1. No category

What is a Martingale?
Author(s): J. L. Doob
Source: The American Mathematical Monthly, Vol. 78, No. 5 (May, 1971), pp. 451-463
Published by: Mathematical Association of America
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WHAT IS A MARTINGALE?
of Illinois
J. L. DOOB, University
1. Introduction.Martingale theoryillustratesthe historyof mathematical
probability:the basic definitionsare inspiredby crude notionsof gambling,but
the theoryhas become a sophisticated tool of modern abstract mathematics,
drawing fromand contributingto other fields. Martingales have been studied
systematicallyfor about thirtyyears, and the newer probabilitytexts usually
devote some space to them, but the applications are so varied that thereis no
one place wherea fullaccount can be found.References[1 ] and [2 ] are the most
completesources.
The followingaccount of martingaletheoryis designed to give a feelingfor
the subject with a minimumof technicality.The basic definitionsare given at
two levels, of which the firstis more intuitiveand elementaryand sufficesfor
some of the examples. The examples illustrateonly the immediateapplications
requiringa minimumof background.
We recall that in probabilitytheoryone starts with a set called the sample
space, that events are subsets of this space, and random variables are functionls
on thisspace. Suppose forsimplicitythat the sample space 2 has only countably
many points wi, W2, ... to which are assigned probabilitiespi, P2, * * * respectively,withp3_0 and Es pi 1. If xi, . . . , xkare randomvariables, we write
{xi
=
al,
*
, xJk-
ak
f
= nm
{ Cj:xm(.wi)=
am,}
forthe set of points wherethe random variables have the indicated values. The
probabilityP {A } of the event A is definedas Epj, wherethe sum is over those
values ofj withwj in A. If P {B } >O, the conditionalprobabilityof the event A
relative to B is definedby P { A B }P {AflB }/P {B }. If x is a random variable, its expectationis definedas
(1.1)
Et{x}
= E
x(oj)pj,
(where it is supposed that the series convergesabsolutely) and the conditional
expectationof x relative to B is definedcorrespondinglywhen P { B } >0 as
(1.2)
E{x I B}
(,w)p1P{B
=
Doob receivedhis HarvardPh.D. underJ. L. Walsh,spentthe nexttwoyearsat
Professor
a thirdyearthereon a CarnegieCorp.grantto HaroldHotelColumbiaon an N.R.C. Fellowship,
of Illinoisin 1935.He is a memberofthe
ling,and beganhislongassociationwiththeUniversity
oftheAmericanMathematical
Societyin 1963NationalAcademyofSciences,servedas President
in 1945and as President
in 1950.He is the
64,and servedtheInst.ofMath.Stat.as Vice-President
authorof Stochastic
Processes(Wiley,1953) and of numerous
important
researchpaperson probabilitytheoryand applications.Editor.
451
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452
J. L. DOOB
[May
where the prime indicates that the sum is over the values ofj with cj in B. If
P {B } = 0 the precedingconditionalprobabilityand expectationcan be defined
arbitrarilywithoutaffecting
laterwork.
It has been found useful to make conditional expectations into functions,
as follows.Let {B., n> 1 } be a partitionofQ, that is, a countable class ofdisjoint
sets with union U. This partitiongenerates,and is in turn determinedby, a oalgebra Y,namelythe class of all unionsof sets of the partition.If x is a random
variable with an expectation,defineE {xj 5 }, the conditionalexpectationof x
relativeto C, as therandomvariablewiththeconstantvalue E {x IBn} on each
set B,n.This definitionis unambiguous except on the partitionsets (if thereare
any) of probability0. In particular if yi, * * *, ykare random variables, they
induce the partition3feach ofwhose sets is determinedby a conditionof the form
, yk =ak }; in thiscase E {xI }, also denoted by E {xl yi, * *
},
{yi=a,,
is the functionwiththe value E { x Iy =a,,
,y; = akI on theset {y = al,
Yk=ak
1.
Let TiCT2C . . be a finiteor infiniteincreasing sequence of a-algebras
(generated by partitionsof the sample space as just described). The intuitive
pictureto keep in mind is that TX representsthe class of all relevantpast events
up to and includingtime n. The monotoneityrelation correspondsto the idea
that the past to time n?1 includes more events than the past to time n. Let
xl, x2, * * be a sequence of random variables. We consider xn as part of the
relevanthistoryto time n, interpretingthis statementto mean that each event
of the form {C -a } is a set in the class 5n. The sequence of randomvariables is
are specifiedand an is the past as
to be analyzed. In some applicationsxi, x2,
determinedentirelyby xi, * * *, xn,that is Tnis generatedby the partitionof Q
induced by xi, - *, x.. This choice of 5. will be called minimal (relative to a
specifiedsequence ofrandomvariables).
