copies of Y. We wat to prove that there exist c > 0 ad d such that: Y Y d = c Y + d By Theorem.6, if such a c exists, the c = /α. We have: [ ϕ Y+...+Y (u) = E e iu(y+...+y)] = [E(e iuy )] = [ϕ Y (u)] = e ψ Y (u) ad [ ϕ cx+d (u) = E e iu(cy +d)] = e iud E[e iucy ] = e iud ϕ Y (c u) = e iud+ψ Y (c u) It suffices to show that: ψ Y (u) = iud + ψ Y ( /α u) If α, we have by (3) : [ ( πα )] ψ Y (u) = iuδ u α γ α isg(u)β ta [ ( πα = iu( )δ + iuδ /α u α γ α isg( /α u)β ta )] with = iud + ψ Y ( /α u) The calculatio for α = is similar. d := iu( )δ (iii) Suppose that Y has a stable distributio. By (0), Y belogs to its ow S domai of attractio, i.e. b d /α Y, where S = Y Y ad (Y i ) i are i.i.d. copies of Y. By (3) ad (7), it follows that, for ay x > 0, [ F ( /α x)] M + (x) = Cp α α x α

13 ad [F ( /α x)] M (x) = Cq α α x α. Hece, for ay x > 0, ad Deote /α x = x. We get: x α P (Y > /α x) Cp α α ( /α x) α P (Y > /α x) Cp α α ( /α x) α P (Y < /α x) Cq α α. ad x α P (Y < /α x) Cq α α If lim x x α P (X > x) the lim x x α P (X > x) = lim x α P (X > x ) for ay sequece (x ), such that x. Take x = /α x. We still eed to see whe the limit exists ad why Feller makes these coclusios whe it does. 4 Coditios for X belogig to the domai of attractio of Stable(α) Lemma 4.. Let X be a arbitrary radom variable. For ay x > 0, defie U(x) = E ( X { X x} ) ad V (x) = P ( X > x). Suppose that E(X ) =, i.e. lim x U(x) =. (i) If U is regularly varyig with idex α for some 0 < α or V is regularly varyig with idex α, for some 0 < α, the x P ( X > x) lim = α x U(x) α. (8) (ii) Coversely, if (8) holds for some 0 < α <, the there exists a slowly varyig fuctio L 0 such that: U(x) αx α L 0 (x) ad P ( X > x) ( α)x α L 0 (x). If (8) holds for α =, the U is slowly varyig. Proof. Apply Theorem A.8 with a = ad b = 0. Theorem 4. (Theorem XVII.5. of []). Let X be a radom variable with law F ad U(x) = E ( X { X x} ). (i) If X belogs to some domai of attractio the: U(x) = x α l(x) (9) 3

14 for some 0 < α ad a slowly varyig fuctio l. (ii) Assume that X is o-degeerate ad U is slowly varyig (i.e. (9) holds for α = ). The X belogs to the domai of attractio of ormal distributio. (iii) Assume that (9) holds for some 0 < α <. The X belogs to the domai of attractio of a stable law of idex α if ad oly if the tails are balaced, i.e. lim x for some p, q 0 with p + q =. P (X > x) P (X < x) = p ad lim P ( X > x) x P ( X > x) = q (30) Proof. (i) By Defiitio 3., there exist ( ) ad (b ) such that S d b Y, where S = i= X i ad (X i ) i are i.i.d. copies of X. By Theorem 3.3, Y ID F eller (b, M) where M = M α or M = Cδ 0 ad M α is give by (4). Assume M = M α with 0 < α <. By (3) ad (7): P ( X > x) M + (x) + M ( x) = C α α x α. By Lemma A.7, it follows that V (x) = P ( X > x) is regularly varyig with idex α. By Lemma 4..(i), (8) holds. By Lemma 4..(ii), U is regularly varyig with idex α, i.e. (9) holds. Assume that M = Cδ 0. I this case, M + (x) = M ( x) = 0 for ay x > 0 ad Y has ormal distributio. By (3), P ( X > ) 0. (3) We ow prove that U is slowly varyig. Sice S d b Y, it follows that ( ) S does ot coverge i probability to 0. Moreover, ( Xk P ) > for some k 0. (3) This statemet eeds to be prove i details. We will prove that there exist C > 0 ad 0 such that U( ) ( ) X a = E k a { Xk } > C for ay 0, N (33) k= where N N is a subsequece. Usig (3) ad (33), it follows that Hece U( ) a P ( X > ) = U() P ( X > ). a a P ( X > ) U( ) 4 0

