By Eugene Lukacs

This quantity reports attribute functions--which play a necessary function in likelihood and statistics-- for his or her intrinsic, mathematical curiosity.

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S s = 'P t = = B 55 FUNDAMENTAL PROPERTIES the functions F (x) and G(x) have discontinuities only at the points xv (v 0, + 1 , + 2, . . ; xv+ l > xv) and there exists a constant L > 0 such that lnf (xv+ l - xv) � L (v) I G ' (x) I � A for all x xv ( v 0, + 1 , +2, . . ) . Then to every nuntber k > 1 there correspond two finite positive numbers c1 (k) and c2 (h), depending only on k, such that jF(x) - G(x) l � k2n + c1 (k)AT, provided that TL � (k). (iv) = # == s c2 The limit theorems of probability theory give approximations for the distributions of normalized sums of random variables.

P x 0 f P dyF(y+x) � e x J = F(x) lvl < ho holds. 9) f sin lv l > ho -+ oo . 10) h0 > h1 > 0. h1 J yJ < hl d11F( y+x)- Px lvl < ho d11F(y+x) � Px + s �e while 0�f d11F(y+x) � 2s. We are still free to choose h1, subject only to the restriction (3. 10). We select h1 so small that sin Ty 1 - 8 � Ty for I y I < h1• According to the law of the mean there exists a real 0( I I < 1) such that sin Ty sin T6h1 f d ) F(x d11F(x +y +y) y f Ty TOh1 (1 - 01s) f d11F(x+y) where 0 � 01 � 1 . From this we see immediately that sin Ty f Ty dy F(x+y)- Px f d11F(x+y)-Px - 02 s with 0 � 0 2 � 1.

Let {Fn(x)} be a sequence of distribu tion functions and denote by {fn (t)} the sequencee of the corresponding charac teristic functions. sequence {Fe (x)} econv rges weakly to a distribution function F (x) if, tznd only if, the s quenc {fn ( t ) } converges for every to 1,he n t a 49 FUNDAMENTAL PROPERTIES function f(t) which is continuous at t = 0. The limiting function is then the characteristic function of F(x). This theorem indicates that the one-to-one correspondence between distribution functions and characteristic functions is continuous.