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Gambling Systems and Multiplication–Invariant Measures
Advances in Applied Mathematics 22, 303᎐311 Ž1999. Article ID aama.1998.0630, available online at http:rrwww.idealibrary.com on
Gambling Systems and Multiplication᎐Invariant Measures Jeffrey S. Rosenthal* Department of Statistics, Uni¨ ersity of Toronto, Toronto, Ontario, Canada M5S 3G3
and Peter O. Schwartz† Department of Mathematics, Ohio State Uni¨ ersity, Columbus, Ohio 43210 Received May 13, 1998; accepted September 19, 1998
1. INTRODUCTION This short paper describes a surprising connection between two previously unrelated topics; the probability of winning certain gambling games, and the invariance of certain measures under pointwise multiplication on the circle. The former has been well studied by probabilists, notably in the book of Dubins and Savage w3x. The latter is the subject of an important and well-studied conjecture of Furstenberg w6x. But to the best of our knowledge, the connection between them is new. We shall show that associated with the gambling games are certain measures Žwhose cumulative distribution function values F Ž x . are equal to the probability of winning the gambling game when starting with initial fortune x and using the strategy of ‘‘Bold Play’’.. We shall then show that these measures have interesting properties related to multiplication invariance; in particular, they give rise to a collection of measures which are invariant under multiplication by 2, and which have a weak limit which is also invariant under multiplication by 3. These measures provide some candidates for possible counterexamples to Furstenberg’s conjecture. * E-mail address: [email protected]. Supported in part by NSERC of Canada. † E-mail address: [email protected]. 303 0196-8858r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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Necessary background about gambling is presented in Section 2. Necessary background about multiplication-invariant measures is presented in Section 3. The connection between the two is discussed in Section 4. Some further observations are in Section 5.
2. GAMBLING GAMES AND BOLD PLAY We shall consider gambling games which involve modifying our ‘‘fortune’’ by repeatedly making ‘‘bets.’’ the goal of the game is to reach a fortune of 1 Žin which case we win., before we reach a fortune of 0 Žin which case we lose.. The rules of the gambling game are as follows. We begin with some initial fortune between 0 and 1. We also have a biased coin that comes up heads with probability p and tails with probability 1 y p Žfor some 0 p - 1.. We are further told some fixed positive number r Žwhich represents the ‘‘payoff ratio’’.. We then repeatedly make bets as follows. If at some time we have a fortune x, then we may choose any value y F min x, Ž1 y x .rr 4 as our next value to bet. ŽThe choice of y may depend on the outcomes of previous bets, but not on the outcomes of future bets.. Given the bet value y, we flip the biased coin. If it comes up heads, we win and add r y to our fortune; if it comes up tails, we lose and subtract y from our fortune. The game ends when we either reach a fortune of 1 Žin which case we win., or reach a fortune of 0 Žin which case we lose.. It is clear from this description that the probability of winning this game depends upon the strategy we employ, i.e., on how much we choose to bet for each turn. It was shown by Dubins and Savage w3, pp. 90, 101x; see also w1, Theorem 7.3x that in the subfair case Ži.e., when rp - 1 y p ., the probability of winning is maximized when the strategy employ is Bold Play. By Bold Play, we mean the strategy that, given fortune x g w0, 1x, chooses a bet value of min x, Ž1 y x .rr 4 Ži.e., that bets the largest possible amount at each turn.. In the sequel, we shall write F Ž x . s Fr, p Ž x . for the probability of winning the gambling game under the bold strategy, given parameters r and p and initial fortune x. Remark 1. Clearly, in the superfair case Ži.e, when rp ) 1 y p ., Bold Play instead minimizes the probability of winning; this can be seen immediately by considering the game from the opponent’s point of view. ŽIn the fair case rp s 1 y p, our strategy is irrelevant; all strategies give us probability x of winning, assuming only that with probability 1 the game eventually terminates..
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Remark 2. To see intuitively why Bold Play is the best strategy in the subfair case, note the following. If X n is our fortune at time n, and Bn is the total amount bet up to time n, then X n y Ž pr y Ž1 y p .. Bn is a semi-bounded martingale. Thus, letting n ª ⬁, we see that for strategies which terminate with probability 1, Px Ž win . s x q Ž pr y Ž 1 y p . . E x Ž total amount bet . . In particular, for subfair games, maximizing the probability of winning is equivalent to minimizing the expected total bet.