The sequence {Ixn,n ? 1} is called a martingalerelativeto {15U;,n 1} if
each xn has an expectation,and if form <n the expected value of xn given the
past up to timem is Xm,that is
(1.3)
E{Xn
= Xm
I nIm}
This is a relationbetween functionson the sample space and is to hold almost
everywhere,that is everywhereexcept perhapson a subset ofthe sample space of
probability0. If everypi is strictlypositive,the exceptionalset is empty.If xn is
thought of as the fortuneof a gambler at time n, the definingequality (1.3)
correspondsto the idea that the game the gambler is playing is fair. If '=' in
or '<, the sequence of random variables is called a
(1.3) is replaced by '
submartingale or supermartingalerespectively,and the correspondinggames
are then respectivelyfavorable or unfavorable to the gambler. Trivially (for
ifand onlyif { -Xn, n ? 1 }
specified
a-algebras){Xn,n : 1 } is a supermartingale
is a submartingale,and is a martingaleifand only ifit is both a supermartingale
and a submartingale.The definitionsimply that E {xn} increases with n in the
submartingalecase, decreases with n in the supermartingalecase, and does not
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19711
453
WHAT IS A MARTINGALE?
vary with n in the martingale case. (All monotoneitystatements are to be
interpretedin the wide sense.)
Equation (1.3) impliesthat
(1.4)
E{Xn I XI;
*
Xm} = XM
equivalentlythat
(1.4')
E{xn I xi = a,,
.. *Xm = am}
am,
whenever the conditioningevent has strictlypositive probability,and in fact
(1.4) is the same as (1.3) whenever every 5k is minimal. In other words, a
martingalerelative to a given sequence of u-algebrasis also one relative to the
minimal sequence of a-algebras. A correspondingremark is valid for submartingales and supermartingales.If the sequence of cr-algebrasis not mentioned,
the minimalsequence is to be understood.
The definitionof a martingaleis applicable to complex-valuedrandom variables, and we shall consider certain complex martingalesbelow. Trivially, the
real and imaginaryparts of a complex martingaleare real martingales.
2. Definitionsin the general case. The definitionsin Section 1 assumed
countabilityof the sample space, a condition not satisfiedforsome of the applications to be described below. In this section definitionswill be given in the
general case, in non-probabilisticlanguage to convince cynical readers that
probabilitytheorydoes not need an admixtureof non-mathematicaltermslike
coin, event,gambler,urn, * * *, even though the ideas behind these termshave
inspiredmuchof the theory.
Let {Q, C, P } be a measurespace: Q is a set, 5fis a a-algebra of subsets of Q,
and P is a measure definedon W.Assume furtherthat PI{Q} = 1. (Some nonprobabilists accuse probabilists of seeking to mystifyoutsiders by disguising
measurable functionsand theirintegralswith the aliases randomvariablesand
expectations.Note however that probabilistswere dealing with the integralsof
functionson abstract sets beforeotheranalysts dreamed of measuretheory.It is
sardonic that,dually, some probabilistsaccuse othersof obfuscatingprobability
with measure theory.)Let Y1C52C . . . be an increasingsequence of u-algebras
of 5fsets. Let {X., n > 1 } be a sequence of complex functionson Q satisfyingthe
followingconditions:
(a) xXis measurablerelativeto 5i;
(b) xXis integrable;
(c) If m <n and ifA is any set in Jmnthen
(2.1)
fdP=
xmdP.
Then the sequence of random variables is said to be a martingalerelative to
1 }. If the randomvariables are real and if' =' is replaced in (2.1) by ' '
nt?,n a
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454
J. L.
[May
DOOB
or '_', the sequence of functionsis said to be a supermartingaleor submartingale, respectively,relative to the sequence of a-algebras. Conditional expectationsrelative to a a-algebra (in the present generalcontext) are definedin
such a way that (2.1) and (1.3) (to hold P almost everywhereon Q) are equivalent. Thus the present definitionsinclude the earlier ones and the collateral
remarksabout martingaletheoryin Section 1 are valid in the generalcase also.
The definitionshave been given for the parameter set 1, * * *, k or 1,
2, * * *, ordered as usual, but they are obviously extendable to any simplyorderedparameterset.