15 ad relatio (8) holds with α =. By Lemma 4..(ii), ( ) U is slowly varyig. It S remais to prove (33). To see this, ote that sice does ot coverge i probability to 0, there exists ɛ 0 > 0, C 0 > 0 ad a subsequece N N such that P ( S > ɛ 0 ) > C 0 for ay N. (34) Writig X k = X k { Xk } + X k { Xk >}, we obtai that S X k { Xk a } + X k { Xk >a } k= ad hece ( ) ( ) ( ) S X k P > ɛ 0 P { Xk a } > ɛ 0 X k +P { Xk >a } > ɛ 0. k= (35) By Chebyshev s iequality (ad assumig that X k has a symmetric distributio) ( ) X k P { Xk a } > ɛ 0 4 X k ɛ E k= 0 { Xk a } = 4 U( ) ɛ k= 0 a. (36) O the other had, for the secod term o the right had side of (35) we have { } X k { Xk >a } > ɛ 0 { X k > for some k } k= sice if X k < for all k the greater tha ɛ 0. Hece, by (3) ( ) lim P X k { Xk >a } > ɛ 0 k= k= k= k= X k { Xk >} = 0, which caot be lim P ( X k > for some k ) = 0. Usig (34), (35), (36) ad (37) it follows that ( ) S C 0 < lim, N P > ɛ 0 4 U( ) ɛ lim, N 0 a (37) Hece there exists a iteger 0 such that U() 0, N. This cocludes the proof of (33). a C 0 ɛ 0 4 =: C for ay (ii) This will follow from Theorem 4. (below). 5

16 (iii) Assume first that X belogs to the domai of attractio of a stable law of idex α <. By (3), (4) ad (7) { P (X > x) M + (x) = Cp α α x α ad P ( X > x) M + (x) + M ( x) = C α α x α. We the get: P (X > x) P ( X > x) M + (x) M + (x) + M =: p for ay x > 0 ( x) P (X > x) If lim x P ( X > x) exists, the lim x for ay x. We take x =. P (X > x) P ( X > x) = lim P (X > x ) P ( X > x ) Assume ext that (30) holds. The fact that X belogs to domai of attractio of a stable law will follow by Theorem 4. below. We still eed to kow whe the limit exists. Theorem 4. has two importat corollaries. Corollary 4.3. A o-degeerate radom variable X belogs to the domai of attractio of the ormal distributio (ad we write X DAN) if ad oly if U is slowly varyig, or equivaletly: x P ( X > x) lim = 0 (38) x U(x) Proof. We first show that X DAN if ad oly if U is slowly varyig. Assume X DAN. By Theorem 4..(i), U is slowly varyig. Assume that U is slowly varyig. By Theorem 4..(ii), X DAN. Note that, by Lemma 4., U is slowly varyig if ad oly if (38) holds. Corollary 4.4. A radom variable X belogs to the domai of attractio of a stable law of idex α, with 0 < α <, if ad oly if the followig coditios hold: (i) P ( X > x) x α L(x), for some slowly varyig fuctio L. P (X > x) (ii) P ( X > x) p ad P (X < x) q for some p, q 0, p + q =. P ( X > x) Coditio (i) is equivalet to (8). Proof. Assume first that X belogs to the domai of attractio of a stable law of idex α, with 0 < α <. The U(x) x α l(x) (by Theorem 4..(i)). By Lemma 4..(i), (8) holds. By Lemma 4..(ii), with αl 0 = l, P ( X > x) α α x α l(x). Hece coditio (i) holds with L = α α l. Coditio (ii) holds by Theorem 4..(iii). Assume (i) ad (ii) hold. By (i), V (x) = P ( X > x) is regularly varyig with idex α. By Lemma 4..(i), (8) holds. By Lemma 4..(ii), (9) holds. 6

17 By Theorem 4..(iii), (9) ad (ii) imply that X belogs to the domai of attractio of a stable law of idex α, with 0 < α <. The fact that (i) is equivalet to (8) follows by Lemma 4.. The followig result is kow i the literature as Karamata Theorem. Lemma 4.5. (i) Assume that U(x) = E ( X { X x} ) is regularly varyig of idex α, where 0 < α. The, for ay β < α, E( X β ) < ad x β lim x U(x) E ( X β ) α { X >x} = α β. (39) (ii) Assume that V (x) = P ( X > x) is regularly varyig of idex α, where 0 < α <. The, for ay β > α, E( X β ) = ad E ( ) X β { X x} lim x x β = α (40) P ( X > x) β α Proof. (i) elatio (39) follows usig Theorem A.8 with a = ad b = β, i.e. U(x) = U (x) = E ( ) X { X x} ad Vβ (x) = E ( ) X β { X >x}. Because E ( ) X β { X x} x β < ad E ( ) X β { X >x} <, we get: E( X β ) = E ( ) ( ) X β { X x} + E X β { X >x} < (ii) elatio (40) follows usig Theorem A.8 with a = β ad b = 0, i.e. U β (x) = E ( ) X β { X x} ad V0 (x) = P ( X > x). From (40), we get that: E ( ) X β α { X x} β α xβ P ( X > x). But lim E ( X β ) { X x} = E( X β α ) ad lim x x β α xβ P ( X > x) =. It follows that E( X β ) =. The followig result shows that if covergece () holds the the ormalizig sequece ( ) must satisfy a very importat coditio. Lemma 4.6. Suppose that () holds, where S = i= X i ad (X i ) i are i.i.d. radom variables, with commo distributio F. The the ormalizig sequece ( ) has to satisfy: for ay x > 0. I particular U( x) Cx α, (4) a U( ) C. (4) a Proof. I the proof of Theorem 3.3 the case whe F is arbitrary, we have see that ( ) has to satisfy: P ( X > x) M + (x) + M ( x) = C α α x α for some 0 < α <, (43) 7

