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Independent And Stationary Sequences Of Random Variables - Chapter 10

Chapter 10 MONOMIAL ZONES OF LOCAL ATTRACTION TO CRAMER'S SYSTEM OF LIMITING TAILS 1 . Formulation The theorems to be discussed in this chapter are important generalisations of those of the last chapter . There only elementary theorems (Taylor's theorem and elementary results in complex analysis) were used | Chapter 10 MONOMIAL ZONES OF LOCAL ATTRACTION TO CRAMER S SYSTEM OF LIMITING TAILS 1. Formulation The theorems to be discussed in this chapter are important generalisations of those of the last chapter. There only elementary theorems Taylor s theorem and elementary results in complex analysis were used here we shall make use of the method of steepest descents. Theorem 7.1.1 shows that for variables of class C d i.e. d variables satisfying the very stringent condition of Cramer the limiting relations 7.1.3 and 7.1.4 are satisfied in the ranges 0 and 1 4 0 so long as These relations involve the Cramer series 2 z defined at 7.2.20 . Now let 7r z n0 7i1z zr2z2 . 10.1.1 be any given power series with real coefficients with non-zero radius of convergence. Let Xj be variables of class d and let Sn Zn a and pn x be as in the last chapter. We shall be interested in the possibility of limiting relations of the form P x 2tt exp - x2 3 X 7 X 71 1 - 1 n2 n2 10.1.2 and Pn -x 2n 2 exp - x2 x3 -----r 7l n2 10.1.3 in O x n . We shall see that as before it is sufficient to consider a . We remark that if a 3 Bn3 - o l n2 n2 10.1. FORMULATION 191 so that relations like 10.1.2 and 10.1.3 imply local normal convergence. In the last chapter the zones of local normal convergence were characterised so that we may take this case as having been dealt with. Suppose therefore that 10.1.4 For 0 x na 3 y 00 vs 3 En E n-i 10.1.5 n2 s o Let s be the unique non-negative integer with i l a iU2. 10.1.6 It is easily seen that oo t 3 E nt iT Bn S 10.1.7 t s 1 n where e s 2 a s 4 0 and thus z may be replaced in 10.1.2 and 10.1.3 by the truncated series 7iw z nvz2 10.1.8 v 0 s being determined by 10.1.6 . Theorem 10.1.1. Let and define s by 10.1.6 . Let p n - co as n tends to infinity and suppose that E exp JQ 4a 2a 1 oo . 10.1.9 Then uniformly in 0 x na p n as n cQ 3 x x - x2 - 1 10.1.10 and 3 z v . - x2 - _ 4 - 1 10.1.11 n2 n2 J where A z is Cramer s series. 192 CRAMER S SYSTEM OF LIMITING TAILS Chap.