3. Example: expectationsknowingmore and more. Let 5f1C52C * be an
increasingsequence (finiteor infinite)of u-algebras,as usual, and let x be a random variable with an expectation. Then if xn=E {xJI }, the sequence xi,
x2, * * * is a martingalerelative to the given u-algebrasequence. That is, successive conditional expectations of x, as we know more and more,yield a martingale. More generally,the parameter set 1, 2, * * *can be replaced in this
example by any simply orderedset. The essential condition is that 53tincrease
with t. Every martingalewhose parameterset has a last elementis of this type,
withx identifiedwiththe last element.
This example suggests the possibilityof applications to statistics and informationtheoryand induces such probabilisticextravagancesas the statement
that 'the game man plays with nature as he learns more and more is fair.'
(Perhaps this statementindicates how realisticallya martingalereproducesthe
idea ofa fairgame.)
4. Example: sums of independent random variables. Let yi, Y2, * be
independent random variables with expectations, and let x =y1?+ * * +y.
Then it is intuitivelyobvious and easily proved that xi, x2, * is a martingaleif
0 forj> 1, a submartingaleif E {yI} 0 forj> 1, a supermartingaleif
E =y
E y; } <Oforj>1.
5. Example: averages of independent random variables. In the preceding
example suppose that yl, Y2, * * * have a common distributionwhich has an expectation.Then a symmetryargumentshows that
X3
X2
-
2
,
xl
is a martingalein the indicated order (left to right). This example suggests
possible applications to the law of large numbers and thereforesuggests ties
between martingaletheoryand ergodic theory.In fact thereare close relationships between the two theories,and sometimesit is said that one contains the
other.Which is said to contain the otherdepends on the speaker. At any rate, it
is true that in a reasonable sense there are only two qualitative convergence
theoremsin measure theory(aside fromtheoremsof the form"convergenceof
type I impliesconvergenceof type II"), the ergodictheoremand the martingale
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1971]
WHAT IS A MARTINGALE?
455
convergencetheorem.The latterwill be discussed below. Each is involved with
finerand fineraveraging.
6. Example: harmonicfunctionson a lattice.Let S be any subset ofthe set S'
of points with integralcoordinates in d-dimensionalcoordinate space, d 2 1. A
pointt of S will be called an interiorpoint of S if S contains all 2d of the nearest
neighborsin S' of t. Otherwiset will be called a boundary point of S. Unless
S S' therewill be boundary points. A functionu on S will be called harmonic
(superharmonic)if u at each interiorpoint of S is equal (at least equal) to the
average of u on its 2d nearest neighbors. For example, a linear functionis
harmonic,a concave functionof a harmonicfunctionis superharmonic.Define a
walk on S, that is, a sequence x0,xi, * * - of random variables with values in S,
as follows.Prescribesome initial point in S and set x0 identicallythis point. If
xo= a0, ... * X, =an and ifan is an interiorpoint of S, thenXn+1is to be (conditional probability) one of the 2d nearest neighborsof an, with (conditional)
probability1/(2d) foreach one. If an is a boundary point of S, Xn+1is to be a".
Thus the walk proceeds until the boundaryis reached,if ever, and sticks at the
firstboundary point reached. It can be shown that such a walk exists. If u is
harmonic (superharmonic) on S, the sequence of random variables u(xo),
u(xi), ...**is a martingale(supermartingale).If we are to consider the infinite
sequence x0oxi, ... , the sample space mustbe uncountable.
7. Example: classical harmonic and analytic functions.A variation of the
idea of Section 6 is the following.Let S be an open subset of d-dimensional
coordinatespace, d _ 1. A functionu on S is said to be harmonicifu is continuous
and if,whenevert is a point of S, and B is a ball with centert whose closure
lies in S, the value ofu at t is the average of its values on the boundaryofB. For
example linearfunctionsare harmonicforall d, and are the only harmonicfunctions if d =1 and S is an interval. The harmonic functionsare the infinitely
differentiablefunctionswhose Laplacians vanish. If d 2, the real part of an
analytic functionis harmonic.If t is in S and if S is the whole space, denote by
B(t) the boundary of the ball with centert and radius 1. If S is not the whole
space, denote by B(t) the boundaryof the ball with centert and radius half the
distance fromt to the boundary of S. If d > 1 and A CB(t), let p(t, A) be the
(d- 1)-dimensional"area" of A divided by that of B(t). If d = 1 let p(t, A) be
one-halfthe number of points in A. Now definea walk on S, that is, random
variables xo, xi, .. . withvalues in S, as follows.Prescribesome initialpointin S
and set x0 identically this point. If xo=ao, * Xn =an, then Xn+1 is to be on
B(an), and in fact the conditional probability that xn+1 is in the subset A of
B(an) is to be p(an, A). With this definition,ifu is real and harmonic,or ifd=2
and u is complex and analytic on S, the sequence of random variables
{u(xn), n 0} is a martingale.If superharmonicand subharmonicfunctionsare
defined as usual in this context, there is a correspondingrelation between
superharmonic(subharmonic)functionsand supermartingales(submartingales).