18 (see relatios (3) ad (7)), or P ( X > x) 0. The measure M is give by M = M α, respectively M = Cδ 0, where M α is give by (4). To prove (4), i the case of α <, let V (x) = P ( X > x). By Lemma A.7 ad (43), V is regularly varyig of idex α. By Lemma 4..(i), (8) holds. This a gives us: x P ( X > x) α a for ay x > 0, i.e: P ( X > x) U( x) α U( x) α x. By multiplyig the omiator ad deomiator by, we get α P ( X > x) a U(a x) α α x. elatio (4) follows by (43). I the case α =, relatio (4) follows from (6) (assumig that F is symmetric) ad the fact that M = Cδ 0, ad hece M([ x, x]) = C. (We still do t kow why (4) holds i the o-symmetric case). The followig result gives a explicit recipe for costructig a sequece ( ) satisfyig (4). Lemma 4.7. Assume that is U is regularly varyig of idex α, with 0 < α. i.e. Let C > 0 arbitrary. Defie { } U(x) = if x > 0; x C where we use the covetio if =. The: (i) is fiite, (ii) ( ) satisfies (4), (iii) lim =. { U(x) Proof. (i) We have to prove that the set A = x > 0; x C } is o empty. To see this, say U(x) = x α l(x) for a slowly varyig fuctio l. We have to fid some x > 0 such that x U(x) = x x α l(x) = x α l(x) C. l(x) This follows sice lim x x α = 0. { U(x) (ii) Note that + sice x; x C } By the defiitio of, we get: we get U( ) a > C { x; U(x) x C }. + U(x) x > C for ay x <. Takig x =, for ay. We apply this result with +. We obtai: U( ) a > C +. (44) 8

19 O the other had, by the defiitio of : U( ) a C (45) By combiig (44) ad (45), we get: which is equivalet to C + < U() a C C + < U() elatio (4) follows from the fact that a C. +. (iii) Sice ( ) is o-decreasig sequece, lim =: C 0 [0, ] exists. Suppose C 0 <. By (4), we kow that U( ) Ca. Note that U( ) U(C 0 ) as. Hece U( ), which is a cotradictio. emark 4.8. Note that if U is regularly varyig of idex α, ad ( ) is chose to satisfy (4), the it also satisfies (4). To see this, ote that l(x) (because l is slowly varyig ad ) l( ) ad a α l( ) = a α l( ) a = U() a C. Hece a U( x) = a a α x α l( x) = (a α l( )) l(x) l( ) x α Cx α. The followig result specifies a more coveiet way of choosig the sequece ( ), i the case α (0, ). Lemma 4.9. Suppose that α (0, ) ad ( ) satisfies: P ( X > ) C (46) for some C > 0. If V (x) = P ( X > x) is regularly varyig of idex α (i.e. coditio (i) of Corollary 4.4 holds), the ( ) satisfies (4) with C = α C α. 9

20 Proof. By hypothesis, the fuctio V (x) = P ( X > x) is regularly varyig of idex α. By Lemma 4..(i) (8) holds. By Lemma 4..(ii) with L = ( α)l 0 we obtai: U(x) α α x α L(x), i.e. U is regularly varyig of idex α. By (8) a P ( X > ) U( ) U( ) a = P ( X > ) U() a α C = C, i.e. (4) holds. α α. Usig (46), we obtai: α emark 4.0. Assume α (0, ). A example of a sequece ( ) satisfyig (46) is: { = if x > 0; P ( X > x) C } { = if x > 0; P ( X x) C }. This ca be proved similarly to Lemma 4.7. We deote by F 0 the law of X. The { = if x > 0; F 0 (x) C } ( =: F0 C ), where F 0 deotes the geeralized iverse of the o-decreasig fuctio F 0. If F 0 is a cotiuous distributio, the this is equivalet to sayig that: F 0 ( ) = C, i.e. is the C -quatile of F 0. (ecall that z α is called the α-quatile of a radom variable Z if P (Z > z α ) = α, i,e, P (Z z α ) = α.) I the remaiig part of this sectio, we cocetrate o provig that relatio () holds, uder suitable coditio imposed o the uderlyig distributio of the sequece (X i ) i of i.i.d. radom variables. For this, we will assume that ( ) is o-decreasig sequece of costats with lim =, which satisfies (4). We begi with a prelemiary result. Propositio 4.. Let F be the law of X ad M (dx) = x F (dx). Let U be defied by (4). Assume that U is regularly varyig of idex α, 0 < α. Let ( ) be a sequece satisfyig (4), ad. If α <, we assume i additio that the tails of X are balaced, i.e. (30) holds. The M M properly where M = Cδ 0 if α =, ad if α (0, ), M = M α, where M α is give by (4) with costats c +, c satisfyig (6). Moreover, e iux iu si x e iux iu si x x M (dx) x M(dx). (47) 0

