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

By B. Gyires

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5) (V=d(k)-I=[I] - I ) . ° ~" a But [~]-I 2 ~ p. j=1 where the follows. 5) By induction the Corollary is proved. 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.

I I) lim r-'+ oo (2. ryIISE r xii 2 II ;'ll (dy) n n The relation (2. II) holds if pnd only if to any there lln be the "conjugate" of such that \ - 37 - O. E > 0 (2. 13) f ~ *~ 06 yO>£ n n < (dy) for all k£ n r n. 2 we conclude that vectors v chosen such that the measures n,r =~ n (. x -x n,r may be n,r ) satisfy the relation f v 06 xO>£ n,r < (dx) r and (hence also for r=r(£) may be chosen such that (2. 14) lim x ..... +QO for any m n,r 6 x r > r n,r =x n r(£». Note that n,r x n,r Hence lim sup k f v (dx) n_+oo n06 xO>£ n,r o r > o.

2J Z. , 43(1978), Wahrschein- 205-221. L. Chung, On the maximum partial sums of sequences of independent random variables, Trans. Amer. Math. , 64(1949),205-233. [3J W. Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. , [4J 22(1951), 427-432. W. Feller, An introduction to Probability Theory and Its Application, Vol. [5J II, Wiley, New York, 1966. V. Gnedenko, Kriterien fUr die Unveranderlichkeit der Wahrscheinlichkeitsverteilung von zwei unabhangigen Stichprobenreihen, Math.

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