Sections 6 and 7 indicate connections between martingale theory and
,
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J. L. DOOB
456
[May
potential theory. In fact the probabilitytheoryof Markov processes and abways of lookingat
stract potential theoryare to a'considerable extentdifferent
the same subject, and martingale theory is an essential tool of probabilistic
potentialtheory.
8. Two basic principles.We shall need the concept of a stoppingtime, also
called a Markov time and optional time. If ifoC5iC * is an increasingsequence of cr-algebrasand ifv is a random variable whose range is the set of positive integerstogetherwith + oo,the random variable v is called a stoppingtime
(relative to the sequence of a-algebras) if {v < k } is a set in Yikfork = 0, 1,
that is, in intuitivelanguage, if the conditionv < k is a conditioninvolvingonly
what has happened up to and includingtime k. For example,if {X,, n ?0 1 is a
martingalerelative to the given sequence of a-algebras and if v(co) is the first
integerj forwhich xj(X) > 0, or v(X) = oo if there is no such integer,thenrv is a
stoppingtime.
Two basic principlesare embodied in the followingroughstatements,which
will be given exact versionsand applied in various contexts.Let xo, xi, .*. be a
supermartingalerelativeto go,5.
* * are finite stopping times which are not too large (with
it is a
reference
toxo,xl, .), thenthesequencex.1,x^2, ... is a supermartingale;
martingaleif {xt&,n > 0 is a martingale.
Pl. If
?v2 ?
V
P2. If supnE {
{xo(co), x1(@),
oscillatory.
xX
} is not large,thensupnIxn I is not large,and thesequences
} of possible values of the supermartingaleare not strongly
The principleP1 is suggested by the fact that a gambler playing an unfair
(fair) game will still consider it unfair (fair) if he looks at his money not after
every play, but only after plays number v1, P2, - - . The followingversion of
P1, in which all stopping times are finite,will be called the STOPPINGTIME
THEOREMbelow:
(martingale)and if each v; is bounded,then
If {xn,n 01} is a supermartingale
Withnorestriction
(martingale).
thesequence{IX,,j_ I } is also a superinartingale
if thefirstwas,
on thefinitestoppingtimes,thesecondsequenceis a supermartingale
and ifalso xi Oforallj.
The principleP2 is exemplifiedby the followingtheorem,whichwill be called
CONVERGENCE
THEOREMbelow:
the MARTINGALE
If {X., n >1 } is a supermartingalewith sup.E xn| }<oo, thenlimn,o: xn
exists and is finitealmost surely.If { . . , X_, xQ} (orderedleftto right)is a
then(almostsurely)limn_ = x- existswith- to x_,< co.
supermartingale,
Here almost surely means everywhereon the sample space except possibly
fora set of probabilityzero. (One of the most noticeable distinctionsbetween
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19711
WHAT IS A MARTINGALE?
457
probabilityand measure theoryis that a probabilistfrequentlywrites "almost
surely"wherea measuretheoristwrites"almost everywhere.")
The Martingale Convergence Theorem should be contrasted with the
ERGODIc THEOREM, whichwe state in the followingversion:
Let zi, Z2, * * * be randomvariableswithexpectationsand havingtheproperty
thatfor everyn > 0 thejoint distributionof zi,
zl+j does notdependon j> 1.
Thenalmostsurely
,
lim (z1 +
(8.1)
+ Z")In
existsand is finite.
Denote by z' the ratio in (8.1). Then z' is a weighted average of zi, Z2,
. Thus z'+. is a coarser
, i,
*
and z+1 is a weightedaverage of z',
averagethanz,. Nowthedefining
equalityofa martingale{Xnc n > 1 } makesxna
partial average of x.+i. Thus if there is any relation between the Martingale
ConvergenceTheorem and the Ergodic Theorem,one would conjecturethat the
Ergodic Theorem correspondsto the Martingale Convergence Theorem with
decreasingindex.The application in Section 11 verifiesthisconjecturein a special
case.
9. Continuation of Section 3. In the example in Section 3, E {xj }
<5Et I }. Thus according to the Martingale Convergence Theorem, liM
EF{x Tn exists almost surely.If 5:. is definedas the smallesta-algebra containing everyset of U.T., then the limitcan be identifiedas E { x 9 }. We have obtained a kind of continuitytheorem for conditional expectations. There is a
correspondingtheoremrelative to a decreasingsequence of a-algebras.