21 Proof. Note that M is caoical measure o because M (dx) = x F (dx) < ad x M (dx) = x x M is also a caoical measure. By emark 4.8, (4) holds. We claim that x x M ([ x, x]) M([ x, x]) for ay x > 0. (48) This follows from (4) ad the defiitio of M sice [ ] U( x) X a = E a { X ax} = [ x,x] (y)y F (dy) = M ([ x, x]) ad M([ x, x]) = Cx α. From (4), we also obtai that, for ay α (0, ], P ( X > x) M + (x) + M ( x) for ay x > 0 (49) where M + ad M are give by Defiitio A.. To see this, ote that sice U is regularly varyig of idex α, x P ( X > x) U(x) α α, as x by Lemma 4.. Sice x, it follows that for ay x > 0, which is equivalet to sayig that: a x P ( X > x) U( x) α α a U( x) P ( X > x) α α x. x x F (dx) <. By (4), we have: a U( x). We the obtai: Cx α P ( X > x) C α α x x α which is equivalet to (49), sice due to the defiito of M, M + (x) + M ( x) = C α α x α for ay x > 0. (50)

22 I particular, whe α =, M + (x) = M ( x) = 0 for all x > 0 (sice M = Cδ 0 ), ad relatio (49) becomes: P ( X > x) 0 for ay x > 0. Whe α <, the tails are balaced, (i.e. (30) holds) by hypothesis. Due to (49) ad (50), we obtai that P (X > x) Cp α α x α ad P (X < x) Cq α α x α (5) To see this, ote that P (X > x) = P (X > x) P ( X > x) P ( X > x) = Cp α α x α. The secod part of (5) is obtaied similarly. Due to the defiitio of F ad (7), (5) ca be writte as: M + (x) = x y y F (dy) x y M(dy) = M + (x) x M x ( x) = y y F (dy) y M(dy) = M ( x) Puttig together (48) ad (5) we get: (5) M ([ x, x]) M([ x, x]), M + (x) M + (x), M ( x) M ( x) i.e. M M properly (accordig to Defiitio A.) Defie ψ (u) = ψ(u) = e iux iu si x x M (dx) e iux iu si x x M(dx). Sice M M properly (i the sese of defiitio A.), by Theorem A.3, it follows that ψ (u) ψ(u). This proves (47). The followig result is called the Stable Limit Theorem, which is the focus of the preset project. Theorem 4. (Theorem XVII.5.3 from []). Let Y be a radom variable with characteristic fuctio ϕ Y (u) = e ψ Y (u), where u α Γ(3 α) C α( α) cos πα ( isg(u)(p q) ta πα ), α ψ Y (u) = u C π + isg(u)(p q) π ) l u, α = (53)

23 for some C > 0, p, q 0, p + q =, i.e. Y S α (γ, β, 0), where Γ(3 α) C γ α = α( α) cos πα, α C π ad β = p q (54), α = Let (X i ) i be i.i.d. radom variables whose distributio F satisfies either oe of the followig coditios: (a) U(x) = E(X { X x} ) is slowly varyig ad X is o-degeerate (b) U is regularly varyig of idex α for some 0 < α <, ad the tails of X are balaced, i.e. (30) holds. Let ( ) be a sequece of costats satisfyig (4). (i) If 0 < α <, the S d Y. (ii) If < α ad we assume that E(X ) = whe α =, the S E(X) d Y (iii) If α =, the S b [ ( )] d X Y, where b = E si. Corollary 4.3. (0 < α < ) Assume that P ( X > x) = x α L(x) where L is slowly varyig ad the tails are balaced, i.e. (30) holds. Choose ( ) such that (46) holds for some C > 0. The U(x) = x α l(x), where l = α L ad the α α coclusio of Theorem 4. holds with ψ Y (u) give by (53) with C = C α. Proof. We apply Theorem 4.. The fact that ( ) satisfies (4) follows by Lemma 4.9. Proof of Theorem 4.. Defie M such that M([ y, x]) = Cpx α + Cqy α, i.e. M = Cδ 0 if α = or M(dx) = ( α)cpx α+ (0, ) (x)dx + ( α)cq( x) α+ (,0) (x)dx if α <. Assume first that α <. The M = M α where M α is give by (4) with c + = ( α)cp ad c = ( α)cq. Note that c + + c α = C ad c + c c + + c = p q (55) Let W be a ifiitely divisible radom variable with ϕ W (u) = e ψ W (u) where e iux iu si x ψ W (u) = x M(dx) 3

24 Accordig to Lemma.8, W S α (γ, β, δ) where (γ, β, δ) are give by (5), (6), (8). Due to (55), formulas (5) ad (6) are exactly the same as (54). Hece, W δ S α (γ, β, 0), i.e. Y d = W δ. Therefore W d = Y + δ ad ψ W (u) = iuδ + ψ Y (u) Let F be the law of X ad M = x F (dx). (i) Assume that α <. We claim that i this case, e iux e iux x M (dx) x M(dx) (56) To see this, recall that M M properly (by Propositio 4.). Sice f(x) = e iux x is cotiuous o {x; x > δ}, e iux e iux lim x >δ x M (dx) = x >δ x M(dx) for ay δ > 0. (57) This still eeds to be show i more details. We see that e iux e iux lim δ 0 x M(dx) = x M(dx). x >δ Hece, to prove (56), it suffices to show that lim lim e iux e iux δ 0 x M (dx) x >δ x M (dx) = 0. (58) Note that, e iux x M (dx) x δ = x δ x δ e iux x x F (dx) = e iux F (dx) u = u E [ ] X { X aδ}. x δ (e iux )F (dx) x δ x F (dx) where we used the iequality e ia = (cos a ) + si a a for ay a. Usig Lemma 4.5.(ii) with β = (sice α < ), we obtai that E [ ] α X { X aδ} α (δ)p ( X > δ). 4

