Quansheng Liu

Xingang Liang

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 347–352Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comProbability TheoryTail behavior of laws stable by random weighted meanVariation régulière des lois stables par moyenne pondérée aléatoireXingang Liang a,b, Quansheng Liu a,ba LMAM, Université de Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, Franceb Université Européenne de Bretagne, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 13 January 2011Accepted 31 January 2011Available online 18 February 2011Presented by the Editorial BoardLet (N, A1, A2, . . .) be a sequence of random variables with N ∈ N ∪ {∞} and Ai ∈ R+.We are interested in asymptotic properties of solutions of the distributional equation Z =∑Ni=1 Ai Zi , where Zi are nonnegative random variables independent of each other andindependent of (N, A1, A2, . . .), each has the same distribution as Z which is unknown.For a solution Z  0 with finite mean, we show that under a natural moment condition,the regular variation of P(Z > x) (x → ∞) is equivalent to that of P(Y1 > x), where Y1 =∑Ni=1 Ai . The results generalize the corresponding theorems of Bingham and Doney (1974,1975) [1,2] and de Meyer (1982) [6] on Galton–Watson processes and Crump–Mode–Jirinaprocesses, and improve those of Iksanov and Polotskiy (2006) [7] on branching randomwalks.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoit (N, A1, A2, . . .) une suite de variables aléatoires avec N ∈ N ∪ {∞} et Ai ∈ R+. Nousnous sommes intéressés aux propriétés asymptotiques des solutions de l’équation en dis-tribution Z = ∑Ni=1 Ai Zi , où Zi sont des variables aléatoires non-négatives, mutuellementindépendantes et indépendantes de (N, A1, A2, . . .), chacune a la même loi que Z quiest inconnue. Pour une solution Z  0 de moyenne finie, nous montrons que sous unecondition de moment naturelle, la variation régulière de la probabilité de queue P(Z > x)(x → ∞) est équivalente à celle de P(Y1 > x), où Y1 = ∑Ni=1 Ai . Les résultats généralisentles théorèmes correspondants de Bingham et Doney (1974, 1975) [1,2] et de Meyer (1982)[6] sur les processus de Galton–Watson et de Crump–Mode–Jirina, et améliorent ceux d’Ik-sanov et Polotskiy (2006) [7] sur les marches aléatoires branchantes.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeSoit (N, A1, A2, . . .) une suite de variables aléatoires avec N ∈ N ∪ {∞} et Ai ∈ R+ , où N = {0,1,2, . . .} et R+ = [0,∞).Supposons que E∑Ni=1 Ai = 1, et considérons l’équation en distributionZ =N∑i=1Ai Zi, (0.1)E-mail addresses: xingang.liang@univ-ubs.fr (X. Liang), quansheng.liu@univ-ubs.fr (Q. Liu).1631-073X/$ – see front matter © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2011.01.029348 X. Liang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 347–352où Zi sont des variables aléatoires non-négatives, mutuellement indépendantes et inépendantes de la suite (N, A1, A2, . . .),chacune a la même loi que Z qui est inconnue. En termes de la transformée de Laplace φ(s) = Ee−sZ , l’équation (0.1) s’écritφ(s) = EN∏i=1φ(sAi), s  0. (0.2)L’étude de l’équation (0.1) est intéressante grâce aux applications dans des modèles divers, comprenant en particulier desprocessus de branchement, des cascades multiplicatives et des marches aléatoires branchantes. Ici nous étudions la variationrégulière de la probabilité de queue P(Z > x) lorsque x → ∞, pour une solution Z de moyenne finie. Dans les cadres duprocessus de Galton–Watson (où Ai = 1/EN ∈ ]0,1[) et du processus de Crump–Mode–Jirina (où Ai  1), Bingham et Doney[1,2] ont prouvé que quand α > 1 n’est pas un entier, P(Z > x) est une fonction