10. Continuationof Section 4. Suppose in Section 4 that E { yj }O0 forall j.
Then the sequence {Xn, n 21 } is a martingale,and accordingto the Martingale
Convergence Theorem, the series Ej yj converges almost surely if supn
y } < oo. Suppose for example that E{y2} < 00 for all j, so that the
E
series Ej yj is a seriesof orthogonalrandomvariables, and as such convergesin
<0
.
the mean if and only if Ei E {yy} < oo, that is, if and only if sup,E{x2}
But the finitenessof this supremum means that the hypothesisof the Martingale ConvergenceTheorem is satisfied,and we have proved that convergence
in mean of a sum of independentrandom variables with zero expectationsimplies almost sureconvergence.
To obtain a second application of martingaletheoryto sums of independent
random variables, let zi, Z2, . . . be mutually independent random variables
with a commondistributionhaving expectationa. If xo= 0 and xn(zj-a)
forn> 0, the sequence { Xn, n > 0 } is a sequence of sums of independentrandom
variables with zero expectations and as such is a martingale. If v is a finite
stoppingtime (relative to the minimala-algebra sequence) with an expectation,
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458
[May
J. L. DOOB
and if V1=0, v2=v, it can be shown that a version of P1 yields the fact that
x.1,x2 is a martingalewith two randomvariables, having commonexpectation0.
The factthatE { x"}O0 means that
(10.1)
E{
z} }
aE{v}.
This equality is Wald's FUNDAMENTAL THEOREM
whichhas many applicationsin statistics.
OF SEQUENTIAL
ANALYSIS,
11. Continuationof Section 5. An applicationofthe MartingaleConvergence
Theorem to the martingalein Section 5 yields the almost sure existenceof the
limit
(11.1)
lim (yi+
+yn)/n.
.-
The limitcan be shown to be E {yi } . This convergenceresultis known to probabilists as the STRONG LAW OF LARGE NUMBERS FOR INDEPENDENT RANDOM
VARIABLES WITH A COMMON DISTRIBUTION, and the resultcan also be obtained
as an application of the Ergodic Theorem. (See the discussion of the relation
between the Ergodic Theorem and the Martingale Convergence Theorem in
Section 8.)
12. Continuationof Section 6. We suppose that S is bounded and show first
that almost everywalk path froma point t of S reaches the boundary.There are
elementaryproofsof this fact and the followingproofis given only to illustrate
martingaletheory.If u is definedand harmonicon S, then {u(xn), n_O} is a
bounded martingaleand is thereforealmost surelyconvergent.In particularifu
is a coordinatefunction,it is trivialthat u on a walk sample sequence cannot be
convergentunless u on the sequence is finallyconstant.Then almost everywalk
sample sequence musthit the boundary (whereit sticks), as was to be proved.
If t is not a boundarypoint,thereis an integer-valuedrandomvariable v (the
firsthittingtime of the boundary) such that xvis a boundarypoint,but xj is not
forj <v. The random variable v is a stopping time relative to the sequence of
minimala-algebras. If u is harmonicon S, if vP=0 and V2= V, the stopping time
theorem (slightlyextended) asserts that u(a), u(x,) is a martingalewith two
randomvariables. But then
(12.1)
u(Q) = E{u(0)}
=
E{u(x)}.
Denote by /(q, 77)the probabilitythat x-=7, that is, the probabilitythat a walk
startingat t hits the boundary at X. The distributionu(, *) is called harmonic
measure relativeto t. In termsofharmonicmeasure,(12.1) becomes
(12.2)
u( )
= E
u(n1)L( , X),
where the sum is over all boundary points . Thus a harmonicfunctionon S is
determinedby its values on the boundary,in fact,is the weightedaverage using
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1971]
459
WHAT IS A MARTINGALE?
harmonic measure of its values on the boundary. The rules for manipulating
conditionalexpectationsyield the fact that if u is an arbitraryfunctiondefined
on the boundary of S, and if u is definedat interiorpoints by (12.2), then u is
harmonicon S. We have now shown the existenceand uniquenessof a harmonic
functionwith a specifiedboundary function.We omit the correspondingtreatment of more general harmonicitydefinedusing weightedaverages, not necessarilyat nearestneighbors.When d 1, the resultobtained is particularlyintuitive. In this case ifS is the set of points -a, - * *, b, wherea and b are strictly
positive integers,the walk is the random walk in whicha step is either1 or -1,
with probability 1/2 each, independent of previous steps, until -a or b is
reached. In gamblinglanguage: a gambleris playinga fairgame in whichat each
play he can win or lose a dollar with probability 1/2 each and the plays are
independent;he startswitha dollars, his opponent with b dollars,and the game
ends when he or his opponent has lost all his money; x,,is the gambler's total
winnings(positive or negative) afterthe nth play. Since the play starts at time
0, we definexo=0. If we take the harmonicfunctionu to be the identityfunction,u(s) = on S, the sequence {x., n0>} is seen to be a martingale,so (12.1)
becomes
E{u(x )} c 0,
(12.1')
that is, the expected finalgain is 0 as it should be. Equation (12.1') can also be
obtained as a special case of (10.1). However obtained, this equation yields the
standard result that the probabilitythe gambler wins, that is, the probability
thatxvis b,is a/(a+b).
