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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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J=I, ... ,n) ~J will be positive, and this phenomenon is persistent with the decrease of dimension. In this sense the recursion in (4. I) But one also notes that in is "homogeneous". e. the ball reaches two faces at the same time, and, a bit later, it reaches the other faces again at the same time). This "regularity" disappears after the step, as seen starting out from three dimensions. - 64 - Much work has been done to compile tables of percen2 wand similar statistics, in partic- tage points for n ular by STEPHENS.

J n! n S nB n p(x} = V(x- =max(O,y}. dx n , I~n)+. vol [s nB (c ,p(x») n n n n c Einterior(s ), n n faces of S n faces, equally and the distance of I from two c n Z-, and from the other n I n+3 - - . , we have to subtract the volume of two I2n n \ - 55 - ball-hats from the volume of the ball. the situation becomes extremly V (x). n we do not know exactly For larger complicated, If > n - 3' x x this is why t h en (c ,p(x», and vol [5 ] ;y At the present n- n n n n stage we are interested only in smoothness properties of S CB V (x).

M; joint distribution of by SMIRNOV [9J functions (see also TAKAcs [IOJ). 8) ill e \ --; 2c 2 a has been determined In terms of theta this can be expressed by M where MB and The > 0, b > 0, = -2a 2 3 ~) ill II lll2))' a , -\}3 ( ;- - 2 . 2 (2ac -\)-- 2c c=a+b. The joint distribution of R and B is given by THEOREM 2. 9) 2 2 + L n=1 2 n(n-I) e- 2u (n-v) _ L n-v (0 -2u n n (n+l) n-tv n=1 < u < By putting 8 2 '" 2 -2u n - vu '" n e n=1 2 e -2u (n+v) o < 00 v 2 < I). has been determ- The marginal distribution of ined by GNEDENKO [5J.

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