3. MULTIPLICATION-INVARIANT MEASURES AND FURSTENBERG’S CONJECTURE We let n : w0, 1x ª w0, 1x denote the function nŽ x . s nx mod 1; and for a Borel measure on w0, 1., we let (n be defined by (nŽ a, b x s H 1Ž a, bxŽnŽ x .. d Ž x .. These functions are related to a conjecture of Furstenberg w6x. Say that is ‘‘=n invariant’’ if (n s , i.e., if the measure is unchanged upon multiplying the circle by a factor of n. Furstenberg conjectured that any probability measure on w0, 1. which is simultaneously both =2 and =3 invariant must be a convex combination of Lebesgue measure and a purely atomic measure. This conjecture has received a great deal of attention w4᎐8x. In particular, it has been shown that under additional hypotheses the conjecture is true. However, the original conjecture remains unsolved. In what follows, we shall use the gambling games to construct a measure which is simultaneously both =2 and =3 invariant, but which is not obviously a combination of Lebesgue and atomic measures. If it could be shown definitely to not be a convex combination of Lebesgue measure and an atomic measure, then that would provide a counterexample to Furstenberg’s conjecture. ŽUnfortunately, we are unable to show this.. 4. GAMBLING MEASURES AND =2 =3 INVARIANCE We recall that F Ž x . s Fr, p Ž x . stands for the probability of winning the gambling game under the bold strategy, with parameters r and p and initial fortune x. We note that F Ž0. s 0, that F Ž1. s 1, and that F is nondecreasing on w0, 1x. Furthermore, it is straightforward to show that F is a continuous function. Thus, we can define probability measures s r, p by the formula Ž a, b x s F Ž b . y F Ž a.. It is these ‘‘gambling measures’’ which will provide the connection to Furstenberg’s conjecture. A key computation is the following.
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PROPOSITION 1. that
For the case where r s n is a positi¨ e integer, we ha¨ e
n , p (nq1 s p n , p q Ž 1 y p . Ž n , p (n . . In words, composing the measure n, p with the function nq1 results in a con¨ ex combination of the measure itself, and the measure composed with n . Proof. Set F Ž x . s n, p w0, x x. Elementary arguments show that the proposition is equivalent to the following equation involving F: n
Ý js0
žž F
xqj nq1
yF
/ ž
j
// . Ý ž ž
nq1
ny1
s pF Ž x . q Ž 1 y p
F
js0
xqj n
yF
j
/ ž // n
.
Ž 1.
Now, given a current fortune x, there are two possibilities arising from the alternatives in Bold Play. We see by inspection that if x - 1rŽ n q 1., then F Ž x . s pF ŽŽ n q 1. x .. Similarly, if x ) 1rŽ n q 1., then F Ž x . s p q Ž1 y p . F Ž x y Ž1 y x .rn. s p q Ž1 y p . F ŽwŽ n q 1. x y 1xrn.. These observations imply that for any x g w0, 1x, we have F Ž xrŽ n q 1.x s pF Ž x . and F ŽŽ x q j .rŽ n q 1.. y F Ž jrŽ n q 1.. s Ž1 y p .w F ŽŽ x q j y 1.rn. yF ŽŽ j y 1.rn.x for j s 1, 2, . . . , n. Summing these equations over j establishes Ž1., and hence also establishes the proposition. In particular, applying this proposition with n s 1, we see that 1, p ( 2 s 1, p , i.e., 1, p is =2 invariant Žfor any 0 - p - 1.. Unfortunately, the measures 1, p are not also =3 invariant Žunless p s 1r2, in which case we obtain Lebesgue measure.. Applying the proposition with n s 2, we see that 2, p ( 3 s p 2, p q Ž1 y p .Ž 2, p ( 2 .. It follows that if 2, p were =2 invariant, then it would also be =3 invariant Žand hence a candidate for a counterexample to Furstenberg’s conjecture.. Unfortunately, this is not the case; for example, it is straightforward to show that 2, 1r2 w0, 16 x / Ž 2, 1r2 ( 2 .w0, 16 x. However, note that since Ž (m .(n s Ž (n .(m , we have that ( 3 s p q Ž1 y p .Ž ( 2 ., whenever s 2, p ( k for any positive integer k. If follows immediately that this equation also holds when is any weak* limit of any convex combination of these measures. This suggests that we try to find such a measure which is also =2 invariant. For concreteness, we shall focus on the case p s 1r2. LEMMA 2. We ha¨ e that 2, ( 3 m s Ž1r2 m .Ý mjs0Ž mj .Ž 2, (s j .. 1 2
1 2
Proof. The previous proposition proves the case where m s 1. As an inductive hypothesis, assume the formula is established for all m F M. For
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GAMBLING SYSTEMS
convenience, denote by the measure 2, . Using the inductive hypothesis and applying the case m s 1 to the measures ( 2 j ( 3 s ( 3 ( 2 j , we compute as follows: 1 2
( 3 Mq 1 s s s
M
1 2
M
ž /
js0
1 2
M 1 ( 2 k . q Ž ( 2 kq 1 . . j 2 ŽŽ
Ý
Mq1
M
q Ž ( 2 Mq 1 . q
ž
js1
Mq1
1
Ý
2 Mq1
Ý
js0
ž
M M q j jy1
ž / ž /Ž
( 2 j .
/
Mq1 Ž ( 2 j . , j
/
M . 1. ŽMq since Ž Mj . q Ž j y . Thus the formula holds for m s M q 1 and 1 s j the lemma is proved.
