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February 12, 1998: NOTE on HISTORY Ville is most likely long dead. My vague memory is that he went into business soon after his thesis  Levy told me. It does not seem quite "fair" to says as Snell recorded in the interview, that he did not give a formal definition of martingales name concocted by him mean ing a s tem of bets. His definition is given on p. 99 in the discrete case, on p. 111 in the continuous case. He referred to Levy for condi tional probability and expectations and to Doob (*937) in the continu ous for reinforcing. It is true axH~ xhkaxtxxxraixxxt that he adopt ed a peculiar notation for random variable Xn taking values xn, and in nutilement superposed nonnegave twkytl functions sn = Sn(x 1X2,...xn). nis main definition MxIs...xnl{8n(Xl*...*xnl' Xn = anl(xl...xnl) is somewhat redundantly mysterious. But we must dsume a man innocent until proven guilty and give Ville the benefit of doubt that he merely confused his symbolism. Those dispensable functions sn are meant by hitl to representiehi n the "selection rules" in the frequency theory of Miees (von or demand Wald* which is to a large extent Ville's colletif is "all about"i see his final chapter VI: Conclusions. What struck me as odd is that Ville did not cite Doob's 1936 "Note on probability" which is extremely relevant to that theory, and has cleAr Lale.d cirt~j 4. H. otn a'.70h4i "4emmpv lydotowith a selection rule which he called optional sampling. Did(~ Ville know Doob's result? %fd Another thing: although Ville referred to Levyis Addition profusely 1 did not mention the latter's concrete reituAtof a sequence of random variables {Xn) with the property: i { x xE ..., xI = o0 . which we all know is a la aa l case of a mart kngale* Thus he proved "e the maximum inequality (due to Kolmogorov in the independentt case) and something about the law of iterated logaritA without seeming to realize that Levy could have done similar things in his context (and probably did in fact). p. 2 of HISTORY It is also odd that when it comes to a true stopping time, Ville was just as casual as if random times were real times. He gave the famous gambler's ruin problem and wrote down its solution on p. 99, and "vaunted" a little on p. 100 that his martingale solution did not depend on the specifics of the game as Bertrand's solution. In our +Zs h, A 7 A notation, 4A amounts to obsey'v(' A S ) x = X0 = E (X where T = the ruin time, H .^ U~aD h V;t 0 a r"0 ^ (I  Vvlle allowed T to be infinite with positive probability, The theo ry of martingale yields the equation above for any "fair game" provid ed the stopping time satisfies certain conditions. The me~db trial A easrqst condition is that T be bounded with probability one, which is not true even in cointossing. NO other neat condition is available since we now have necessary and sufficient conditions for its validity, which is book, definitely messy (see p. ,oTof Doob's 1953 though he did not attempt necessity; cf. Neveu's book. p/. 76 wh&O1a stated in roundabout ways [Fe habitually this author gives definitions "re gu er" in two am&ng s&e statements of results  a vile habit that should be forbid den by Napoleanic codel] but presumably ( i. e. "let us hope") does produce a NaS set of conditions). The heuristic ide was actually 3 already given in Khintchine's 1933 book who seemed f aware of its V 0 requirement of proof. Yet Ville f4reat gave hint of 1ite fundamen tal notion of "a martingale stopped at random times to remain a martin A A gale". The latter is the main idea in Doob's 1936 Note although there it is a trivial situation : an IID sequence_ remains so when optionally sample~{(XT).. Strictly speaking this .be regarded as a particu n { outi L .4h lar case of (1): firstly because we do not need a martingale(E(Xn) need not be u secondly and moVe seriously, the Tn required is not a general e .ual no optional time but is one (or two, or 17) plus an optional time. When and who first thought of a general case of (1)? Doob might have some thing unpublished till 1953, but Wald had an excellent special case p. 3 of H. (1944). He did not bother with martingale and dealt with an III se u if quence with finite mean) ihen the mean is zero the cu2nulative sums jYr form a martingale. If T is an optional time with E(T) nite. he proved (2) E(YT) = E(T).E(X1). Actually when E(X1)=0 the stronger result >at the pair (YI YTm forms a martingale. There is a less elegant generalization of the result see Doob, p. 303, 0 Wald also has a iarer result dealing with *srjn; (a,' the characteristic functions of the cumulative sims stopped~teart ep tLunal ltte, see Doob, p. 350ff. It is called the fundamental theorem of sequential analysis, presumably very practical though not partiou larly pretty  just as G. H. Hardy said long ago about Littlewood's ballistic research. As mentioned above