By Kai Lai Chung

This publication starts off with a historic essay entitled "Will the sunlight upward thrust Again?" and ends with a normal handle entitled "Mathematics and Applications". The articles conceal a fascinating diversity of subject matters: combinatoric chances, classical restrict theorems, Markov chains and tactics, capability thought, Brownian movement, Schrödinger–Feynman difficulties, and so on. They comprise many addresses awarded at overseas meetings and detailed seminars, in addition to memorials to and recollections of well-known modern mathematicians and stories in their works. infrequent previous images of lots of them brighten up the booklet.

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Soit ( X v } une suite de variables aleatoires independantes ay ant la mime fonction de repartition* Alors enposant S„ = ^y X v , la probability P ( S „ € M pour une infinite de valeurs de n) est egale a o ou i suivant que la serie n= l est convergente ou divergente. Dimonslration. — Le cas de convergence est une consequence immediate du theoreme classique de Borel-Cantelli. Considerons done le cas de divergence. Designonspar E„ l'evenement S „ e M et en general par E' le complement de E. Nous avons, d'une m a n u r e generate, E „ = ( E , + E ; E2 + .

For the sake of brevity we introduce the following symbolic notation: ( 1, if Si is favorable to E2 Ei/Ei = \ 0, if Ex is indifferent to E2 [—1, if Ei is unfavorable to E2. Then by (ii) and (iii) we have Ei/E, = E2/Ex, E[/E2 = E2/E[ = Ei/Et = ft/Ei. = E[/E2 = E'2/E[ = -{EYfE2), Ei/E,, analogous to the rules of signs in the multiplication of integers. i is favorable to E3 ; in fact, it may happen that Ei is unfavorable to E3. For instance, imagine 11 identical balls in a bag marked respectively with the numbers - 1 1 , - 1 0 , - 3 , - 2 , - 1 , 2, 4, 6, 11, 13, 16.

For k = 1, • • • , n — 1 and 1 g m ^ k we have PROOF. Substitute (13) and a similar formula for & + 1 into the two sides respectively. After this substitution we observe that the number of terms is the same on both sides, since (n — m\f \k -m)\k n \ A + l \ _ / n —m + l)\ m ) ~ \k + 1 - \ A \ A \ m)\k)\m)' Also, the number of terms with a given U = (MI , • • • , Mm) unaccented is the same, since — m\ / n — m \ _ / n — m \ / n — m\ n— — mj \k + 1 — m) \k + 1 — mj \k — m)' k Let the sum of all the terms with U unaccented in the two summations be denoted by ak+i =

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