PROPOSITION 3. Fix r s 2 and p s 12 . Then lim m ª⬁ 5 2, 1r2 ( 3 m y 2, 1r2 ( 2 ⭈3 m 5 s 0, where 5 ⭈⭈⭈ 5 denotes total ¨ ariation distance. ŽThat is, the measures 2, 1r2 ( 3 m are asymptotically =2 in¨ ariant, as m ª ⬁.. Proof. Using the previous lemma, noting that 2, 1r2 ( 2 ⭈3 m s Ž 2, 1r2 ( 3 m .( 2 , and recalling that 5 ⭈⭈⭈ 5 is a metric, we see that
2, 1r2 ( 3 m y 2, 1r2 ( 2 ⭈3 m F F s s
1 2
m
js0
2 2m 2 2
m
1 2
m
Ý
m
q
m j
ž / 1
2, 1r2 ( 2 j y 2, 1r2 ( 2 jq 1
m
m m y j jy1
ž / ž / Ý ž / Ý ž /
q
2m 1
2
mq1 js0
Ý
js1 m
m
js1
1
mq1
mq1 j
1q
mq1 j 2j
mq1
mq1y2j
.
Now, this expression is simply twice the expected value of <1 y 2 Xr Ž m q 1.<, where X is a random variable having distribution Binomial Ž m q 1, 1r2.. But then EŽ X . s 12 Ž m q 1., whence EŽ1 y 2 XrŽ m q 1.. s 0. Furthermore, 1 y 2 XrŽ m q 1. is bounded, and may be represented
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ROSENTHAL AND SCHWARTZ
as the average of m q 1 different mean-0 i.i.d. terms, viz. 1y
2X mq1
s
1 mq1
mq1
Ý Ž 1 y 2 Xk . , ks1
with X k 4 i.i.d ; BernoulliŽ1r2.. Hence, by the strong law of large numbers Žsee, e.g., w1, Theorem 22.1x., we have 1 y 2 XrŽ m q 1. ª 0 as m ª ⬁. Then by the bounded convergence theorem Žsee, e.g., w1, Theorem 16.5x., we have that E <1 y 2 XrŽ m q 1.< ª 0 as m ª ⬁. Thus our upper bound converges to 0, and the proposition is shown. Now, the set of all probability measures is compact under the weak* topology. Hence, there exists at least one weak* limit point of the measures 2, 1r2 ( 3 m 4 . If is such a measure, then we necessarily have that ( 3 s p q Ž1 y p .Ž ( 2 .. Furthermore, by the above proposition, we also have that ( 2 s , i.e., that is =2 invariant. We thus obtain the following. THEOREM 4. There exists at least one weak* limit point of the set of measures 2, 1r2 ( 3 m 4⬁ms1; and all such limits are simultaneously both =2 and =3 in¨ ariant. Furthermore, if one such limit is not a con¨ ex combination of Lebesgue measure and a purely atomic measure, then this measure pro¨ ides a counterexample to Furstenberg’s conjecture. Unfortunately, we are unable to establish the existence of such a counterexample. We believe that any such weak* limit would be nonatomic; however, it is quite possible that all such weak* limits are simply Lebesque measure on w0, 1x. 5. FURTHER PROPERTIES OF THE MEASURES r, p The measures r, p are interesting in their own right. We make a few further observations about them here. We shall have occasion to consider Bold Play on the interval w a, b x Žwhere 0 F a - b F 1.. By this we mean the betting strategy that, given fortune x g w a, b x, chooses a bet value of min x y a, Ž b y x .rr 4 ; and then repeats this process until either the fortune a or the fortune b is achieved. For x g w a, b x, let Fw a, bxŽ x . denote the probability of increasing our fortune from x to b, under Bold Play on the interval w a, b x. We observe that if Ž x y a.rŽ b y a. s Ž x⬘ y a⬘.rŽ b⬘ y a⬘., then clearly Fw a⬘, b⬘xŽ x⬘. s Fw a, bxŽ x .. We shall use this in our calculations below. We next observe that if x g w0, 1rŽ r q 1.x, then ordinary Bold Play Ži.e., on w0, 1x. is equi¨ alent Žin the sense of giving the same overall probability
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GAMBLING SYSTEMS
of winning the gambling game. to Bold Play on w0, 1rŽ r q 1.x followed by Bold Play on w0, 1x. This fact is well known Žcf. w3x., and in any case follows from the observation that, for 0 F x F 1rŽ r q 1., we have Fw0, 1xŽ x . s pFw0, 1xŽŽ r q 1. x . s pFw0, 1rŽ rq1.x Ž x .. Similarly, if x g w1rŽ r q 1., 1x then ordinary Bold Play is equivalent to Bold Play on w1rŽ r q 1., 1x followed by Bold Play on w0, 1x; this follows simply by considering the game from the opponent’s point of view. We can inductively apply this observation, as follows. Let S0 and S1 be two operators which act on intervals, by keeping the first 1rr or the last rrŽ r q 1. of the interval, respectively. That is, S0 a, b . s a, a q
bya rq1
/
S1 a, b . s a q
,
bya rq1
/
,b .
By induction, we have the following. PROPOSITION 5.