Ville was intent on Wald's theory of collectives* one wonders if he got into sequential analysis after Paris fell in 1940, and Wald invented Se sequential stopping (of quality control of armynavy supplies) to same war expenses? Vip master Emile Borel served as minister of the Navy, as well as minister of Science or Research  I no longer remember. In the particular case where p=a/b, q=(ba)/b, so that E(Xi)=0 for all i, it is a famous theorem due jo Polya (1921/2?) that P(S2n=0 for infinitely many values of n) = 1; in particular PX{T0 (any integer). Emile Borel called this "retour a l'equilibre" and discoursed on it at le ngth (see his Valeur pratique et philosophies . ID this case the exact distribution of TO (fxrm for x=0, or for x=l) is given by the combinatorial formula of Andrd generalized by Aeppli The question arises tatxx whether the exact distribution of TO may be obtained in the more general case of b>a>A, at least when nd render E(Xi )=. Although there vas a huge literature on the subject, known as "duration of play" in Todhunter's History ([ 1), and numerous analytical formulas in terms of generating functions and their power series, due to DeMoivre, Montmart, Lagrange, Laplace (...), such a result SEEMS to be missing. Whereas Andre's "reflection principle" or "counting ballots reverse order" has been greatly extended by RotheHagen..... Takacs ..... they do not yield an ex plicit expression for the Fn in (4), except in one special case. This is the case b>a=l. Let us repeat: let P(X.=bl) = 1/b, P{Xi=}) =(bl)b, where b is any integer >2. In this case we can establish the formula F = P(T =nb) 0=6. N 0 n IbV n1 ) In6b1 This formula was found experimentally by Chung c. 1940 (when the Japs were bombing Kunming) and verfied by induction. P. L. Hsu then summed the series EF to confirm it to be one, as it should. It was .=, n announced in the Amer. Monthly (see [1]) in the algebraic form ( ). H. Gould gave a solution using the identities of RSH.... without reference to its probabilistic o'~igin or meaning. WHO would care for such a formula (out of million others) without the meaning? As Borel. .~ . \^\Wjqq Around 1980, Chung gave a lecture to the pisigma society in the University of Floroda at Gainsville on the topic, and showed his very old graph for b=., a=l, and demonstrated the inductive proof of the result. Tis old document was lost. Early in 1993, in the effort to recover the material he observed a peculair numerical equality that led to a logical proof of the result. Several experts wre consukted (incl. the French author Com$e whaf aielittle book contains the formula) but nobody knew. Finally the result was communi cated to Takacs who gave a different proof. Since both proofs are brief but based on rather diverse principles" one on the renewal ideahmd the other on the finclusitfexclusi" idea (Poincareis expres sion for the probability of the union of a number of arbitrary events, much used also in elementary number theory (HardyWright) such as Mobius .. .). they are presented here. DYSON, F. J. 1953 CAN. J. MATH. Vol. 5, pp. 554558 JLIb'&AAV ~PU~A. O Le co't, 69 1^r A UJ0 Z stY C A *,V6AA%AX S COURIER TRANSFORMS OF DISTORTION FUNCTIONS FOURIER TRANSFORMS OF DISTRIBUTION FUNCTIONS F. J. DYSON rD)A'S A&u\Y.~cW 0&t Co...M. A. W w oo*'.*" , .GAN fr, k"A *p% %JX. YbuLr Yw Reprinted from Canadian Journal of Mathematics Iwa p, (\ "s iVeQJb. CcrruJ' R 0,V\r ** UL{I & 9, ay~ ~,;SJi Vi Lu e} tt~u~t~ IUL~CO~ FOURIER TRANSFORMS OF DISTRIBUTION FUNCTIONS F. J. DYSON A distribution function O (x) is assumed to have the following properties: (1) q(x) is nondecreasing (2) lim 4(x) = 0, lim O(x) = 1, cozco (3) 4(x) = lim k(y) for every x. V*z+O The Fourier transform of 4(x) is defined by the Stieltjes integral (4) (t)= f `zd(x). Let 41 and 02 be two distribution functions. Let a positive real number 8 be given. We consider the question, does there exist a positive E such that the condition (5) u1i(t) e2(t) < e for all t implies (6) Il(x) ?(x)\ < ? There are three separate problems here. (i) We may allow e to depend on 6, 01, and x. Then our question is, does the uniform convergence of 42 to $1 imply a pointwise convergence of 02 to 01?. The answer to this question is yes, as is well known; in fact L6vy [1, p. 49] proves a theorem which states considerably more than is needed for our problem. (ii) We may allow e to depend on 8 and 01, but not on x. Then our question is, does uniform convergence of 12 to C1 imply uniform convergence of 02 to 01? The answer to this question is also yes; we prove this in Theorem 1 below. (iii) We may allow e to depend on 8 only. In this case the answer is no, as we shall show by an example. Counterexample for case (iii). Let a and b be real numbers with b > a > 0. We consider the distribution functions (7) 1(X) = log x+ )/log < 0 1, x > 0. (8) S2(x) = 1 1(x). Received June 10, 1952. FOURIER