13. Continuationof Section 7. If d= 2, the propertiesof the random walk of
Section 7 are intimatelyrelated to the propertiesof harmonicand analytic functions. We shall see this firstin a simple application whereS is the whole plane.
It can be shown in this case that almost every sample walk startingfromany
point t is dense in the plane. This fact correspondsto Liouville's theoremthat a
bounded complexfunctionwhichis analytic on the plane is a constantfunction,
and we shall now prove Liouville's theoremprobabilistically.Since the real and
imaginaryparts of an analytic functionare harmonic,we shall obtain a stronger
result if we prove that a functionharmonicand bounded on the plane is constant. In fact we shall do even betterand prove that a harmonicfunctionon the
plane which is bounded fromabove or below is a constant. By linearitywe can
suppose that the function u is positive, and we consider the martingale
{u (x.), n 0}. Trivially,
E{u(xo)}
=
E{u(xn)} =
E
u(x| )
j}
According to the Martingale ConvergenceTheorem, this martingaleis almost
surelyconvergent.If we choose any sample walk whichis dense and on whichu
has a limit,say c, the functionu must be identicallyc by continuity,as was to be
proved.
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460
[May
J. L. DOOB
We proceed to the analog of the work in Section 12, assumingfromnow on
that S is bounded and d arbitrary.The almost sure convergenceof the bounded
martingale {u(x) n_0 }, when u is a coordinate function,impliesthat limn+OO
xn=x0o exists almost surely. This is impossible unless x., has its values on the
boundary of S. If t is the initial point of the walk and ifA is a Borel boundary
set, let g(, A) be the probabilitythat x,Ois in A. Then u(t, *) is a measure of
boundary sets, harmonicmeasurerelativeto t. If u is a bounded harmonicfunction on S, or even if u is merelybounded above or below, the Martingale ConvergenceTheorem impliesthat lim.oO u(x.) existsalmost surely.That is, u has a
limitat the boundaryof S along almost everyone of the sample walks. We shall
continue this aspect of the discussion in Section 16. Suppose now that u is
actually defined and continuous on the closure of S. Then trivially lim,
to show that the orderedset
u(x") u((x.) almost surelyand it is straightforward
ofrandomvariables
i(xo),
u (x.)
U(Xi),
is a martingale.Since the expectationsof the firstand last randomvarlables are
equal,
(13.1)
u()
=E{u(x.O)}
-
fu(?)A(t, dq),
wherethe integrationis over the boundaryof S, that is, u(t) is the average of its
values on the boundary, weighted by harmonic measure. Conversely,if u is a
functiondefinedand continuouson the boundaryofS, and ifu is definedon S by
(13.1), it can be shown,usingthe ideas in Section 16, that u is harmonicon S and
has the assigned boundaryfunctionvalue as a limitat each boundarypoint near
which the boundary is well-behaved (more preciselyat each regularboundary
point in the potential theoreticsense). Thus martingaletheorysolves the first
boundary value problem for harmonic functions.This kind of analysis is applicable not only to classical harmonic functions,but also to the solutions of
equations.
generalellipticand parabolic partial differential
14. Example: application to derivation. The close relation between martingale theoryand derivationtheoryis illustratedby the followingexample. Let
Q be the unit interval [0, 1 and let the probabilityof a subset A be its Lebesgue
measure, denoted by IA I. For each n > 1, let A., ,
A,nkbe a partitionof
the
into
that
(n+l)th partitionis a re[0, 1]
disjoint intervals,and suppose
finementof the nth, that is, we suppose that each Aj is a union of sets in the
is the class of unions of sets in the nth partition,then
(n+l)th partition.If
Iffis an integrablefunctionon [0, 1], definethe randomvariable x, by
9.C5Fnf.
,
in
x?=
tn
f(c.)dw/lAn}|
onAAj, j
1
nt
relativeto
togeta martingale
n_ 1 } forwhich
nX
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1971]
461
WHAT IS A MARTINGALE?
El Ixn} < f
(14.1)
f(w)I d@.
