6o) by the requirement that radii map linearly onto radii. Then c1? = g,1 (D") is. [0, 1 ]), where h* is a C'-diffeomorphism except on (Sni, 1/2).

Transcription

1 330 MATHEMATICS: CHOW AND ROBBINS PROC. N. A. S. 6o) by the requirement that radii map linearly onto radii. Then c1? = g,1 (D") is a suitable extension of d)j, hence of AD, over N(s, Qo). In the general case, the sets T ' are fibered, one by one, in order of decreasing values of m. Each fibering of a Tmi is an extension of the previous fiberings. In those steps for which m = n - k, the inductive hypothesis for a (k -1)-sphere on a k-sphere is used. LEMMA 4. The shell h(sn-i X [0, 1 ]) can be approximated by a shell h*(sn-i X [0, 1 ]), where h* is a C'-diffeomorphism except on (Sni, 1/2). Each of the manifolds rt-' admits a transverse vector field. This is easily seen, with the aid of the fact that c' _i intersects cr-' if and only if so and sj are incident, so that at most n of the (n - 1)-cells of { c have any given vertex in mmon. Hence, there is a transverse vector at each vertex, and the extension into a field on the entire manifold F"-1 offers no difficulty; similarly for An-i. One can therefore approximate to rf-' and An-' by equipotential manifolds, rresponding to the parameters (6o,..., 6, 1). As the 6's simultaneously approach zero, the equipotentials nverge uniformly to Pn"'. Details of the proofs outlined above will be submitted elsewhere for publication. Lemma 4 brings us to a point where Morse's methods can be applied to mplete the proof of the rollary to the theorem. The result is easily adapted to the spherical form of the theorem. The case n = 3 was treated, from another viewpoint, by Alexander.3 Some procedures due to Noguchi4 were suggestive in nnection with part of the above proof. * This work was mmenced at the Institute for Advanced Study in with support from ONR Grant 2989(00) and NSF Grant G It was mpleted under NSF Grant G and was presented to the American Mathematical Society on December 19, t (Note added in proof.) The author is informed that a related result by M. H. A. Newman is appearing in the Proceedings of the London Mathematical Society. I Morse, Marston, "Differentiable mappings in the Schoenflies theorem," Compositio Mathematica, 14, (1959). 2 Huebsch, William, and Marston Morse, "An explicit solution of the Schoenflies extension problem," J. Math. Soc. Japan, 12, (1960). 3 Alexander, J. W., "On the subdivision of 3-space by a polyhedron," these PROCEEDINGS, 10, 6-8 (1924). 4 Noguchi, Hiroshi, "The smoothing of mbinatorial n-manifolds in (n + 1)-space," Annals of Math. (2), 72, (1960). ON SUMS OF INDEPENDENT RANDOM VARIABLES WITH INFINITE MOMENTS AND "FAIR" GAMES* BY Y. S. CHOW AND HERBERT ROBBINS INTERNATIONAL BUSINESS MACHINES CORPORATION AND COLUMBIA UNIVERSITY Communicated by Paul A. Smith, January 25, Introduction. At the origin of this investigation is the well-known result of Feller' ncerning the Petersburg game, in which the player receives $2V if heads first appears at the ith toss of an unbiased in (i = 1, 2,..). Since the expectation of the player's gain x is infinite, a problem arises in deciding on the proper

2 VOL. 47, 1961 MATHEMATICS: CHOW AND ROBBINS 331 fee for the privilege of playing the game; Feller shows that if the player pays variable entrance fees with cumulative fee bn = n log2 n for the first n games, and if sn = x1 + + Xn denotes his total gain, then the game bemes "fair" in the sense that lim n = 1 in probability. (1) n-cc bn For a game with finite and positive expectation Ex =,u, and nstant entrance fee ji for each game, the strong law of large numbers asserts that ratio of total gain to cumulative entrance fee tends to 1 in the strong sense, P( lim S = 1) = 1. (2) n-a bn It is a natural to ask whether the same holds of (1). It does not; for the Petersburg game with bn = n log2 n, PQirn = 1) =0, (3) n-o-c bn an interesting example in which nvergence in probability does not imply nvergence with probability 1. To show this, we note that for the Petersburg random variable x, Hence, for any nstant c > 1 and n > 2, P(x > a) - for a > 1. a P(x > cbn) 2 1 cbn cn log2 n and therefore, E P(x > cbn) = (a n=1 which implies by the Borel-Cantelli lemma that p Xn > c infinitely often) = 1 and thus, P= = 1 and hence, P lim. = b -1 (4) which implies (3). Since (1) holds, it follows moreover that (3) holds for every sequence of nstants bn. We shall show in the next section that (3) holds for every game in which the player's gain is such that EjxI = and for every sequence of nstants; thus no game with E~xI = o can be made "fair" in the strong sense (2) even by allowing variable entrance fees.

