# Chance & choice: memorabilia by Kai Lai Chung

By Kai Lai Chung

This publication starts with a historic essay entitled "Will the solar upward thrust Again?" and ends with a normal handle entitled "Mathematics and Applications". The articles hide an attractive variety of issues: combinatoric chances, classical restrict theorems, Markov chains and strategies, capability idea, Brownian movement, Schrödinger–Feynman difficulties, and so on. They comprise many addresses awarded at overseas meetings and distinctive seminars, in addition to memorials to and recollections of admired modern mathematicians and reports in their works. infrequent outdated photographs of a lot of them liven up the publication.

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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).