A note on asymptotics of linear combinations of iid random variables
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A NOTE ON ASYMPTOTICS OF LINEAR COMBINATIONS OF IID RANDOM VARIABLES ´ter Kevei1 Pe Analysis and Stochastics Research Group of the Hungarian Academy of Sciences Bolyai Institute, University of Szeged, Aradi v´ertan´ uk tere 1, H–6720 Szeged, Hungary E-mail: [email protected] and Centro de Investigaci´ on en Matem´ aticas, Callej´ on Jalisco S/N, Mineral de Valenciana, Guanajuato 36240, Mexico (Received June 2, 2009; Accepted January 13, 2010) [Communicated by Istv´ an Berkes]
Abstract Let X1 , X2 , . . . be iid random variables, and let an = (a1,n , . . . , an,n ) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination San = a1,n X1 + · · · + an,n Xn under the natural negligibility condition limn→∞ max{|ak,n | : k = 1, . . . , n} = 0. We prove that if San is asymptotically normal for a weight sequence an , in which the components are of the same magnitude, then the common distribution belongs to D(2).
1. Introduction Let X, X1 , X2 , . . . be iid random variables with the common distribution function F (x) = P{X ≤ x}. For each n ∈ N = {1, 2, . . .} consider the random variable San = a1,n X1 + a2,n X2 + · · · + an,n Xn , where an = (a1,n , . . . , an,n ) is an arbitrary sequence of weights. We investigate the asymptotic behavior of the weighted sum San , therefore it is reasonable to assume that each component is asymptotically negligible, that is for every ε > 0 lim
sup P{|ak,n Xk | ≥ ε} = lim P{|X| ≥ ε/an } = 0,
n→∞ 1≤k≤n
n→∞
Mathematics subject classification number : 60F05. Key words and phrases: linear combinations, asymptotic normality, domain of attraction. 1
Work supported in part by the Hungarian Scientific Research Fund, Grant T-048360.
0031-5303/2010/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
26
P. KEVEI
where an = max{|ak,n | : k = 1, 2, . . . , n}, which holds if and only if an → 0, as n → ∞. Therefore from now on we assume that an → 0. In the sequel asymptotic relations are meant as n → ∞, unless otherwise specified. Since the possible limiting distributions of San are necessarily infinitely divisible, we need the well-known representation of their characteristic functions. Let Y be an infinitely divisible real random variable with characteristic function φ(t) = E(eitY ) in its L´evy form ([4, p. 70]), given for each t ∈ R by 0 ∞ σ2 2 φ(t) = exp itθ − t + βt (x) dL(x) + βt (x) dR(x) , 2 −∞ 0 where βt (x) = eitx − 1 −
itx 1 + x2
and the constants θ ∈ R and σ ≥ 0 and the functions L(·) and R(·) are uniquely determined: L(·) is left-continuous and nondecreasing on (−∞, 0) with limx→−∞ L(x) = L(−∞) = 0 and R(·) is right-continuous and nondecreasing on 0 2 ε 2 (0, ∞) with limx→∞ R(x) = R(∞) = 0, such that −ε x dL(x) + 0 x dR(x) < ∞ for every ε > 0. As usual, we say that the distribution F is in the domain of attraction of the α-stable law W , α ∈ (0, 2], written F ∈ D(α), if for some centering and norming sequence An and Cn n 1 D Xk − An −→ W , Cn k=1
where X1 , X2 , . . . are iid ra
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