By Kai Lai Chung

This e-book starts off with a ancient essay entitled "Will the solar upward thrust Again?" and ends with a common handle entitled "Mathematics and Applications". The articles hide an attractive diversity of themes: combinatoric possibilities, classical restrict theorems, Markov chains and strategies, power thought, Brownian movement, Schrödinger–Feynman difficulties, and so forth. They comprise many addresses provided at foreign meetings and certain seminars, in addition to memorials to and memories of well-known modern mathematicians and reports in their works. infrequent previous images of a lot of them brighten up the ebook.

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Trans. Amer. Math. Soc. Vol. 64 (1948), pp. 205-233. This page is intentionally left blank R e p r i n t e d from t h e P r o c e e d i n g s of t h e N A T I O N A L A C A D E M Y O F S C I E N C E S , Vol. 35, N o . 10, p p . 6 0 5 - 6 0 8 . O c t o b e r , 1949 ON FLUCTUATIONS IN COIN-TOSSING* B Y K A I LAI CHUNG AND W. FELLER CORNELL UNIVERSITY, ITHACA, N E W YORK Communicated by J. von Neumann, August 4, 1949 In the classical coin-tossing game we have a sequence of independent random variables Xv, v = 1, 2, .

R,,„ T x 1 /2r\/2n-2r\ THEOREM 2A. P(NU = 2r\Sin = 0) = — | - n +1 THEOREM 2. P{Nin = 2r\Sin = 2m) = ml \n — m) 53 £ . t = m V — m/ \ n — i / 1 i(n — i + 1) • 54 606 MATHEMATICS: CHUNG AND FELLER PROC. N. A. S. From these theorems it is easy to derive the following limiting forms lim o P(N„ < an) = - arc sin dh 0< a< 1 (a) (with the corresponding density ir~l(a — a 2 )~' / ! ), and lim P(Nn < an\Sn = \n;') = T'A MI - awa (x*(« - i) + <*y)y-ih e-y dy (b) for 0 < a < 1 and for every fixed X such that Xre'/!

By (11) this is equal to the sum (m + b\ _ /m\/m + b\ V m ) \\)\m + l)^\ (m + l V m + b\ _ 2 )\m + 2) since this number is the coefficient of (— l)mxm in the expansion of (1 - x)m+b (l - ^T Thus by (10) we have (3). = (-mx m (1 - xf. 23 332 KAI LAI CHUNG THEOREM (6) 2: For 21 ^ n — m and 21 ^ n — m respectively, we have m+i r V s ^ ( 1 ' • • • - n) ^£(-D l ( m+ r 1)5»H- E1 (-iy( t-0 \ & i-0 / \ * / By the reasoning in the previous proof, it is sufficient (in fact also necessary) to show that PROOF: & (m - 1 + *Vm + b \ y^1 / m - 1 + A / m + ^ > < 1.