Analytic Function Methods in Probability Theory: Colloquium by B. Gyires

By B. Gyires

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The case ~ ~ of formula c=o = H. ( ~ T": ) ~ for H. :E. 2 ••••• n+I). (3. I) follows with lim vol [TnpB n ]. p"O n i. e •• if where cone T n is the oET. oo)XT. But then the lemma also follows in the stated generality. Viewing now the volume as a function of the radius we immediately have the following COROLLARY. 0":). 1 ~ ~ Hence. considering an isometry E~ space E~npBn where ~ and radius here can be written in the ~ T~n(E~npBn) form ] d(n) uously differentiable. - 62 - times contin- PROOF. For continuity of some k ~ 2 t h at t h e f n=l, AI (p), is true.

M. Hirsch, A strong law for the maximum cumulative sum of independent random variables, Comm. Appl. , [7J 18(1965), Pure 109-127. W. Magnus - F. Oberhettinger, Formeln und Satze fur die speziellen Funktionen der Mathematischen Physik, Springer, Berlin, [8J A. R~nyi, 1943. On the distribution function Hungarian), Publ. Math. Inst. Hung. L(z) (in Acad. , 2(1958),43-50. V. Smirnov, An estimate of divergence between empirical curves of a distribution in two independent samples 14. (in Russian), Bull.

In KNOTT [3J. A survey and comparison can be found In fact, the most accurate. Knott's results are proved to be All these results, on some kind of approximation of formula Lemma statistics, Knott. Vn(x). 1) ulation. , M2n for the for other similar statistic of Durbin and seems to be accessible on a computer. n=20 Unfortunately, our computer facilities here are not adequate at present to do this work. REFERENCES [I J S. Csorg5, On an asymptotic expansion for the von Mises w2 statistic, Acta. Sci.

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