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Independent And Stationary Sequences Of Random Variables - Chapter 7

Chapter 7 RICHTER'S LOCAL THEOREMS AND BERNSTEIN'S INEQUALITY 1 . Statement of the theorems The theorems of this chapter do not have a collective character, and are related to Theorem 6 .1.1 . We shall consider a sequence of independent, identically distributed random variables XX | Chapter 7 RICHTER S LOCAL THEOREMS AND BERNSTEIN S INEQUALITY 1. Statement of the theorems The theorems of this chapter do not have a collective character and are related to Theorem 6.1.1. We shall consider a sequence of independent identically distributed random variables Xj with X 0 V Xj 72 0 7.1.1 satisfying Cramer s condition C E exp u XJ oo 7.1.2 where a is a positive constant. We shall call such variables those of class C and distinguish the subclass C d of variables with a bounded continuous probability density g x and the subclass C e of lattice variables i.e. those taking only the values b kh k 0 1 . h being maximal. Assuming as before that Zn X1 X2 . X ani we remark that for C d variables Z has a probability density p x while for C e variables Zn takes only the values x xnk kh bn on . The local theorems of Richter 147 148 treat the asymptotic behaviour of p x and P Z x fc respectively. We shall consider only the simplest formulation of these theorems in order to make the proofs reasonably simple c 4.2 4.3 . Theorem 7.1.1. If the variables Xj belong to C d then for x 1 x o n as n- co we have 7.2. A LOCAL LIMIT THEOREM FOR PROBABILITY DENSITIES 161 3 X ix I T . n2 exp -r- A PoW Ln1 7.1.3 t exp-7- A PoW L x3 x . X n 7.1.4 l o Here p0 x 2tc ie 2x2 and A z Aq A i z A2 zf . is Cramer s power series convergent for z e a where e a depends only on a cf. 6.1.11 . The construction of this power series will be detailed later. Theorem 7.1.2. If the variables Xj belong to C e and x xnk kh bn on2 then for x l x o n2 as n- co we have P Zn x p x exp 7.1.5 For x 1 x o n2 we have ffP Zn xnk Po x exp 1 oft l . 7.1.6 . yr J The symbols p0 x A z have the same meanings as before. Theorems 7.1.1 and 7.1.2 will be proved by the method of steepest descents. 2. A local limit theorem for probability densities Let the Xj belong to C d and denote their characteristic function by t M it i eltxg x dx . J 00 We remark that t eL2 co 00 i.e. that i f t 2dt co 7.2.1 oc Indeed 0 t 2 is the .