25 This gives us: x δ e iux x M (dx) α u α δp ( X > δ). Takig the upper limit whe teds to ifiity, we get: e iux lim x M (dx) α u δ lim α P ( X > δ) x δ Usig (49) ad (50) we obtai lim P ( X > δ) = C α α δ α. This gives us e iux lim x M (dx) u α α Cδ α. x δ Usig the fact that α > 0, we get that whe δ 0, the term o the right had side of the previous iequality coverges to zero. This cocludes the proof of (58) ad the proof of (56). Note that e iux x M (dx) = ad e iux x M(dx) = = E e iux x [ exp x F (dx) = ] (iu Xa ) e iux iu si x x = (e iux )F (dx) ( ) ] [ϕ X ua M(dx) + iu si x x M(dx) = ψ W (u) iuδ = ψ Y (u). ( ) ] Usig (56), we obtai: [ϕ X ψ Y (u), which is equivalet to the ua fact that S i the case α <. d Y (by emark 3.). This cocludes the proof of the theorem (ii) Assume that < α. We claim that, i this case e iux iux e iux iux x M (dx) x M(dx). (59) To see this, we use agai the fact that M M properly, so e iux iux e iux iux lim x t x M (dx) = x t x M(dx) (60) 5

26 for ay t > 0. We still eed to show this i more details. We see that e iux iux e iux iux lim t x M(dx) = x M(dx). x t Hece to show (59), it suffices to prove that: lim lim e iux iux e iux iux t x M (dx) x t x M (dx) = 0. Note that, x >t e iux iux x where we used the iequality M (dx) = = x >t x >t x >t e iux iux x x F (dx) (e iux iux)f (dx) ( + ) u e iux iux F (dx) x >t x F (dx) = ( + ) u E[ X { X >at}] e ia ia e ia + a ( + ) a. Usig Lemma 4.5.(i) with β = (sice α > ), we obtai that This gives us: x >t E[ X { X >at}] α α t U(t). e iux iux x M (dx) ( + ) α α u t Takig the upper limit whe teds to ifiity, we get: e iux iux lim x M (dx) ( + ) α α u t lim x >t U( t) a. U( t) a. U( t) Usig (4) we obtai lim a = Ct α. This gives us e iux iux lim x M (dx) ( + ) α α u Ct α. x >t 6

27 Usig the fact that α < 0, we get that whe t, the right had side of the previous iequality coverges to zero. This fiishes the proof of (59). Assume first that E(X) = 0. The e iux iux e iux iux x M (dx) = x x F (dx) [ ) ] = E exp (iu Xa iu Xa ( ) = [ϕ X i ua u ] E(X) ad e iux iux e iux iu si x x M(dx) = x = ψ W (u) iuδ = ψ Y (u). ( ) ] Usig (59), we obtai [ϕ X ψ W (u), i.e. ua M(dx) + iu S 3.). This fiishes the proof i the case α (, ] ad E(X) = 0. Assume ext that E(X) is arbitrary. We prove that ( ) ] = [ϕ X ua si x x x M(dx) d Y (by emark S E(X) d Y. (6) Writig E(X) = E(X { X >a}) + E(X { X a}), ad lettig b = E(X { X a}), we have We claim that S E(X) = S E(X { X a}) E(X { X >a}) = S b E(X { X >a}). (6) E [ ] X { X >a} x x M(dx) := δ. (63) To see this we use (57) with δ =. We get: e iux x M (dx) x x x e iux x M(dx). By takig the differece betwee (59) ad (60) (with t = ), we obtai: e iux iux e iux iux x M (dx) x M(dx). 7 x

28 Takig the differece betwee these two equatios, we obtai: iu x M (dx) = iu xf (dx) iu x M(dx). x x Due to (6) ad (63), (6) is equivalet to x S b d Y := Y + δ. (64) From (59), we kow that ( ) [ϕ X iu ua E(X) ] = = e iux iux x M (dx) e iux iux x M(dx) e iux iu si x x M(dx) si x x +iu M(dx) x = ψ W (u) iuδ = ψ Y (u), (65) usig defiitio (8) of δ ad the fact that Y = d W δ. Note that (64) is equivalet to ( ) } ϕ S a b (u) = {ϕ X e ua iub e ψ Y (u) = ϕ Y (u) To prove this, it suffices to show that { ( )} ( ( ) ]) u ϕ X exp [ϕ X ua sice the ( ) } ( ( ) ]) {ϕ X e ua iub exp [ϕ X iub e ua ψ Y (u) = e iuδ +ψ Y (u). For the last covergece, we used the fact that: ( ) ] ( ) [ϕ X iub = [ϕ X iu ua ua E(X ] { X }) ( ) = [ϕ X iu ua E(X) ] + iu E(X { X >}) = ψ Y (u) + iuδ, due to (65) ad (63). (66) 8