régulière d’ordre α si et seulement s’ilen est de même pour P(∑Ni=1 Ai > x) ; de Meyer [6] a montré que le résultat de Bingham et Doney [1] reste valablequand α > 1 est un entier. Leur argument était fondé sur l’équation (0.2) et un théorème Tauberien sur la transformée deLaplace. Récemment, la suffisance a été étendue par Iksanov et Polotskiy [7] au cas des marches aléatoires branchantes enutilisant un argument de martingale. L’argument de Bingham et Doney [1,2] utilise fortement la condition que les Ai sontbornés, tandis que l’argument de Iksanov et Polotskiy [7] ne montre pas la nécessité. Nous allons montrer que les résultatsde Bingham et Doney [1,2] et de Meyer [6] restent vrais dans le cas général, de sorte qu’ils peuvent être appliqués auxmartingales de Mandelbrot et aux marches aléatoires branchantes. Nos résultats améliorent ceux d’Iksanov et Polotskiy [7]au sens que nous montrons une condition nécessaire et suffisante, pas seulement une condition suffisante, pour la variationrégulière de P(Z > x).Posons pour x  0,ρ(x) = EN∑i=1Axi et ρ′(x) = EN∑i=1Axi ln Ai (0.3)si l’espérance existe dans [−∞,∞], ρ ′(x) étant une expression formelle qui coîncide avec la dérivée de ρ en x lorsqueρ est finie dans un voisinage de x. Par Lyons [8], on voit que lorsque E∑Ni=1 Ai ln+ Ai < ∞ (ln+ x = max(ln x,0)), alorsl’équation (0.1) a une solution Z  0 avec EZ = 1 si et seulement si EY1 ln+ Y1 < ∞ et ρ ′(1) < 0 ; par Liu [9], on sait quepour α > 1, EZα < ∞ si et seulement si EY α1 < ∞ et ρ(α) < 1. SoitR0 ={ : [0,∞) → [0,∞):  est measurables et limx→∞(λx)(x)= 1 ∀λ > 0}la classe des fonctions à variation lente en ∞. Comme d’habitude, nous écrivons f (s) ∼ g(s) si f (s)/g(s) → 1 (quand x → 0ou ∞ selon le contexte), et f (s) = O (g(s)), s ∈ I (resp. s → 0), si | f (s)|  C |g(s)| pour s ∈ I (resp. pour s > 0 assez petit),où C > 0 est une constante ; nous soulignons que C > 0 est une constante déterministe lorsque g est une fonction aléatoire.Dans les théorèmes suivants, nous posons Y1 = ∑Ni=1 Ai , et nous supposons que Z  0 est une solution de l’équation(0.1) avec EZ = 1 :Théorème 0.1. Supposons que ρ(α) < 1 pour un certain α > 1 et que ρ(α + δ) < ∞ pour un certain δ > 0. Soit  ∈ R0 . Alors lorsquex → ∞,P(Y1 > x) ∼ x−α(x) si et seulement si P(Z > x) ∼(1 − ρ(α))−1x−α(x). (0.4)Lorsque α > 1 n’est pas un entier, le résultat a été démontré par Bingham et Doney [1,2] d’abord pour le processus deGalton–Watson, et ensuite pour le processus de Crump–Mode–Jirina ; lorsque α > 1 est un entier, il a été montré par deMeyer [6] pour le processus de Galton–Watson. Dans le cas général, la suffisance a été montré par Iksanov et Polotskiy [7]par des techniques de martingale ; la nécessité est nouvelle. Notre approche utilise des théorèmes Tauberiens comme dansBingham et Doney [2] et de Meyer [6], et donne une nouvelle démonstration du résultat de Iksanov et Polotskiy [7].Lorsque α = 1, la situation est différente. Dans ce cas nous avons le résultat suivant qui est une extension des résultatscorrespondants de Bingham et Doney [1,2] pour les processus de Galton–Watson et de Crump–Mode–Jirina :Théorème 0.2. Supposons que ρ(1 + δ) < ∞ pour un certain δ > 0 et que μ := −ρ ′(1) ∈ (0,∞). Soit  ∈ R0 telle que∫ ∞1 (x)dx/x < ∞. Si EY11{Y1>x} ∼ (x) (x → ∞), alorsEZ1{Z>x} ∼ 1μ∞∫x(t)dt/t (x → ∞). (0.5)X. Liang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 347–352 3491. Introduction and resultsLet (N, A1, A2, . . .) be a sequence of random variables with N ∈ N ∪ {∞} and Ai ∈ R+ , where N = {0,1,2, . . .