Let I be an inter¨ al of the form I s S a1 S a 2 ⭈⭈⭈ S a n w 0, 1 . ,
for some n g N and some a1 , a2 , . . . , a n g 0, 14 . Let x g I. Then the strategy of using Bold Play on I, followed by Bold Play on w0, 1x, is equi¨ alent to the strategy of ordinary Bold Play on w0, 1x. Remark. By applying the lemma repeatedly, we see that it is also equivalent to Žsay. apply Bold Play first on I, then on S a1 S a 2 S a 3 w0, 1., then on S a1w0, 1., and then on w0, 1x. Indeed, any nested sequence of intervals of the form S a1 ⭈⭈⭈ S a k w0, 1. may be used. The above lemma suggests writing numbers x g w0, 1. in terms of their r-ary expansion, by which we mean the Žunique. sequence a i 4⬁is1 , with a i g 0, 14 for each i, such that x g S a1 S a 2 ⭈⭈⭈ S a n w0, 1. for all n g N. ŽFor x s 1, we instead assign the special r-ary expansion a1 s a2 s ⭈⭈⭈ s 1.. Equivalently, this means that xs
⬁
Ý is1
1 rq1
iy1
ai Ł
js1
ž
1 rq1
q
ry1 rq1
aj
/
Žwhere we take Ł 0js1Ž ⭈⭈⭈ . s 1 if it occurs.. Such expansions are related to betting using Bold Play. Specifically, we have the following proposition Žwhose proof we omit.. PROPOSITION 6. Let x g w0, 1x ha¨ e r-ary expansion a i 4⬁is1 , as defined abo¨ e. Let Fr, p Ž x . denote the probability of winning the gambling game with
310
ROSENTHAL AND SCHWARTZ
parameters r and p, with initial fortune x. Then Fr , p Ž x . s
⬁
Ý is1
iy1
pai
Ł Ž p q Ž 1 y 2 p. aj . . js1
That is, we can compute Fr, p Ž x . by writing x in its r-ary expansion, and then e¨ aluating the resulting sequence as a ŽŽ1 y p .rp .-ary expansion. Remark. We note that Ž1 y p .rp s r precisely for fair games, in which case Fr, p Ž x . s x Žas it must.. This proposition is related to Žand, indeed, is implied by. another way of constructing the measures r, p . Define an operator Tr, p , acting on measures on w0, 1x, by
Ž Tr , p . w0, x x s
¡p 0, Ž r q 1. x
~
x-
,
¢p q Ž 1 y p . 0, x y
1yx r
,
xG
1 rq1 1 rq1
, .
Then we have the following Žwe again omit the proof.. PROPOSITION 7. The measure r, p is in¨ ariant under Tr, p , i.e., Tr, p r, p s r, p . Furthermore, as n ª ⬁, we ha¨ e for any nonatomic measure that Tr,n p ª r, p , in the sense that lim
sup
nª⬁ 0Fa-bF1
Trn, p Ž a, b x y r , p Ž a, b x s 0,
i.e., the corresponding cumulati¨ e distribution functions con¨ erge uniformly. ŽThis is stronger than weak* con¨ ergence, but weaker than con¨ ergence in total ¨ ariation distance.. In light of this proposition, the measures r, p may be seen as special cases of measures whose Fourier transforms satisfy curious functional equations. These measures are obtained by partitioning the interval into finitely many pieces according to some fixed procedure while performing some fixed action on each piece Žlike our transformation Tr, p above.; and then continuing inductively on each piece. It is easily seen that by this method, for any ␣ i 4 and n i 4 with ␣ i G 0 and n i g N, satisfying Ý ␣ i s 1 and Ý n i F n, one can construct measures whose Fourier transforms ˆ satisfy
Ý ␣ i ˆ Ž n i . s ˆ Ž n . . i
Some related matters are considered by de Rham w2x.
GAMBLING SYSTEMS
311
ACKNOWLEDGMENTS We thank Nathalie Ouellet, Jan Pajak, Gareth Roberts, Peter Rosenthal, and Tom Salisbury for discussing these matters with us.
REFERENCES 1. P. Billingsley, ‘‘Probability and Measure,’’ 3rd ed., Wiley, New York, 1995. 2. G. de Rham, Sur quelques courbes definies par des ´ equations fonctionelles, Rendiconti del Seminario Matematico dell’Univerita ` e del Politecnico di Torino 16, 101᎐113, 1956᎐1957. 3. L. E. Dubins and L. J. Savage, ‘‘How to Gamble if You Must,’’ McGraw-Hill, New York, 1965. 4. J. Feldman and M. Smordinsky, Normal numbers from independent processes, Ergodic Theory Dynam. Syst. 12 Ž1992., 707᎐12. 5. J. Feldman, A generalization of a result of R. Lyons about measures on w0, 1., Israel J. Math. 81 Ž1993., 281᎐87. 6. H. Furstenberg, Math. Syst. Theory 1 Ž1967., 1᎐49. 7. R. Lyons, On measures simultaneously 2- and 3-invariant, Israel J. Math. 61 Ž1988., 219᎐224. 8. D. J. Rudolph, =2 and =3 invariant measures and entropy, Ergodic Theory Dynam. Syst. 10 Ž1990., 395᎐406.