3 332 MATHEMATICS: CHOW AND ROBBINS PROC. N. A. S. 2. Some General Theorems.-In this section, x, xi, x2,... will denote any sequence of independent random variables with a mmon distribution, s, = x x,, and bi, b2,... will denote any sequence of nstants. LEMMA 1. If b1/n is positive and non-decreasing, then (a) E ( IxI > bn) = 0 implies P)liml [ = a 1 (b) EP(IxI > bn) < implies P(lim, = o) = 1. Proof: Since n < b2n < b2n+1 n -2n -2n + 1' it follows that 2b6, < b2n < b2n+1 (5) Now suppose that E P(IxI > bn) = 00; then either 1 (i) EP(IxI >b2n) = o or (ii) EP(IxI > b2n,+1) = O If (i) holds, then by the first inequality of (5), ZP(dxI >2b,) = I; (6) while if (ii) holds, then by the send inequality of (5), (i) also holds, so that (6) holds in either case. In other words, EP(IxI > bn) = o implies ZP(Ixf > 2bn) = c, (7) 1 1 and hence by induction for every k = 1, 2,... Z P(IxI > bn) = o implies E P(Jxj > 2kb) =o (8) 1 1 provided only that bn/n is non-decreasing. Co Now to prove part (a) of the lemma, suppose that E P(jxj > bn) = o. By (8) and the Borel-Cantelli lemma, it follows that so that, since k was arbitrary, P( > 2 i.o.) = 1, P lim m suppo b at To prove part (b) of the lemma, suppose that

4 VOL. 47, 1961 MATHEMATICS: CHOW AND ROBBINS 333 then for some k = 1, 2, P(lim n = ) < 1; (bn 2k ) and hence by the Borel-Cantelli lemma, By (8), this implies that EP~~l >2kn)=O EP(Ixj > b) =. It follows that (b) holds. LEMMA 2. If ba/n is positive and non-decreasing, then (a) E P(IxI > bn) = X implies P(lim lb = 0c) = 1, and if, in addition, Ejxi = c, then (b) 1 (I > bn) < C implies P lim = 0) = 1. Proof: To prove part (a), suppose that E P(IxI > bn) = a; then by Lemma 1, P~lim lb = a) = 1. (9) Suppose now that P(lim l [= b )<1. (10) Then for some finite nstant c, P flm I <c) > 0 and hence, P b is bounded) > 0. (11) JXn4 _ Jn - Sn-11 < l9n+i_~ i < l'%4 sn-1l But since - b+l b < b+ lb-1' (12) it follows from (10) that P (IX7n_ is bounded) > 0,

5 334 MATHEMATICS: CHOW AND ROBBINS PROC. N. A. S. which ntradicts (9). Hence (10) cannot hold, which proves (a). The proof of part (b) of the lemma is ntained in a paper of Feller,2 to which the reader is referred. We can mbine Lemmas 1 and 2 to obtain LEMMA 3. If ba/n is positive and non-decreasing, then (a) E P(jxj > b,) = o implies P (limb = P blim 1 and if moreover E Ix = c, then (b) 1 (IxI > bn) < implies P (limibi ) P(llinb = O)= 1. From Lemma 3 we have at once THEOREM 1. If bn/n is positive and non-decreasing, then (a) P X(nim1l < a)>o.eii = imply P (lim = 0)= 1, (b) Ejxl =o implies P(O<iim' l< c)o=. In the case of the Petersburg game with be = n log2 n, it follows from (1) that for some sequence of integers n1 < n2 <..., we have P (lim- = = 1. k--- bnk Define b = bnk for nk-1 <j< nk- Then, P b- ) =1. This example shows that in Theorem 1 (b), the ndition "bn/n non-decreasing" cannot be replaced by "bn non-decreasing, bn/n -." We can now prove our principal result: THEOREM 2. If EjxI = X, then for any sequence of nstants bn, either P (lir =O) =1 (13) or P (lim = U) = 1 (14) Proof. Suppose that the theorem were false; then, there would exist a sequence of nstants bn and an x with E1xI = o such that P (lim 8n >0)> 0 and P lim < ) >. (15)

6 VOL. 47, 1961 MATHEMATICS: S. S. CAIRNS 335 Case 1. Suppose that limvibni<:~00 II<O n Then from (15), P (lim A< x) > 0O and hence by Lemma 2(a) for bn = n, 2P(Ixl > n) < oo and hence EIx < A. Thus Case 1 is impossible. Case 2. Suppose that IbnI lim. =Czo. n Set atn = nmax [I >libnl.i>b 0. (16) Then the sequence an/n is positive and non-decreasing, and there exists a sequence of integers ni < n2 <... such that ant = Ibnkl- (17) From (15) and (16), it follows that and hence by Lemma 3, p (-i i I < 0) >, a. P (lim M - o) = 1. (18) a. From (17) and (18), it follows that P (lim = 0O) = 1, which ntradicts (15). Thus Case 2 is also impossible, and the proof of the theorem is mplete. COROLLARY. If Ejx = X, then for any sequence of nstants bn, P (lim n = 1). (19) \n...1 b / * Work sponsored in part by National Science Foundation grants at Columbia and Stanford Universities. I Feller, W., Introduction to Probability Theory and its Applications (New York: John Wiley and Sons, Inc., 1957), Vol. 1, 2nd ed., p Feller, W., "A limit theorem for random variables with infinite moments," Am. J. Math., 68, 260 (1946).