# Characteristic functions by Eugene Lukacs

By Eugene Lukacs

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

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5) yield the estimate J: f (x) dF (x) - J: f(x) dFk (x) � 2(1 + C1 )e The last three inequalities and (3 . 6 ) = Ce if k � K. 47 FUNDAMENTAL PROPERTIES But this means that b b dF1c(x) f(x) f(x) dF(x), J J k�oo a a lim = which is the statement of the theorem. 2 (extension of Belly's Second Theorem). Let f(x) be continuous and bounded in the infinite interval - x and let {F1c (x)} be sequence of non-decreasing, uniformly bounded functions which converges weakly to, some function F (x). �� r ' ,/(x) dF, (:�) r ' oo f(x) dF (x).

Is = k i1 k 1 + i2 k 2 + . . + is ks = p . 3. F U N D A M E N T A L P R O P E RT I E S O F C HARACTE R I S T I C F U N C T I O N S In Sections 3 . 6 we discuss the most significant theorems which des cribe the connection between characteristic functions and distribution functions. These properties account for the importance of characteristic functions in the theory of probability. 3. 1 The uniqueness theorem a e Theorem 3 . 1 . 1 . Two distribution functions F1 (x) and F2 (x) r identical if, and only if, their characteristic functions f1 ( t) and f2 ( t) are identical.

Let {Fn(x)} be sequence of distribution functions and denote by {fn ( t the sequence of the corresponding characteristic functions . The sequence {Fn (x)} converges weakly to distribution function F(x) if, and only if, the sequence {fn(t) } converges uniformly to a limiting function f(t) in every finite t-interval T, + T ]. The limiting function f(t) is then the characteristic function of [F(x). 2 follows immediately from the continuity theorem and from the corollaries 1 and 4. a a We conclude this section with two remarks concerning the weak con vergence of a sequence of distribution functions to a limiting distribution function.