29 For the proof of (66), we refer to part (iii) below. i the case α (, ]. This cocludes the proof (iii) Assume that α =. Note that e iux iu si x x M (dx) = e iux iu si x x x F (dx) = (e iux iu si x)f (dx) ( ) )] = [ϕ X iue (si ua Xa ( ) ] = [ϕ X iub ua ad, sice i this case δ = 0, we have: e iux iu si x x M(dx) = ψ Y (u) = ψ W (u). By Propositio 4., it follows that ( ) ] [ϕ X iub ψ W (u). (67) ua To show that S b ϕ S a b = d W, we have to show that: {ϕ X ( ua ) e iub } e ψ W (u) = ϕ W (u). (68) To show (68), it suffices to prove (66), sice the ( ) } ( ( ) ]) {ϕ X e ua iub exp [ϕ X iub ua ad by (67), this last term coverges to e ψ W (u). The remaiig part of the proof is dedicated to (66). Note that relatio (66) ca be writte as ( + z ) e z, where z = ϕ X ( u ) 0, z C, z. We claim that z 0 implies ( + z ) e z. This implicatio still eeds to be prove. Hece it suffices to prove that ( ) u ϕ X 9 0 (69)

30 To prove (69), we use the fact that E X β < for β < (by Lemma 4.5.(i), sice α ). First ote that ( ) [ )] ( u ϕ X = E exp (iu ua exp iu x ) F (dx). Now we use the fact that e it t β, for ay t, β (0, ]. We get: ( ) u ϕ X β ux F (dx) u β This gives us ( ) u ϕ X a β 4 u β (E X β ) x β F (dx) = u β a β E X β. a β We claim that for ay δ > 0, there exists N δ such that C a α δ ( ) = O a β. (70) C a α+δ for ay N δ (7) To see this, recall that U(x) = x α l(x), where l is slowly varyig. By emark A.6, this implies that for ay δ > 0, there exists x δ such that x δ l(x) x δ for ay x x δ. The fact that implies that there exists N δ such that x δ for ay N δ. We the get: a δ U( ) a α+δ for ay N δ. a α δ By (4), C U() a U( ) a C U() a C U() a l( ) a δ for ay N δ. The C. This implies that there exist C, C > 0 such that C, for ay > 0. Therefore, for ay N δ a α δ a a α+δ a cocludes the proof of (7). From (7), we have that: a α+δ Hece, a β C β/(α+δ) ad a β = a α δ, which implies that C a α+δ = a α+δ, which implies that C a α δ ad. This ( ) /(α+δ), which gives us /(α+δ). C = C C β/(α+δ) β/(α+δ) 0 as (7) if < β α + δ i.e α + δ < β. Sice we also eed β <, we must choose δ such that α + δ <, i.e. 0 < δ < α. elatio (69) follows from (70) ad (7). This cocludes the proof i this case α =. C 30

31 I the case α (0, ), there is a alterative formulatio of Theorem 4., based o the represetatio of the characteristic fuctio of a stable radom variable give by Lemma.0. Theorem 4.4. Let (X i ) i be i.i.d. radom variables such that P ( X > x) = x α L(x) where L is slowly varyig ad 0 < α <, ad the tails are balaced i.e. (30) is satisfied for some p, q 0, p+q =. Let S = X i. Choose ( ) such that (46) holds for some costat C > 0. Let Y be a ifiitely divisible radom variable with characteristic fuctio { } ϕ Y (u) = exp (e iux iux { x } )ν α (dx) where ν α (dx) = C α[px α (0, ) (x) + q( x) α (,0) (x)]dx. The S d b Y, where b = E(X { X }) α Proof. By Lemma 4.9, ( ) satisfies (4) with C = C α. Hece C = α α C. Note that the measure ν α has the form (9) with i= c + = C pα = ( α)cp ad c = C qα = ( α)cq. By Lemma.0, Y S α (γ, β, δ ) where (γ, β) are give by (5), (6) ad δ is give by (0). Sice c + + c α = C ad c + c = p q, formulas (5) ad (6) c + + c are exactly the same as (54). X Let F be the law of ad M (dx) = x F (dx). We will use some of the results show i the proof of Theorem 4. with M α (dx) = x ν α (dx). We cosider three cases. ) Suppose that α <. By Theorem 4..(i), S d Y Sα (γ, β, 0). We claim that i this case, E(X { X a}) To see this, by (60), with t =, we get (e iux iux)f (dx) x 3 x x x ν α (dx) = δ. (73) (e iux iux) ν α (dx). (74)

32 Takig the differece betwee (56) ad (57) with δ =, we obtai (e iux )F (dx) (e iux ) ν α (dx). (75) x x Takig the differece betwee (74) ad (75), we get [ ] X E { X a} = xf (dx) This proves (73). Hece x x S E(X { X a}) δ + Y = Y. x ν α (dx) = δ. ) Suppose that α >. By relatio (64) i the proof of Theorem 4. S E(X { X a}) δ + Y = Y. 3) Suppose that α =. By Theorem 4..(iii), S ] E [si Xa d Y. We claim that ] E [si Xa E(X { X }) δ = (si x x { x } )ν α (dx). (76) To see this, ote that by (57) with δ =, (e iux )F (dx) x > (e iux )ν α (dx). By (60) with t =, (e iux iux)f (dx) x (e iux iux)ν α (dx). Takig the sum, we obtai: (e iux iux { x } )F (dx) (e iux iux { x } )ν α (dx). By (47), (e iux si x)f (dx) (e iux si x)ν α (dx). Takig the differece betwee the two previous equatios, we obtai: (si x x { x } )F (dx) (si x x { x } )ν α (dx). This proves (76). This gives us S E(X { X a}) δ + Y = Y. 3

