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.

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