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Independent And Stationary Sequences Of Random Variables - Chapter 11

Chapter 11 NARROW ZONES OF NORMAL ATTRACTION 1 . Classification of narrow zones by the function h We retain the notation of the last two chapters, and record here some new terminology. The narrow zones [0, A (n)] and [ -A (n), 0], where A (n) is continuous and increasing and A (n) = o (n b), | Chapter 11 NARROW ZONES OF NORMAL ATTRACTION 1. Classification of narrow zones by the function h We retain the notation of the last two chapters and record here some new terminology. The narrow zones 0 Z n and -Z n 0 where A n is continuous and increasing and A n o ni will be described in 2 by means of a function h x non-decreasing and continuous in x 2. It will turn out to be natural to distinguish three classes of possible function h. Class I This consists of functions h satisfying for some Co 0 log x 2 Co h x x . 11.1.1 If we write h x exp H log x then H z is required to be monotonic and differentiable with H z l H z - 0 z oo 11.1.2 H z expH z cxz1 il . 11.1.3 Class II This consists of the non-decreasing continuous functions with p0 x log x i x log x 2 h x M x log x N log x log x 11.1.4 N z - 0 z co . As before p with affixes denotes a function tending to infinity at infinity. 11.2. STATEMENT OF THE THEOREMS 199 Class III The functions h with 3 log x h x M log x where M 3 is a constant. 2. Statement of the theorems We shall investigate the narrow zones of local and integral normal convergence for variables of the class d defined in Chapter 9. In terms of the function h x we define A n by the equation h n l n zl n 2 . 11.2.1 We shall show that in a weak sense the condition for 0 L n to be a zone of local normal attraction is related to the condition E exp h Xj oo . 11.2.2 Theorem 11.2.1. If h x belongs to Class I then 11.2.2 is necessary for 0 l n p n A n p n 0 to be zones of local normal attraction and sufficient for 0 Zl n p n and A ri p ri 0 to be zones of local normal attraction. Theorem 11.2.2. The statement of Theorem 11.2.1 remains valid if the word local is replaced by integral throughout. If h belongs to Class II we define A n h n P M n log n 11.2.3 since under the conditions defining this class this differs only by a slowly varying function from that determined by 11.2.1 . Theorem 11.2.3. If h x belongs to Class II the statement of Theorem 11.2.1 .