Gambling Systems and Multiplication᎐Invariant Measures Jeffrey S. Rosenthal* Department of Statistics, Uni¨ ersity of Toronto, Toronto, Ontario, Canada M5S 3G3
and Peter O. Schwartz† Department of Mathematics, Ohio State Uni¨ ersity, Columbus, Ohio 43210 Received May 13, 1998; accepted September 19, 1998
1. INTRODUCTION This short paper describes a surprising connection between two previously unrelated topics; the probability of winning certain gambling games, and the invariance of certain measures under pointwise multiplication on the circle. The former has been well studied by probabilists, notably in the book of Dubins and Savage w3x. The latter is the subject of an important and well-studied conjecture of Furstenberg w6x. But to the best of our knowledge, the connection between them is new. We shall show that associated with the gambling games are certain measures Žwhose cumulative distribution function values F Ž x . are equal to the probability of winning the gambling game when starting with initial fortune x and using the strategy of ‘‘Bold Play’’.. We shall then show that these measures have interesting properties related to multiplication invariance; in particular, they give rise to a collection of measures which are invariant under multiplication by 2, and which have a weak limit which is also invariant under multiplication by 3. These measures provide some candidates for possible counterexamples to Furstenberg’s conjecture. * E-mail address: [email protected]. Supported in part by NSERC of Canada. † E-mail address: [email protected]. 303 0196-8858r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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Necessary background about gambling is presented in Section 2. Necessary background about multiplication-invariant measures is presented in Section 3. The connection between the two is discussed in Section 4. Some further observations are in Section 5.
2. GAMBLING GAMES AND BOLD PLAY We shall consider gambling games which involve modifying our ‘‘fortune’’ by repeatedly making ‘‘bets.’’ the goal of the game is to reach a fortune of 1 Žin which case we win., before we reach a fortune of 0 Žin which case we lose.. The rules of the gambling game are as follows. We begin with some initial fortune between 0 and 1. We also have a biased coin that comes up heads with probability p and tails with probability 1 y p Žfor some 0 p - 1.. We are further told some fixed positive number r Žwhich represents the ‘‘payoff ratio’’.. We then repeatedly make bets as follows. If at some time we have a fortune x, then we may choose any value y F min x, Ž1 y x .rr 4 as our next value to bet. ŽThe choice of y may depend on the outcomes of previous bets, but not on the outcomes of future bets.. Given the bet value y, we flip the biased coin. If it comes up heads, we win and add r y to our fortune; if it comes up tails, we lose and subtract y from our fortune. The game ends when we either reach a fortune of 1 Žin which case we win., or reach a fortune of 0 Žin which case we lose.. It is clear from this description that the probability of winning this game depends upon the strategy we employ, i.e., on how much we choose to bet for each turn. It was shown by Dubins and Savage w3, pp. 90, 101x; see also w1, Theorem 7.3x that in the subfair case Ži.e., when rp - 1 y p ., the probability of winning is maximized when the strategy employ is Bold Play. By Bold Play, we mean the strategy that, given fortune x g w0, 1x, chooses a bet value of min x, Ž1 y x .rr 4 Ži.e., that bets the largest possible amount at each turn.. In the sequel, we shall write F Ž x . s Fr, p Ž x . for the probability of winning the gambling game under the bold strategy, given parameters r and p and initial fortune x. Remark 1. Clearly, in the superfair case Ži.e, when rp ) 1 y p ., Bold Play instead minimizes the probability of winning; this can be seen immediately by considering the game from the opponent’s point of view. ŽIn the fair case rp s 1 y p, our strategy is irrelevant; all strategies give us probability x of winning, assuming only that with probability 1 the game eventually terminates..
GAMBLING SYSTEMS
305
Remark 2. To see intuitively why Bold Play is the best strategy in the subfair case, note the following. If X n is our fortune at time n, and Bn is the total amount bet up to time n, then X n y Ž pr y Ž1 y p .. Bn is a semi-bounded martingale. Thus, letting n ª ⬁, we see that for strategies which terminate with probability 1, Px Ž win . s x q Ž pr y Ž 1 y p . . E x Ž total amount bet . . In particular, for subfair games, maximizing the probability of winning is equivalent to minimizing the expected total bet.