TRANSFORMS OF DISTRIBUTION FUNCTIONS Then (9) 41(x) 2(x) = log / log all x, and in particular (10) 1(0) 42(0) = 1. However, by (9) we have (11) )1(t) s2(t) i= [el e bl]/log , (12) 11(t) )2( < 7/log(). Since b/a may be arbitrarily large, we see that we can satisfy (5) for any e > 0 and still have (6) false for 6 = 1. Statement of theorem for case (ii). THEOREM 1. Let a positive 6 and a distribution function 01 be given. Then we can find e > 0, depending only on 6 and 01, such that (5) implies (6) for all x and for all 02. Let h,(x) be the function defined by (13) h,(x) = max (0, 1 Ix/[). Then (4) gives (14) h,(x w) do(x) = r 4 sin27 e t) dt, both sides being absolutely convergent integrals. If e is chosen so that (5) is satisfied, then (14) gives, for every I and w, (15) h,(x w)[d4i(x) d2(x)] < e. Since 41 is nondecreasing and (3) holds, (16) 41 (w) lim 4)(y) = lim f h(x w) d4(x), V4a0 ij>0 the limits on both sides necessarily existing. Similarly (16) holds for 42. Therefore letting 7  0 in (15), we have, for all w, (17) j(41(w) lim 4)(y)) (42(w) limn 2(y))l < e. y.)WO yto0 That is to say, at every point the discontinuities in 41 and 42 differ by at most e. Another consequence of (15) is obtained by writing in turn w + r, w + 2i, ... , w + NtI for w and adding the resulting inequalities. From the definition of h, (x), N h,(x w mro) = 1, w +  <(ax < w + NI), F. J. DYSON and N 0< E h,(x w mJ) < 1, m1 w N values of w therefore gives pw+(N+1) l w+N1 (18) dJ (x) > di((x) Ne. We write for brevity a = 18. We. can divide the whole line ( oo, + om) into a finite set of intervals II,. Im with the following properties. (i) Each I, is closed on the left and open on the right. (ii) The total variation of 41(x) on I,, is less than a. Let L. and R, be the limits to which 41(x) tends as x tends to the left and right endpoints within I,. Similarly let L: and R: be the limits of 02. By (17) we have (19) Ri R! < L2+, Li+ + e. Now let X be the length of the shortest I,, let A be the combined length of 2I, Ir,, and let N be an integer greater than (2 A/X). The choice of N and of the I, depends only on 8 and 41 and is independent of e. Given any I, with 1 < n < m, we can choose two points x, x' inside I, such that (20) x' x > jX. Then we apply (18) with w = x, w + 7t = x', giving (21) 41(x') + 4f2(x' + N7) > 02(x) + 01(X + N) Ne. By the definition of N, the point (x + Ny) belongs to Im and so OI(x + NV) > 1 a, 02(x' + Nq,) < 1. Hence (21) becomes (22) 41(x') > 42(x) NE a. Again, applying (18) with w = x Ny, w + 1 = x' Ny, 4W(x') + 4x(x' N7y) > 4i(x) + 42(x Nn) Ne, and since (x' Ny7) belongs to Ix this becomes (23) 42(x') > x1(x) Ne a. Let x' and x tend respectively to the right and left to the endpoints of I,. Then (22) and (23) give (24) L2 < R + Ne + a, (25) R: > Ln Ne a. FOURIER TRANSFORMS OF DISTRIBUTION FUNCTIONS These inequalities, (24) and (25), which have been proved for 1 < n < m, are trivially true also for n = 1 and n = m. Writing n + 1 for n in (24) and combining it with (19), we find R: < RI + R +1 L.+1 + (N + 1) e + a (26) < R+(N+ 1) e + 2a. Similarly (25) combined with (19) gives (27) Ln > Ln (N + 1) e 2a. Now R2 and LI are the upper and lower bounds of 02 in In, and R! and L, differ by at most a. Therefore (26) and (27) imply (28) 102(x) 0i(x) < (N + 1)e + 3a = (N + 1)e + 15 for all x in ( c, + co). The choice of N depended only on 8 and 01. Given 8 and 01 we can choose e to be any number less than (6/(4(N + 1))), and then (5) will imply (6). This proves the theorem. Additional remarks. Another theorem can be derived from Theorem 1 by weakening both the hypothesis and the conclusion slightly. Let us define the distance between two distributions 01 and 02 by (29) Ii 021 = max (1{01, 02}l, {02, 01}), where (30) { 1, 42} = max (min (x' x, 4i(x) 42(x'))). This definition of the distance is equivalent to that given by L6vy [1, p. 47]. It is easy to see that [[41 4211 is the side of the largest square that can be inserted between the graphs y = 0i(x) and y = 02(x) when these are plotted in cartesian coordinates in the usual way. Thus the convergence defined by 1102 0111 0 is topologically weaker than uniform convergence of 42 to 01, but topologically stronger than pointwise convergence of 02 to 41. The modified form of Theorem 1 is THEOREM 2. Ldt 8 and 01 be given. Then we can find e > 0 depending only on 8 and 01, such that (31) i(l(t) '2(t) < for all t < implies (32) 112 0111 < 8. The proof is similar to the proof of Theorem 1, only simpler. The counter example given previously also shows that the weaker conclusion (32) does not follow from (5) with e depending only on 6. 558 F. J. DYSON The author is indebted to Dr. K. L. Chung for suggesting this problem to him, and for several stimulating discussions. REFERENCE 1. P. L6vy, Thiorie de addition des variables algatoires (Paris, 1937). Cornell University PRINTED IN CANADA 