} andR+ = [0,∞). Assume that E∑Ni=1 Ai = 1, and consider the distributional equationZ =N∑i=1Ai Zi, (E)where Zi are nonnegative random variables independent of each other and independent of (N, A1, A2, . . .), each has thesame law as Z which is unknown. In terms of the Laplace transform φ(s) = Ee−sZ , Eq. (E) readsφ(s) = EN∏i=1φ(sAi), s  0. (E′)The study of Eq. (E) is interesting due to a large number of applications in a variety of applied probability settings,including branching processes, self-similar cascades, infinite particles systems, branching random walks, quicksort algorithmand random fractals. Here we study the regular variation of the tail probability P(Z > x) when x → ∞, for a solution Z of(E) with finite mean. In the context of the Galton–Watson process (where Ai = 1/EN ∈ (0,1)) and the Crump–Mode–Jirinaprocess (where Ai  1), Bingham and Doney [1,2] proved that when α > 1 is not an integer, P(Z > x) is a regular functionof order α if and only if the same is true for P(∑Ni=1 Ai > x); de Meyer [6] showed that the result of Bingham and Doney[1] remains valid when α > 1 is an integer. Their argument was based on the functional equation (E′) and a powerfulTauberian theorem on Laplace transforms. Recently, the “if” part has been extended by Iksanov and Polotskiy [7] to thecase of branching random walks using an elegant martingale argument. The argument of Bingham and Doney [1,2] dependsheavily on the boundedness of Ai , while the argument of Iksanov and Polotskiy [7] cannot show the “only if” part. We shallshow that the results of Bingham and Doney [1,2] and de Meyer [6] remain true in the general case, so that they can beapplied to Mandelbrot’s martingales and to branching random walks. Our results improve those of Iksanov and Polotskiy [7]in the sense that we show a necessary and sufficient condition, not just a sufficient condition, for the regular variation ofP(Z > x).For x  0, setρ(x) = EN∑i=1Axi and ρ′(x) = EN∑i=1Axi ln Ai (1.1)if it exists in [−∞,+∞], where ρ ′(x) is a formal expression which coincides with the derivative of ρ at x when ρ is finitein a neighborhood of x. By Lyons [8], we see that when E∑Ni=1 Ai ln+ Ai < ∞ (where ln+ x = max(ln x,0)), then Eq. (E) hasa solution Z  0 with EZ = 1 if and only if EY1 ln+ Y1 < ∞ and ρ ′(1) < 0; by Liu [9], we know that for α > 1, EZα < ∞ ifand only if EY α1 < ∞ and ρ(α) < 1. LetR0 ={ : [0,∞) → [0,∞):  is measurable and limx→∞(λx)(x)= 1 ∀λ > 0}(1.2)be the set of functions slowly varying at ∞. As usual, we write f (s) ∼ g(s) if f (s)/g(s) → 1 (where s → 0 or ∞ accordingto the context), and f (s) = O (g(s)), s ∈ I (resp. s → 0), if | f (s)|  C |g(s)| for s ∈ I (resp. for s > 0 small enough), whereC > 0 is a constant; we emphasize that C > 0 is a deterministic constant when g is a random function.In the following theorems, we set Y1 = ∑Ni=1 Ai , and assume that Z  0 is a solution of (E) with EZ = 1:Theorem 1.1. Assume that ρ(α) < 1 for some α > 1 and ρ(α + δ) < ∞ for some δ > 0. Let  ∈ R0 . Then as x → ∞,P(Y1 > x) ∼ x−α(x) if and only if P(Z > x) ∼(1 − ρ(α))−1x−α(x). (1.3)When α > 1 is not an integer, the result was shown by Bingham and Doney [1,2] first for the Galton–Watson process, andthen for the Crump–Mode–Jirina process; when α > 1 is an integer, it was shown by de Meyer [6] for the Galton–Watsonprocess. For branching random walks, the implication from Y1 to Z was shown by Iksanov and Polotskiy [7] by martingaletechniques; the implication from Z to Y1 is new. Rösler, Topchii and Vatutin [10] showed conditions under which Z belongsto the domain of attraction of a stable law with index 1 < α  2. Our approach uses Tauberian theorems as in Bingham andDoney [2] and de Meyer [6], and gives a new proof of the result of Iksanov and Polotskiy [7].When α = 