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Extra info for Analytic Function Methods in Probability Theory: Colloquium Proceedings (Colloquia mathematica Societatis Janos Bolyai ; 21)
Sample text
Now put 6=6(s,~)=~(s+2)(s+4), and let ~ 1 s+ 1 be the smallest integer for which n 2a, and set be larger than integrating by parts f n V 6(--2(k) n times, ( a- 1 ) -~)+I dk (x)=~ dx n/3 . (1) e~txv (x)dx I/I2n n J (t) n/3 . () e~txv a (x)dx, I/I2n n J (it)a-I whence If n (t) I n/3 J Iv(a)(X)\dX= I/I2n - 56 - n a=a(s,~) ~ a. V (x). n Let Then, Thus I f T (t) f n ()O Idt ::; On t n ::; 2 a-I t a n I 6(a-l) ::; I - ° 2 a-I dt f6 n '2(5+1 )+£ nn that is I B'" B"'(s n n ' - -(5+1)+£ Q O(n 2 ). n £) In this way we have the following variant of Theorem 3 in [I J: THEOREM 3'.
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. 3) is said to be an information I] if function (1. I) [0, defined on f (x,y)ED, where D {(x,y): 0 S x < I, 0 S Y < I, x+y S I}. The following result has been proved by Z.
STACHO I. INTRODUCTION ~s The present note a continuation of [IJ. notation used there will be kept here. review these notations. -s uniformly distributed on Let us denote [0,1 J, The first independent and (t) F n 2 empirical distribution function of this sample. wn 1 2 nJ (F ( t ) - t ) dt is the Cramer-von Mises statistic, V f n n o n (x) = (t) V(x) P (w 2 < x) is its n denotes be the distrib~tion in [IJ, the first complete asymptotic expansion for transform of a natural number (Theorem 3) 1 and then tried of [IJ reads as and positive number \ - 53 - As a the Laplace-Stieltjes in powers of s n author has given a to invert this expansion, without reaching the The last result while 2 w .