=f(co)
Iff is continuousand if limn.+.maxjlAniJ= 0, it is trivialthat limurn0x,(w)
forall w. Without this restrictiononf and on the partitionsthe Martingale Conexists almost everywhereon [0, 1].
vergence Theorem implies that
More generally, [0, 1] can be replaced by any measure space {Q2,C, P} with
P { Q} = 1. In this new contextthe partitionis to be a partitionof Q2into countably many disjoint measurable sets; the (n+ 1)th partitionis to be a refinement
of the nth; fA fdcois replaced by 1u(A),where 4 is any finitesigned measure on
i; x. is definedas A(A I)/P{Ani} on A1j if the denominatordoes not vanish,
definedarbitrarilyon Anyif P { Anl} =0. With this definitionthe sequence of
randomvariables is a martingaleif
(a) P{Anj} >0foralln,j,
limnoocxn
or if
(b) /(A n) =0, wheneverP{AnJ} =0.
Without either (a) or (b) the sequence of random variables is a supermartingale
if
(c) ,u20.
Under (a) or (b) or (c), E { JXnl} is at most the absolute variation of A, and we
conclude that the martingaleor supermartingaleconvergesalmost surely.Since
a finitesigned measure is the differencebetween two finitemeasures, we have
derived the fact that a finitesigned measure has a derivative relative to any
finitemeasure with respectto a net of partitions.(The extensionto allow P Io
to be otherthan 1 is trivial.)
15. Example: functionson an infinitedimensional cube. The definitionof a
martingaleimplies that each martingalerandom variable is a partial averaging
of the next one. The followingexample exhibits this fact very neatly. Let Q be
the coordinate space of two way infinitesequences * * * , -1, t0 , ,* * , 06?t
< 1. Let P be the product measure on Q for which each coordinate measure is
Lebesgue measure, that is, P is infinitedimensional volume. Let f be a measurable integrablefunctionon Q. Then we define,in the obvious notation, the
random variable x,,by
(15.1)
xn=
f(
X
) 101
dt.
...01
n+1
If n,,is the smallest O-algebraof
Here we have integratedout tn+1,tn+2
set
determined
subsets of Q containingevery
by a condition of the form{j<c
=
E
?
that
be
almost
}
shown
forj n, it can
x,,
surely,as should be intui{f in
tively clear: we are calculating the expected value of f knowing all the coordinates up to and includingthe nth. Then (see Sections 3 and 9) the two way
<n < X } is a martingalewhichconvergesin both directions:
sequence {nx, x-, exist almost surely. These limits can be
and liMn-.-.X.=
liMn-,=ax,nx
shown to be (almost surely)f and E {f} respectively.
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462
J. L. DOOB
[May
16. Continuousparametercase. Let {xt,tGI } be a supermartingale,where
I is an intervalon the line. Then PrincipleP2 is illustratedby the fact that each
randomvariablew-Xe(w)
0 in such
can be changedon an coset ofprobability
a way that almost every sample function(that is for almost every w the function t-%..xt(c)) becomes continuous except for nonoscillatorydiscontinuities.
Such a change does not affectthe joint distributionsof finitesets of the random
variables or the fact that the familyof random variables is a supermartingale.
The theoryof continuousparametersupermartingalesis very richand plays an
essential role in probabilistic potential theory. We give only one application
here, a continuationof Sections 7 and 13. Let {t, 0 5 t oo} be Brownian motion in d-dimensionalspace, with initial point : z0 is identicallyt; foreach t_0
the randomvariable Zthas its values in d-space; everysample functionte--o.zt(w)
is continuous.We omit the exact specificationof the distributions.It turnsout
that, in the notation of Section 7, if To= 0, if Ti is the firsttime t when Zt is a
point of B(s), if T2 is the firsttime t> T1 when zt is a point of B(ZT1) and so on,
and ifx. is definedas zT., then the sequence {X,c,n ? 0 } is preciselythe walk discussed in Sections 7 and 13. Thus a walk sample sequence is a sequence of
points on a Brownian path. If S is the whole space and if u is a real harmonic
function(or, if d =2, a complex analytic function) on S and if Iuj is not too
large (we omit the precise restriction),then {u(zt), t>0 } is a martingale.Suppose fromnow on that S is bounded. (The exactly appropriateconditionis more
generallythat the complementof S has zero capacity.) Let T=sup.Tn be the
firsttime t at which Zt is on the boundary of S. Then it can be shown that T is
almost surelyfinite,and of course ZTcan be identifiedwithx0 (definedin Section
13). Then the distributionof ZT is harmonicmeasurerelative to the initialpoint
of the Brownian motion.A refinementof the analysis in Section 13 shows that
if u is harmonicand positive on S, and if u(zt) is definedas 0 when t> T, the
process {u(zt), t> 0 1 is a supermartingale.(This resultis valid even if u is only
superharmonicand positive,excluding the parameter value 0 if u(t) = oo, and
the discussion here can be generalizedcorrespondingly.)From this result it is
then concluded, using the existence of left limits of supermartingalesample
functions,that u has a limitat the boundaryof S along almost every Brownian
path: limtt u(Zt) exists almost surely.The limittheoremwe have obtained is a
probabilisticgeneralizationof the classical FATOU THEOREM that if S is a disk,
then any positive harmonic functionu on S has a limit along almost every
radius: if the disk radius is a and if polar coordinates are used, lim7..4u(reil)
exists Lebesgue for almost every 0. (The result was extended later to balls of
arbitrarydimensionality.) Note that Fatou's Theorem is much more special
because in his theoremS is a ball. On the other hand Fatou's paths are undeniably more pleasant, or at least more traditional, and are individually
identifiable.The probabilitytheorem,however,when applied to a ball, can be
used to deduce Fatou's theorem,and in fact when boundary limit theoremsare
extended to cover superharmonicfunctionsand more general classes of func-.
