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Independent And Stationary Sequences Of Random Variables - Chapter 9

Chapter 9 MONOMIAL ZONES OF LOCAL NORMAL A 11 RACTION 1 . Zones of normal attraction In this chapter it will be assumed that the independent random variables X; satisfy E(X;)=0, V(XX)=62 0, and that Sn = X1 + X2+ . +X,, Z n = Sn/ an-1 . | Chapter 9 MONOMIAL ZONES OF LOCAL NORMAL ATTRACTION 1. Zones of normal attraction In this chapter it will be assumed that the independent random variables Xj satisfy E X 0 F 2Q 72 0 and that Sn X1 X2 . Xn Z Sn ani. We shall also suppose that the Xj belong to the class d of variables having a bounded continuous probability density g x . The method discussed in this chapter may be used under less stringent conditions on g x and also for lattice variables but we restrict attention to d for simplicity of presentation. Let i n be any function increasing to infinity. The segments 0 n will be called a zone of integral normal attraction if uniformly in xe 0 as n- oo P Zn x 27t i i e iu2du- l . 9.1.1 Jx If it is desired to emphasise the uniformity the phrase zone of uniform normal attraction may be used. A similar definition holds for zones of normal attraction of the form n 0 . When Z has a probability density p x we can similarly define a zone of local normal attraction as a sequence of segments 0 i n in which pn x 27i -ie-ix2- l 9.1.2 uniformly in x. It will be seen later that a special role is played by the zones delimited by 178 MONOMIAL ZONES OF LOCAL NORMAL ATTRACTION Chap. 9 o ir 9.1.3 such zones are said to be narrow. Zones of the form 0 na or na 0 are called monomial. In what follows dr 52 . e2 . 7i 72 Co Ci C2 are small and positive each one depending on its predecessors c0 c1 . Co C15 . Ko Kx . are positive constants similarly chosen B is bounded and varies from one expression to the next and p n pr n p2 n are positive functions converging to oo as n- oo. In this chapter we study monomial zones of local normal attraction both narrow and wide. 2. The fundamental conditions Theorem 9.2.1. Let 0 a j. Then the condition E exp X 2a 1 oo 9.2.1 is necessary for 0 nap n nap n 0 to be zones of local normal attraction. Proof Write 4a 2a 1 . Suppose that 9.2.1 does not hold. Then there exists a sequence xm- oo such that P Xj xm exp - 2xpm 9.2.2 for all m or P Xi - xm exp -