1, the situation is different. In this case we have the following result which extends the corresponding onesof Bingham and Doney [1,2] for the Galton–Watson process and the Crump–Mode–Jirina process:Theorem 1.2. Assume that ρ(1 + δ) < ∞ for some δ > 0 and that μ := −ρ ′(1) ∈ (0,∞). Let  ∈ R0 be such that∫ ∞1 (x)dx/x < ∞.If EY11{Y1>x} ∼ (x) (x → ∞), then350 X. Liang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 347–352EZ1{Z>x} ∼ 1μ∞∫x(t)dt/t (x → ∞). (1.4)2. Key ideas of the approachThe proofs of theorems are mainly based on Tauberian theorems and relations between the Laplace transforms of Z andY1. For a nonnegative random variable X , we write φX (s) = Ee−sX for its Laplace transform and μr,X = EXr for its r-thmoment (r = 0,1, . . .). If X has at least finite n-th moment, we setφn,X (s) = (−1)n+1{φX (s) −n∑r=0μr,X (−s)rr!}, n  0. (2.1)Then we have the following two Tauberian theorems which give necessary and sufficient conditions for P(X > x) ∼ x−α(x),as x → ∞: see Proposition 2.1 for α /∈ N, and Proposition 2.2 for α ∈ N:Proposition 2.1. (See [1, Theorem A].) If μr,X < ∞ for r = 0,1, . . . ,n and α = n + β  1 with n ∈ {1,2, . . .} and β ∈ [0,1), then thefollowing statements are equivalent for  ∈ R0:φn,X (s) ∼ sα(1/s) (s → 0);EXn1{X>x} ∼ n!(x) (x → ∞) when β = 0,P(X > x) ∼ [(−1)n/Γ (1 − α)]x−α(x) (x → ∞) when 0 < β < 1.Proposition 2.2. Let n ∈ {1,2, . . .}. Assume that μn−1,X < ∞ and define φ̂n−1,X (s) = φn−1,X (s)/sn (s > 0). Then for  ∈ R0 ,limx→∞xnP(X > x)(x)= 1 if and only if limt→0φ̂n−1,X (t) − φ̂n−1,X (λt)(1/t)/(n − 1)! = lnλ ∀λ > 1. (2.2)Proposition 2.1 was established by Bingham and Doney [1], using Karamata’s theorem (cf. [3]). Proposition 2.2 wasobserved by de Meyer [5] as a consequence of de Haan’s theorem (cf. [3, Theorem 3.9.2], [4] or [5]).Set t(s) = − ln φZ (s) and T (s) = ∑Ni=1 t(sAi), and let M be a positive random variable whose distribution is determinedbyE f (M) = EN∑i=1Ai f (Ai) (2.3)for any positive and measurable function f . Then we have the following comparison theorem on the relations between theLaplace transforms of Z and Y1:Theorem 2.1. Let Z be a solution of (E) with EZ = 1.(i) Assume that ρ(α) < 1 for some α > 1, and that ρ(α + δ) < ∞ for some δ > 0. Assume also that Y1 and Z have finite r-thmoment for all r ∈ [1,α).(ia) If α ∈ (1,∞)\{2,3, . . .}, then writing α = n + β with n ∈ N∗ and β ∈ (0,1), we have for ε > 0 small enough,φn,Z (s) − E{M−1φn,Z (sM)} = φn,Y1(s) + O (sα+ε), s  0. (2.4)(ib) If α = n ∈ {2,3, . . .}, then for ε > 0 small enough and some constant cn−1 ,φn−1,Z (s) − E{M−1φn−1,Z (sM)} = φn−1,Y1(s) + cn−1n! sn + O(sn+ε), s  0. (2.5)(ii) Assume that ρ(1 + δ) < ∞ for some δ > 0. Then for ε > 0 small enough,φ1,Z (s) − E{M−1φ1,Z (sM)} = E{e−T (s) − 1 + T (s)} + O (s1+ε), s  0. (2.6)We mention that what is important in (2.4)–(2.6) is the validity of the equations for s → 0.X. Liang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 347–352 3513. Sketch of proofsProof of Theorem 1.1. We distinguish two cases according to α ∈ (1,∞)\N or α ∈ {2,3, . . .}.(i) Case α ∈ (1,∞)\N. By Proposition 2.1, it suffices to show that, as s → 0,φn,Y1(s) ∼ sα(1/s) if and only if φn,Z (s) ∼sα(1/s)1 − ρ(α) . (3.1)If φn,Z ∼ sα(1/s)1−ρ(α) (s → 0), thenEM−1φn,Z (sM) ∼ ρ(α)1 − ρ(α) sα(1/s) (s → 0). (3.2)Together with (2.4), this implies that φn,Y1 (s) ∼ sα(1/s). Conversely, assume that φn,Y1 (s) ∼ sα(1/s) (s → 0). Defineφ̃n,Z = φn,Z (s)/sα , φ̃Z (s) = φ̃n,Z (s) − E{Mα−1φ̃n,Z (sM)}. So φ̃Z (s) = φ̃n,Z (s) − ρ(α)Eφ̃n,Z (sB), where B  0 is a random vari-able whose distribution is determined by E f (B) = 1ρ(α)EMα−1 f (M) for any positive measurable function f . Let {Bi} beindependent copies of B . Then by iteration we can prove thatφ̃n,Z (s) =∞∑i=0ρ i(α)Eφ̃Z (sB1 · · · Bi). (3.3)By the assumption that φn,Y1(s) ∼ sα(1/s), the equality (2.4) and the fact that EBδ = ρ(α + δ)/ρ(α) < ∞, we have φ̃Z (s) ∼(1/s) and Eφ̃Z (sB1 · · · Bi) ∼ (1/s) as s → 0. Therefore (3.3) yields that φ̃n,Z (s) ∼ (1/s)1−ρ(α) (s → 0).