tions it becomes clear that the probabilisticversionsare intrinsic,not accidents
of the geometryof the domains.
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19711
MATHEMATICAL FOUNDATIONS FOR MATHEMATICS
463
References
1. J.L. Doob, StochasticProcesses,
Wiley,NewYork,1953.
2. Paul A. Meyer,Probability
and Potentials,Blaisdell,Waltham,Mass., 1966.
MATHEMATICAL FOUNDATIONS FOR MATHEMATICS
LEON
HENKIN,1 Universityof California,Berkeley
1. Introduction.Most mathematical papers deal with mathematics"in the
small"-a few definitions,a few theorems,a few proofs. If the author has a
modicum of boldness and compassion he may also include some account of the
intuitiveideas fromwhich these formalparts of his work were fashioned.This
paper, however,will have a different
character.
In wonderingwhat subject to choose for this Chauvenet Symposium, I let
my mind's eye wander over those areas of the foundationsof mathematicsin
which I have worked or dabbled-completeness proofs,applications of logic to
algebra, decision problems,infinitarylogic, algebraic logic . . . somehow none
of themseemed appropriate.I began to wonderwhy. Presentlyit seemed to me
that the answerwas bound up withwhat mightbe called the "sociologicalstructure" ofour contemporaryAmericanmathematicalcommunity.
1 The point of view toward foundationsdeveloped here was firstformulatedby me in the
IBM Lectures which I gave at Swarthmorein December, 1967. This viewpointhas been developed
over an extended period, duringwhich much of my work was supported by the National Science
Foundation (most recently,Grant No. GP-6232X).
This paper constitutes the final paper prepared for the Chauvenet Symposium held at the
U. S. Naval Academy in October 1969 (this Monthly, May 1970).
ProfessorLeon Henkin received his PhD at Princeton Universityin 1947 under the direction
of Alonzo Church. His thesis included a proofof G6del's completeness theorem for the predicate
calculus which has since become the standard proofin almost every presentationof mathematical
logic. In addition ProfessorHenkin developed the theoryof cylindrificationalgebras which is an
algebraic formulationof the theory of quantifiers. His principal work has been in the area of
foundationsand mathematical logic in which he has published many papers and is a recognized
authority.He was awarded the Chauvenet Prize in 1964 forhis paper "Are Logic and Mathematics
Identical?" published in Science, 1962 (vol 138).
ProfessorHenkin was Fine Instructorand JewettFellow at Princetonfrom1947 to 1949, having spent four previous years as a mathematician in industry. In addition he was a Fullbright
Scholar in Amsterdamin 1954-55,aVisiting Professorat Dartmouth in 1960-61,and a Guggenheim
Fellow and memberof the Institute For Advanced Study in 1961-62, and a visitingFellow at All
Souls' College, Oxfordin 1968-69. He taughtat the Universityof SouthernCaliforniaand has been
a memberof the facultyof the Universityof Californiaat Berkeleysince 1953 wherehe has served
as chairman twice. He has been Editor for the Journal of SymbolicLogic and served a three year
termas Presidentforthe Association of Symbolic Logic. He has also been a memberof the Council
of the American Mathematical Society as well as active in CUPM. Besides his papers in foundations he is the author of "Retracing ElementaryMathematics," Macmillan, 1962, whichindicates
his keen interestin the teaching of mathematics.J. C. Abbott.
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