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Extra info for Analytic Function Methods in Probability Theory: Colloquium Proceedings (Colloquia mathematica Societatis Janos Bolyai ; 21)

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Taking the simplex S of Section 2 as an example n we see that the Corollary cannot, ed. 2 in general, be improv- On the other hand, we saw that we have troubles with p's only, where the ball differentiability in those knocks against different dimensional (n-I ,n-2, ••• ,n) faces of the boundary of the In fact, easy to prove that A n (p) I- simplex. it is is piecewise analytic on 63 - (0,00) • 4. A (p(x» n (4. pn(x>{S - ~ n i=1 , An- I , 3=··· =A n- I ,n+ I where I =--; A =A lIn n-I, I n - I =nn/2/ r (I 2 +1), since is easily seen that follows then, points of the c S is an inner point of n sn and n S It n do not have obtuse angles.

S then for A (E; . ) ~ = (t : ( t, lying orthogonally are of less then (i=I, ••• ,m) F:R m - the function = vol n [ KnA ( E; I ) n. F ( E; I ' . . 2) where Fro m her e n ~ ~ vol n _ I [ KnA ( E; I ) n. defined by . nA ( E; n )] , (i=1 , . . ) R n-I . ) ~ . ~+ I) n ... nA (E; n ) ] that t h e . part~al .. (E;I, . . ,E; ] =(t:( t , u / =E;j} and (i=I, . . ,m). ), m ] K(E;I"" ,E;m)=K, Indeed, Rm. E·(E;I, .. 3) Lebesgue measure of the as the intersection of concave and upper semicontinuous ••• ,D.

Of Jozsef A. Aradi vertanuk tere I, University 6722 Szeged, Hungary - 65 - , COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY). 1977. NONNEGATIVE INFORMATION FUNCTIONS Z. DAR6CZY - GY. HAKSA I. INTRODUCTION The notion of information functions has been introduced by Z. DAR6CZY [3J. A summary of the investigations concerning information functions can be found in the book [IJ by J. ACZ~L and Z. DAR6CZY. DEFINITION. 2) = f(I), I f(-Z) = I satisfies the functional equation x f (x) + (I-x) f (-y-) f(y) + (I-y) f(-I- ) -y I-x for all (1.

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