3. MULTIPLICATION-INVARIANT MEASURES AND FURSTENBERG’S CONJECTURE We let n : w0, 1x ª w0, 1x denote the function nŽ x . s nx mod 1; and for a Borel measure on w0, 1., we let (n be defined by (nŽ a, b x s H 1Ž a, bxŽnŽ x .. d Ž x .. These functions are related to a conjecture of Furstenberg w6x. Say that is ‘‘=n invariant’’ if (n s , i.e., if the measure is unchanged upon multiplying the circle by a factor of n. Furstenberg conjectured that any probability measure on w0, 1. which is simultaneously both =2 and =3 invariant must be a convex combination of Lebesgue measure and a purely atomic measure. This conjecture has received a great deal of attention w4᎐8x. In particular, it has been shown that under additional hypotheses the conjecture is true. However, the original conjecture remains unsolved. In what follows, we shall use the gambling games to construct a measure which is simultaneously both =2 and =3 invariant, but which is not obviously a combination of Lebesgue and atomic measures. If it could be shown definitely to not be a convex combination of Lebesgue measure and an atomic measure, then that would provide a counterexample to Furstenberg’s conjecture. ŽUnfortunately, we are unable to show this.. 4. GAMBLING MEASURES AND =2 =3 INVARIANCE We recall that F Ž x . s Fr, p Ž x . stands for the probability of winning the gambling game under the bold strategy, with parameters r and p and initial fortune x. We note that F Ž0. s 0, that F Ž1. s 1, and that F is nondecreasing on w0, 1x. Furthermore, it is straightforward to show that F is a continuous function. Thus, we can define probability measures s r, p by the formula Ž a, b x s F Ž b . y F Ž a.. It is these ‘‘gambling measures’’ which will provide the connection to Furstenberg’s conjecture. A key computation is the following.
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ROSENTHAL AND SCHWARTZ
PROPOSITION 1. that
For the case where r s n is a positi¨ e integer, we ha¨ e
n , p (nq1 s p n , p q Ž 1 y p . Ž n , p (n . . In words, composing the measure n, p with the function nq1 results in a con¨ ex combination of the measure itself, and the measure composed with n . Proof. Set F Ž x . s n, p w0, x x. Elementary arguments show that the proposition is equivalent to the following equation involving F: n
Ý js0
žž F
xqj nq1
yF
/ ž
j
// . Ý ž ž
nq1
ny1
s pF Ž x . q Ž 1 y p
F
js0
xqj n
yF
j
/ ž // n
.
Ž 1.
Now, given a current fortune x, there are two possibilities arising from the alternatives in Bold Play. We see by inspection that if x - 1rŽ n q 1., then F Ž x . s pF ŽŽ n q 1. x .. Similarly, if x ) 1rŽ n q 1., then F Ž x . s p q Ž1 y p . F Ž x y Ž1 y x .rn. s p q Ž1 y p . F ŽwŽ n q 1. x y 1xrn.. These observations imply that for any x g w0, 1x, we have F Ž xrŽ n q 1.x s pF Ž x . and F ŽŽ x q j .rŽ n q 1.. y F Ž jrŽ n q 1.. s Ž1 y p .w F ŽŽ x q j y 1.rn. yF ŽŽ j y 1.rn.x for j s 1, 2, . . . , n. Summing these equations over j establishes Ž1., and hence also establishes the proposition. In particular, applying this proposition with n s 1, we see that 1, p ( 2 s 1, p , i.e., 1, p is =2 invariant Žfor any 0 - p - 1.. Unfortunately, the measures 1, p are not also =3 invariant Žunless p s 1r2, in which case we obtain Lebesgue measure.. Applying the proposition with n s 2, we see that 2, p ( 3 s p 2, p q Ž1 y p .Ž 2, p ( 2 .. It follows that if 2, p were =2 invariant, then it would also be =3 invariant Žand hence a candidate for a counterexample to Furstenberg’s conjecture.. Unfortunately, this is not the case; for example, it is straightforward to show that 2, 1r2 w0, 16 x / Ž 2, 1r2 ( 2 .w0, 16 x. However, note that since Ž (m .(n s Ž (n .(m , we have that ( 3 s p q Ž1 y p .Ž ( 2 ., whenever s 2, p ( k for any positive integer k. If follows immediately that this equation also holds when is any weak* limit of any convex combination of these measures. This suggests that we try to find such a measure which is also =2 invariant. For concreteness, we shall focus on the case p s 1r2. LEMMA 2. We ha¨ e that 2, ( 3 m s Ž1r2 m .Ý mjs0Ž mj .Ž 2, (s j .. 1 2
1 2
Proof. The previous proposition proves the case where m s 1. As an inductive hypothesis, assume the formula is established for all m F M. For
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GAMBLING SYSTEMS
convenience, denote by the measure 2, . Using the inductive hypothesis and applying the case m s 1 to the measures ( 2 j ( 3 s ( 3 ( 2 j , we compute as follows: 1 2
( 3 Mq 1 s s s
M
1 2
M
ž /
js0
1 2
M 1 ( 2 k . q Ž ( 2 kq 1 . . j 2 ŽŽ
Ý
Mq1
M
q Ž ( 2 Mq 1 . q
ž
js1
Mq1
1
Ý
2 Mq1
Ý
js0
ž
M M q j jy1
ž / ž /Ž
( 2 j .
/
Mq1 Ž ( 2 j . , j
/
M . 1. ŽMq since Ž Mj . q Ž j y . Thus the formula holds for m s M q 1 and 1 s j the lemma is proved.
