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Independent And Stationary Sequences Of Random Variables - Chapter 13

Chapter 13MONOMIAL ZONES OF INTEGRAL ATTRACTION TO CRAMER'S SYSTEM OF LIMITING TAILS 1 . Formulation This chapter is devoted to Petrov's theorems on integral convergence, whose local analogues have been proved in Chapter 10 . We keep the notation of that chapter, but do not restrict the variables Xj to belong to the class (d) . | Chapter 13 MONOMIAL ZONES OF INTEGRAL ATTRACTION TO CRAMER S SYSTEM OF LIMITING TAILS 1. Formulation This chapter is devoted to Petrov s theorems on integral convergence whose local analogues have been proved in Chapter 10. We keep the notation of that chapter but do not restrict the variables Xj to belong to the class d . The basic results are the following analogues of Theorem 10.1.1. Theorem 13.1.1. Let p n oo be an arbitrary increasing function and suppose that for some a E exp XJ4a 2a 1 oo . 13.1.1 Then uniformly in 0 x na p n Î3 T s T n2 n2 13.1.2 Î3 i 13.1.3 Here 2 s z is the truncated Cramer series and s in the integer defined by 10.1.6 . Theorem 13.1.2. If for all x in 0 x nap n all n n0 and positive constants n0 a0 we have P Zn x e a x2 P Zn -x e a x2 13.1.4 then 13.1.1 necessarily holds and the conclusion of Theorem 13.1.1 applies. I.- 3.2. THE PROBABILITY OF A LARGE DEVIATION 245 The deduction of 13.1.1 from 13.1.4 is exactly like that given in 8.2. Thus no truncated power series 7t s z other than that of Cramer can possibly appear in formulae of the type 13.1.2 13.1.3 . Further in the collective Theorem 13.1.1 the role of the linear functionals ah is played by the moments of Xj. 2. An upper bound for the probability of a large derivation For the sequel we shall need inequalities like 12.2.3 and 12.2.10 for the probability of large deviations. We cannot however use the inequalities already proved since these depended on the vanishing of the first s 3 cumulants which is not here assumed. Suppose then that Xt X2 . are independent and identically distributed with E X 0 V Xj c2 and suppose that 13.1.1 is satisfied. We prove that for any monotonic function p n - oo there exist positive constants ct and c2 such that P Sn na i p n exp -c2n27p n 2 13.2.1 for all sufficiently large n where S E xs. To prove this assertion we consider the normalised sum Z S an 13.2.2 and a modified random variable Zn Zn Yn an 13.2.3 where Yn is a random variable independent of the