(ii) Case α ∈ {2,3, . . .}. For a random variable X  0 and λ > 1, define Φ̂n−1,X (s) = φ̂n−1,X (s) − φ̂n−1,X (λs), whereφ̂n,X (s) = φn,X (s)/sα+1. Then by (2.5),Φ̂n−1,Z (s) − E{Mn−1Φ̂n−1,Z (sM)} = Φ̂n−1,Y1(s) + O (sε), s  0. (3.4)As in Case (i), using (3.4) instead of (2.4), we can show that as s → 0,Φ̂n−1,Y1(s) ∼ln x(n − 1)!(1/s) if and only if Φ̂n−1,Z (s) ∼ln x(n − 1)!(1 − ρ(n))−1(1/s). (3.5)So the conclusion follows from Proposition 2.2. Proof of Theorem 1.2. By Proposition 2.1 (with n = 1 and β = 0), it suffices to show that as s → 0, φ1,Y1(s) ∼ s(1/s) impliesφ1,Z (s) ∼ μ−1s∞∫1/s(t)dt/t. (3.6)Assume that φ1,Y1(s) ∼ s(1/s)(s → 0). Notice that if 1(x) ∼ 2(x) (x → ∞), then∫ ∞x 1(t)dt/t ∼∫ ∞x 2(t)dt/t (x → ∞).Therefore we can suppose that (1/s) = φ1,Y1(s)/s, s > 0. Let φ̂1,Z (s) = φ1,Z (s)/s, φ̂Z (s) = φ̂1,Z (s) − Eφ̂1,Z (sM) and {Mi} beindependent copies of M . Then as in (3.3), we have for s > 0,φ̂1,Z (s) =∞∑n=0Eφ̂Z (sM1 · · · Mn). (3.7)On the other hand, we can show that for ε ∈ (0,1) and s  0,φ1,Y1(s)  E{e−T (s) − 1 + T (s)} = φ1,Y1((1 − ε)s) + O (s1+δ). (3.8)So by (2.6), we haveφ1,Y1(s)s+ O (sε)  φ̂Z (s)  φ1,Y1((1 − ε)s)s+ O (sε). (3.9)This implies that in (3.7) we can approximately replace φ̂Z (s) by (1/s), to prove that when s → 0,φ̂1,Z (s) ∼∞∑n=0E(1sM1 · · · Mn)∼ μ−1∞∫1/s(t)dt/t, (3.10)where in the last step we use the condition that EMδ = ρ(1 + δ) < ∞. This gives (3.6). 352 X. Liang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 347–352AcknowledgementThe authors are very grateful to the referee for useful remarks and comments.References[1] N.H. Bingham, R.A. Doney, Asymptotic properties of supercritical branching processes I: The Galton–Watson processes, Adv. in Appl. Probab. 6 (1974)711–731.[2] N.H. Bingham, R.A. Doney, Asymptotic properties of supercritical branching processes II: Crump–Mode and Jirina processes, Adv. in Appl. Probab. 7(1975) 66–82.[3] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge, 1987.[4] L. de Haan, An Abel–Tauber theorem for Laplace transforms, J. Lond. Math. Soc. 17 (1976) 102–106.[5] A. de Meyer, J.L. Teugels, On the asymptotic behavior of the distributions of the busy period and service time in M/G/1, J. Appl. Probab. 17 (1980)802–813.[6] A. de Meyer, On a theorem of Bingham and Doney, J. Appl. Probab. 19 (1982) 217–220.[7] A. Iksanov, S. Polotskiy, Regular variation in the branching random walk, Theory Stoch. Process. 12 (28) (2006) 38–54.[8] R. Lyons, A simple path to Biggins’ martingale convergence for branching random walk, in: Classical and Modern Branching Processes, in: IMA Vol.Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221.[9] Q. Liu, On generalized multiplicative cascades, Stoch. Proc. Appl. 86 (2000) 263–286.[10] U. Rösler, V. Topchii, V. Vatutin, Convergence rate for stable weighted branching processes, in: B. Chauvin, P. Flajolet, A. Mokkadem (Eds.), Mathematicsand Computer Science, vol. II, in: Trends Math., Versailles, 2002, Birkhäuser, Basel, 2002, pp. 441–453.

Tail behavior of laws stable by random weighted mean

Liang, Xingang

Liu, Quansheng

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2011.01.029.pdf

Tail behavior of laws stable by random weighted mean

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Mathématique

Numérisation de Documents Anciens Mathématiques