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Chapter 14 INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE 1 . Formulation In the preceding chapters we have studied theorems of a collective type concerning large deviations in zones of the form [0, 0 (n)] and [ - (n), 0], where 0 (n) = o (n 2) . The role of the linear functionals a j, bj was played by moments of the random variables XX . | Chapter 14 INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE 1. Formulation In the preceding chapters we have studied theorems of a collective type concerning large deviations in zones of the form 0 ij n and i r n 0 where o r . The role of the linear functionals bj was played by moments of the random variables Xj. In the case i r n nap n the condition E exp A X 4a 2a 1 oo 14.1.1 appears as a condition for normal attraction this implies that all the moments of Xj exist and the probability of a large deviation in Xj itself falls off very sharply. In this chapter we study theorems in which x is not restricted to any zone but allowed to range over the whole real line. Thus let X2 . be independent and identically distributed with E X 0 F Xj. 72 0. 14.1.2 We shall seek classes of such variables for which collective limit theorems hold which assert that uniformly in x 1 as n- oo P Zn x Ê x a. . ak n - 1 14.1.3 and P Z x tf -x h15 . bh n - 1 . 14.1.4 Here the limiting tails depend on linear functionals a - bj of F x P X1 x . We remark that the restriction x 1 is harmless since in x 1 the classical theorems hold. For simplicity we shall restrict attention to the case in which F is symmetric having a bounded continuous density g x such that for x 1 14.2. PROBABILITY OF VERY LARGE DEVIATIONS ELEMENTARY RESULT 255 - oo 6a j P xt x 7 u du X 4 0 x-6- Jx r a X and thus -x 6a a P X -x p u du X 4 0 x-6 - . oo r a X 14.1.5 14.1.6 Here a 3 since the variance exists the Ar are constants with Aa 0 and 0. The class of such probability densities we call X . Such variables have only a finite number of moments and the role of the linear functionals aj bj is layed by pseudomoments defined in 5 below. Theorem 14.1.1. For x l we have uniformly in x as n co 14.1.7 where r x n is a rational function in both arguments. For x n a i s n n0 e i e- 2du r x n nP X1 oxni 14.1.8 r x n is determined by a finite number of linear functionals of the distribution of Xr called pseudomoments. This theorem has a .
