Ecole d'Ete de Probabilites de Saint-Flour X - 1980. by J.-M. Bismut, L. Gross, K. Krickeberg, P. L. Hennequin By J.-M. Bismut, L. Gross, K. Krickeberg, P. L. Hennequin

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LEMMA 2. Let c > 0, \$„ > 0 and n/ipn T «°, e„ i 0. Then we have 49 134 MATHEMATICS: K. L. CHUNG PROC. N. A. S. In order to evaluate \$(c\pn~l/') we shall consider the special case where each X, — =±= 1 with probability 1/2. In this case we find LEMMA 3. If an = 0 ( \ / « ) . then Pr(5„* < a„V«) = - -L r r — - exp. I 7T>=O2J + 1 \ — I + 8a„ 2 / 0 f-^7=) = T(an) + 0 ( - V ) w/fere T(aB) w defined by the series. The proof of Lemma 3 starts with a combinatorial formula due to Bachelier,5 and makes use of standard approximations together with a Fourier series transformation.

3. Generalization of Frechet's inequalities and related inequalities. Before proving our remaining theorems, we shall give a more detailed account of the general method which will be used. In the foregoing work we have already given two different expressions for the function pm(l, • • • ,n), namely, formulas (3) and (10), but they are not convenient for our later purposes. ,i's are related in magnitudes; while formula (10) has gone so far in the separation of the additive constituents that its application raises algebraical difficulties.

The number of such terms is the same on both sides. (u-i)'u , while those on the right-hand side, being of the form pu[---v{v where 0 ^ X ^ / 7 — 1 and (Ui, • • • , U\) is a combination out of (1, • • • , U — 1), are greater than or equal to it. Hence the result. 4. The p a i . ai]'s of the pi(vi, • •• , nVs. T H E O R E M 5 : For 1 ^ m ^ nwe have \m- l) P l - n = 2 2 p m ( " i , • • • » " » . ) - Hpm(vi, in terms ••• , vm+i) + . . •• PROOF: AS m g k ^ n. Pm(vi, • • • , Vm+i). "'m+i in t h e proof of Theorem 3, consider ck{in , • • • , Mm).
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