33 emark 4.5. Uder the coditios of Theorem 4.4, we have: ] E [si Xa si x x M(dx) = si x ν α (dx) if α (0, ). (77) To see this, ote that ad ψ Y (u) = ψ (u) = = \{0} e iux iu si x x M (dx) e iux x M (dx) iu e iux iu si x x M(dx) = Usig (47) ad (56), we obtai: ] E [si Xa = si x x x F (dx) si x x x F (dx) e iux x M(dx) iu si x x M(dx). si x x M(dx) emark 4.6. Uder the coditios of Theorem 4.4 [ X E si X ] x si x x M(dx) = (x si x) ν α (dx) if α (, ]. To see this, ote that ad ψ (u) = = ψ Y (u) = = Usig (47) ad (59), we obtai: [ ( )] X X E si = e iux iu si x x M (dx) e iux iux x M (dx) iu e iux iu si x x M(dx) e iux iux x M(dx) iu \{0} x si x x x F (dx) x si x x x F (dx) x si x x M(dx) x si x x M(dx). 33

34 5 Simulatios o I this sectio we will use Theorem 4. to verify if certai distributios are i the domai of attractio of a stable distributio. We cosider a radom variable X with Pareto distributio of parameter α with distributio fuctio F. Let (X i ) i be i.i.d. copies of X. Let S = X i. By Corollary 4.3, for 0 < α <, X is i the domai of attractio of a stable distributio, i.e. S b d Y Sα (γ, β, 0), (where the parameters are give i Theorem 4.) if P ( X > x) = x α L(x) where L is a slowly varyig fuctio (78) ad the tails are balaced, i.e. (30) holds. We choose satisifyig (46). By Theorem 4., for α =, X is i the domai of attractio of a stable distributio if U(x) = E(X { X x} ) is slowly varyig ad X is o-degeerate. We choose satisfyig (4). Because the Pareto distributio is always positive, we get that (78) is equivalet to P (X > x) = x α L(x), which is i tur equivalet to F (x) = P (X > x) = x α L(x). Aother cosequece of the positiveess of X is that P (X > x) lim x P ( X > x) = ad lim P (X < x) = 0. So (30) holds with p = x P ( X > x) ad q = 0. Therefore β = p + q =. I this example, for 0 < α < we will = c, where c. The, choose{ L(x) F (x) = cx α. We the obtai: f(x) = cαx α, x > c 0 0, x c 0 for some c 0. { Takig c 0 =, we obtai X P areto(α), i.e. f(x) = αx α, x > 0, x. For α =, we have U(x) = E(X { X x} ) = l(x) for some slowly varyig fuctio l. We have: U(x) = x x y αy α dy = x i= y dy = l x = l(x). (79) We will ow fid the for 0 < α <, i.e. satisfyig (46). Takig C =, ad usig the positiveess of X, we get: P (X > ). We will choose such that P (X > ) =. Usig the fact that, for X P areto(α), P (X > x) = x α, we the obtai P (X > ) = a α =. Hece, = /α. α By Corollary 4.3, C = C α = α α. 34

35 We will ow fid the for α =. Usig (79) ad (4), we get: a = C l(). Takig C = we obtai the followig recurrece relatio: a = l( ) (80) ( Note that = exp ) W ( /) where W is the secod brach of the Lambert W fuctio. To compute the value of W we used the gsl package. With = 0000, we obtai: I the ext examples we will use the parameter α with values 0.5,,.5 ad. I each case, usig, we will simulate k = 0000 Pareto distributio samples of populatio = 0000 usig the VGAM package. We will the calcutate the sum = S(i) of values iside the sample to obtai S (i) for i k. Let W (i) b. Let W be the vector with W (i) as coordiates, the by Theorem 4., whe d k is very large, W Sα (γ, β, 0). To illustrate this result, we compute the desity fuctio of W by the histogram method o, ad the empirical distributio fuctio of W. We the superpose the graphs obtaied with the desity fuctio ad the distributio fuctio of a stable distributio with parameters (γ, β, 0). The graphs should coverge. Case : α = 0.5. ( Γ(3 α) I this case, Theorem 4. gives us b = 0 ad γ = C α( α) cos πα ) /α. We kow α = 0.5, the C = /3, hece γ We the compute W ad draw the two curves. To draw the stable curves, we use a Lévy distributio with parameters µ = 0 ad c = (because as see i [3], Lévy(µ, c) Stable 0.5 (, c, µ)). Therefore, by Theorem 4., we obtai: W = S d Y S 0.5 ( ,, 0) Lévy(0, ). We obtai the graphs (Y : red curve, W :black curve) i Figure Case : α =. I this case, Theorem 4. gives us E ( ( )) X si ad γ = C π. We kow 35