PROPOSITION 3. Fix r s 2 and p s 12 . Then lim m ª⬁ 5 2, 1r2 ( 3 m y 2, 1r2 ( 2 ⭈3 m 5 s 0, where 5 ⭈⭈⭈ 5 denotes total ¨ ariation distance. ŽThat is, the measures 2, 1r2 ( 3 m are asymptotically =2 in¨ ariant, as m ª ⬁.. Proof. Using the previous lemma, noting that 2, 1r2 ( 2 ⭈3 m s Ž 2, 1r2 ( 3 m .( 2 , and recalling that 5 ⭈⭈⭈ 5 is a metric, we see that
2, 1r2 ( 3 m y 2, 1r2 ( 2 ⭈3 m F F s s
1 2
m
js0
2 2m 2 2
m
1 2
m
Ý
m
q
m j
ž / 1
2, 1r2 ( 2 j y 2, 1r2 ( 2 jq 1
m
m m y j jy1
ž / ž / Ý ž / Ý ž /
q
2m 1
2
mq1 js0
Ý
js1 m
m
js1
1
mq1
mq1 j
1q
mq1 j 2j
mq1
mq1y2j
.
Now, this expression is simply twice the expected value of <1 y 2 Xr Ž m q 1.<, where X is a random variable having distribution Binomial Ž m q 1, 1r2.. But then EŽ X . s 12 Ž m q 1., whence EŽ1 y 2 XrŽ m q 1.. s 0. Furthermore, 1 y 2 XrŽ m q 1. is bounded, and may be represented
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ROSENTHAL AND SCHWARTZ
as the average of m q 1 different mean-0 i.i.d. terms, viz. 1y
2X mq1
s
1 mq1
mq1
Ý Ž 1 y 2 Xk . , ks1
with X k 4 i.i.d ; BernoulliŽ1r2.. Hence, by the strong law of large numbers Žsee, e.g., w1, Theorem 22.1x., we have 1 y 2 XrŽ m q 1. ª 0 as m ª ⬁. Then by the bounded convergence theorem Žsee, e.g., w1, Theorem 16.5x., we have that E <1 y 2 XrŽ m q 1.< ª 0 as m ª ⬁. Thus our upper bound converges to 0, and the proposition is shown. Now, the set of all probability measures is compact under the weak* topology. Hence, there exists at least one weak* limit point of the measures 2, 1r2 ( 3 m 4 . If is such a measure, then we necessarily have that ( 3 s p q Ž1 y p .Ž ( 2 .. Furthermore, by the above proposition, we also have that ( 2 s , i.e., that is =2 invariant. We thus obtain the following. THEOREM 4. There exists at least one weak* limit point of the set of measures 2, 1r2 ( 3 m 4⬁ms1; and all such limits are simultaneously both =2 and =3 in¨ ariant. Furthermore, if one such limit is not a con¨ ex combination of Lebesgue measure and a purely atomic measure, then this measure pro¨ ides a counterexample to Furstenberg’s conjecture. Unfortunately, we are unable to establish the existence of such a counterexample. We believe that any such weak* limit would be nonatomic; however, it is quite possible that all such weak* limits are simply Lebesque measure on w0, 1x. 5. FURTHER PROPERTIES OF THE MEASURES r, p The measures r, p are interesting in their own right. We make a few further observations about them here. We shall have occasion to consider Bold Play on the interval w a, b x Žwhere 0 F a - b F 1.. By this we mean the betting strategy that, given fortune x g w a, b x, chooses a bet value of min x y a, Ž b y x .rr 4 ; and then repeats this process until either the fortune a or the fortune b is achieved. For x g w a, b x, let Fw a, bxŽ x . denote the probability of increasing our fortune from x to b, under Bold Play on the interval w a, b x. We observe that if Ž x y a.rŽ b y a. s Ž x⬘ y a⬘.rŽ b⬘ y a⬘., then clearly Fw a⬘, b⬘xŽ x⬘. s Fw a, bxŽ x .. We shall use this in our calculations below. We next observe that if x g w0, 1rŽ r q 1.x, then ordinary Bold Play Ži.e., on w0, 1x. is equi¨ alent Žin the sense of giving the same overall probability
309
GAMBLING SYSTEMS
of winning the gambling game. to Bold Play on w0, 1rŽ r q 1.x followed by Bold Play on w0, 1x. This fact is well known Žcf. w3x., and in any case follows from the observation that, for 0 F x F 1rŽ r q 1., we have Fw0, 1xŽ x . s pFw0, 1xŽŽ r q 1. x . s pFw0, 1rŽ rq1.x Ž x .. Similarly, if x g w1rŽ r q 1., 1x then ordinary Bold Play is equivalent to Bold Play on w1rŽ r q 1., 1x followed by Bold Play on w0, 1x; this follows simply by considering the game from the opponent’s point of view. We can inductively apply this observation, as follows. Let S0 and S1 be two operators which act on intervals, by keeping the first 1rr or the last rrŽ r q 1. of the interval, respectively. That is, S0 a, b . s a, a q
bya rq1
/
S1 a, b . s a q
,
bya rq1
/
,b .
By induction, we have the following. PROPOSITION 5.
