TAILIEUCHUNG - Independent And Stationary Sequences Of Random Variables - Chapter 12

Chapter 12 WIDE MONOMIAL ZONES OF INTEGRAL NORMAL ATTRACTION 1 . Formulation In this chapter, as before, we study the independent, identixally distributed random variables X1, X2, . . . with E (Xi) = 0, V (Xl) = 1 . We shall study the zone [0, n"] where a 6 ; we recall that this is said to be a zone of normal attraction if, | Chapter 12 WIDE MONOMIAL ZONES OF INTEGRAL NORMAL ATTRACTION 1. Formulation In this chapter as before we study the independent identixally distributed random variables Xt X2 . with E X 0 FpQ 1 . We shall study the zone 0 n where a we recall that this is said to be a zone of normal attraction if uniformly in 0 x na as n- oo P Zn x 2n J e- 2du- l . An analogous definition holds for n 0 . As before the symbols p n pr n . pk n will denote functions monotonically increasing to infinity each one usually defined in terms of its predecessors. In this chapter we prove the following theorems. Theorem . I n 0 and 0 na are zones of normal attraction for all a then the variables Xj are normally distributed. It follows that we need only consider values of a . This theorem is a corollary of the following more precise result. Theorem . If consider the series of critical numbers s -U 1 113 i1 i 6 4 10 2 . 0 2 s 3 Let s be the unique integer with j s 1 . s 2 Js 3 a i s 4 . THE PROBABILITY OF A LARGE DEVIATION 227 In order that 0 nap n and nap n 0 be zones of normal attraction it is necessary that E expA Xj 2a i oo 0 4 l and that the moments of Xj up to order s 3 should coincide with those of a normal distribution. These conditions are moreover sufficient for 0 nff p n and nffp n 0 to be zones of normal attraction. The reason why it is necessary to include A in is that we have used a change of scale already to set r l. We remark that the necessity of has already been proved in . Theorem is completely analogous to the corresponding local Theorem but the method used in Chapter 9 is not sufficiently powerful to prove the present theorem except under more restrictive conditions. ore precisely it requires that f t 1 for t 0 and that t c 1 for i C. This will be true for example if F x P Xj x contains an absolutely continuous-component in which case Theorem can be proved by the methods of Chapter 9. 2. An upper bound

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