36 Figure : Illustratio of Stable Limit Theorem for α = 0.5 α =, the C =, hece γ = π. We ow calculate b : ( ( )) X ( x ) b = E si = si αx α dx /α /α ( x = si x ) dx = si y(y) (dy) = / si y dy y usig y = x ad computig the last itegral with = 0000 o Therefore, by Theorem 4., we obtai: W = S d Y Stable ( π,, 0) To draw the stable curve, we use the package stabledist. The fuctio uses the approach of J.P. Nola for geeral stable distributios. We obtai the graphs (Y : red curve, W :black curve) i Figure / Case 3: α =.5. I this case, Theorem 4. gives us b = E(X) ( Γ(3 α) ad γ = C α( α) cos πα ) /α. We kow α =.5, the C = 3, hece γ We ow calculate b : b = E(X) = /α =.5 /3 x.5 dx = 3 /3 36 xαx α dx

37 Figure : Illustratio of Stable Limit Theorem for α = Therefore, by Theorem 4., we obtai: W = /3 S 3 /3 d Y Stable.5 (.8457,, 0) The stable curve is draw usig the package stabledist. We obtai the graphs (Y : red curve, W :black curve) i Figure 3 Case 4: α =. I this case, Theorem 4. gives us b = E(X) ( Γ(3 α) ad γ = C α( α) cos πα ) /α. We kow α = ad we took C =, hece γ =. We ow calculate b : b = E(X) = xαx α dx = x dx = by Theorem 4., we obtai: W = S d Y Stable (,, 0) The stable curve is draw usig the package stabledist. We obtai the graphs (Y : red curve, W :black curve) i Figure 4 37

38 Figure 3: Illustratio of Stable Limit Theorem for α =.5 Figure 4: Illustratio of Stable Limit Theorem for α = 38

39 A Auxiliary esults Defiitio A.. A measure M is called caoical if it attributes fiite masses to fiite itervals ad the itegrals M + (x) = x y M(dy) ad M ( x) = are fiite for some (ad therefore all) x > 0. x M(dy) (8) y Defiitio A.. Let (M ) ad M be caoical measures. We say that (M ) coverges properly to M (ad we write M M properly) if: (i) M ([a, b]) M([a, b]) for ay a, b cotiuity poits of M, (ii)m + (x) M + (x) ad M ( x) M ( x) for ay x > 0 which is a cotiuity poit of M. Theorem A.3 (Theorem XVII.. of []). Let M be a caoical measure o, b ad ψ (u) = iub + + e iux iu si x x M (dx) u. (8) There exists a cotiuous fuctio ψ : C such that ψ (u) ψ(u) u if ad oly if there exist a caoical measure M o ad b such that I this case, ψ(u) = iub + Corollary A.4. Let M M properly ad b b. + e iux iu si x x M(dx) u. (83) ψ (u) = C (ϕ (u) iu b ) (84) where ϕ is the characteristic fuctio of a probability distributio F ad b, C. The there exists a cotiuous fuctio ψ : C such that ψ (u) ψ(u) for ay u if ad oly if C x F (dx) M(dx) ad C (β b ) b for some caoical measure M ad b where β = si xf (dx). I this case, ψ(u) is give by (83). Proof. Note that ψ (u) ca be writte i the form (8) with ad. The result follows by Theorem A.3. M (dx) = C x F (dx) b = C (β b ) 39

40 Defiitio A.5. (i) A fuctio L is slowly varyig (SV) if L(λx) lim = for ay λ > 0. x L(x) (ii) A fuctio U is regularly varyig with idex ρ (V ρ ) if for a slowly varyig fuctio L. U(x) = x ρ L(x) emark A.6. Let L be a slowly varyig fuctio. The for ay δ > 0 there exists x δ > 0 such that x δ < L(x) < x δ for ay x > x δ. Lemma A.7 (Lemma VIII.8.3 of []). Suppose that λ + λ ad. If U is a mootoe fuctio such that: lim λ U( x) = χ(x) exists for ay x D (D is a dese set) χ(x) is fiite ad positive for ay x I (I is a iterval), the U is regularly varyig of idex ρ ad χ(x) = Cx ρ, for some C > 0. Theorem A.8 (Theorem VIII.9..(i) of []). Let a > 0 ad < b < a. Defie the trucated momet fuctios U a ad V b by U a (x) = E( X a { X x} ) ad V b (x) = E( X b { X >x} ) Suppose that U a ( ) =. (i) If either U a or V b varies regularly the u a b V b (u) lim = c (85) u U a (u) for some 0 c. We write this limit uiquely i the form c = a α α b (86) for b α a with α = b if c =. (ii) If (85) holds for some c (0, ) the there exists a slowly varyig fuctio L such that U a (x) (α b)x a α L(x) ad V b (x) (a α)x b α L(x) where α [b, a] is chose such that (86) holds. 40

41 efereces [] Feller, William. A Itroductio to Probability Theory ad Its Applicatios. Joh Wiley, 97. [] Billigsley, Patrick. Covergece of Probability Measures. Joh Wiley, 999. [3] P. Nola, Joh. Stable Distributios: Models for Heavy Tailed Data [4] Gaudreau Lamarre, Pierre-Yves. oots of x = l(x). 0 4