Let I be an inter¨ al of the form I s S a1 S a 2 ⭈⭈⭈ S a n w 0, 1 . ,
for some n g N and some a1 , a2 , . . . , a n g 0, 14 . Let x g I. Then the strategy of using Bold Play on I, followed by Bold Play on w0, 1x, is equi¨ alent to the strategy of ordinary Bold Play on w0, 1x. Remark. By applying the lemma repeatedly, we see that it is also equivalent to Žsay. apply Bold Play first on I, then on S a1 S a 2 S a 3 w0, 1., then on S a1w0, 1., and then on w0, 1x. Indeed, any nested sequence of intervals of the form S a1 ⭈⭈⭈ S a k w0, 1. may be used. The above lemma suggests writing numbers x g w0, 1. in terms of their r-ary expansion, by which we mean the Žunique. sequence a i 4⬁is1 , with a i g 0, 14 for each i, such that x g S a1 S a 2 ⭈⭈⭈ S a n w0, 1. for all n g N. ŽFor x s 1, we instead assign the special r-ary expansion a1 s a2 s ⭈⭈⭈ s 1.. Equivalently, this means that xs
⬁
Ý is1
1 rq1
iy1
ai Ł
js1
ž
1 rq1
q
ry1 rq1
aj
/
Žwhere we take Ł 0js1Ž ⭈⭈⭈ . s 1 if it occurs.. Such expansions are related to betting using Bold Play. Specifically, we have the following proposition Žwhose proof we omit.. PROPOSITION 6. Let x g w0, 1x ha¨ e r-ary expansion a i 4⬁is1 , as defined abo¨ e. Let Fr, p Ž x . denote the probability of winning the gambling game with
310
ROSENTHAL AND SCHWARTZ
parameters r and p, with initial fortune x. Then Fr , p Ž x . s
⬁
Ý is1
iy1
pai
Ł Ž p q Ž 1 y 2 p. aj . . js1
That is, we can compute Fr, p Ž x . by writing x in its r-ary expansion, and then e¨ aluating the resulting sequence as a ŽŽ1 y p .rp .-ary expansion. Remark. We note that Ž1 y p .rp s r precisely for fair games, in which case Fr, p Ž x . s x Žas it must.. This proposition is related to Žand, indeed, is implied by. another way of constructing the measures r, p . Define an operator Tr, p , acting on measures on w0, 1x, by
Ž Tr , p . w0, x x s
¡p 0, Ž r q 1. x
~
x-
,
¢p q Ž 1 y p . 0, x y
1yx r
,
xG
1 rq1 1 rq1
, .
Then we have the following Žwe again omit the proof.. PROPOSITION 7. The measure r, p is in¨ ariant under Tr, p , i.e., Tr, p r, p s r, p . Furthermore, as n ª ⬁, we ha¨ e for any nonatomic measure that Tr,n p ª r, p , in the sense that lim
sup
nª⬁ 0Fa-bF1
Trn, p Ž a, b x y r , p Ž a, b x s 0,
i.e., the corresponding cumulati¨ e distribution functions con¨ erge uniformly. ŽThis is stronger than weak* con¨ ergence, but weaker than con¨ ergence in total ¨ ariation distance.. In light of this proposition, the measures r, p may be seen as special cases of measures whose Fourier transforms satisfy curious functional equations. These measures are obtained by partitioning the interval into finitely many pieces according to some fixed procedure while performing some fixed action on each piece Žlike our transformation Tr, p above.; and then continuing inductively on each piece. It is easily seen that by this method, for any ␣ i 4 and n i 4 with ␣ i G 0 and n i g N, satisfying Ý ␣ i s 1 and Ý n i F n, one can construct measures whose Fourier transforms ˆ satisfy
Ý ␣ i ˆ Ž n i . s ˆ Ž n . . i
Some related matters are considered by de Rham w2x.
GAMBLING SYSTEMS
311
ACKNOWLEDGMENTS We thank Nathalie Ouellet, Jan Pajak, Gareth Roberts, Peter Rosenthal, and Tom Salisbury for discussing these matters with us.
REFERENCES 1. P. Billingsley, ‘‘Probability and Measure,’’ 3rd ed., Wiley, New York, 1995. 2. G. de Rham, Sur quelques courbes definies par des ´ equations fonctionelles, Rendiconti del Seminario Matematico dell’Univerita ` e del Politecnico di Torino 16, 101᎐113, 1956᎐1957. 3. L. E. Dubins and L. J. Savage, ‘‘How to Gamble if You Must,’’ McGraw-Hill, New York, 1965. 4. J. Feldman and M. Smordinsky, Normal numbers from independent processes, Ergodic Theory Dynam. Syst. 12 Ž1992., 707᎐12. 5. J. Feldman, A generalization of a result of R. Lyons about measures on w0, 1., Israel J. Math. 81 Ž1993., 281᎐87. 6. H. Furstenberg, Math. Syst. Theory 1 Ž1967., 1᎐49. 7. R. Lyons, On measures simultaneously 2- and 3-invariant, Israel J. Math. 61 Ž1988., 219᎐224. 8. D. J. Rudolph, =2 and =3 invariant measures and entropy, Ergodic Theory Dynam. Syst. 10 Ž1990., 395᎐406.