One-dependent processes

ONE-DEPENDENT PROCESSES

V. de Valk

TR diss

1641

ONE-DEPENDENT PROCESSES

PROEFSCHRIFT

ter verkrijging van de graad van Doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus,

Prof.dr. J.M. Dirken,

in het openbaar te verdedigen

ten overstaan van een commissie

door het College van Decanen daartoe aangewezen

op dinsdag 14 juni 1988

te 10.00 uur

door

VINCENT DE VALK,

geboren te Amsterdam,

Doctorandus in de Wiskunde.

TR diss

1641

Promo tiecommissie:

Prof.dr. M.S. Keane (promotor, Technische Universiteit Delft),

Prof.dr. R. Burton (Oregon State University, Corvallis, U.S.A.),

Dr. F.M. Dekking (Technische Universiteit Delft),

Prof.dr. D. van Duist (Universiteit van Amsterdam),

Mw. Prof.dr. P.E. Greenwood (University of British Columbia,

Vancouver, Canada),

Prof.dr. C L . Scheffer (Technische Universiteit Delft),

Prof.dr. M. Smorodinsky (Tel Aviv University, Israel).

The investigation presented in this thesis was carried out as a part of

the research program of the Netherlands Foundation for Mathematics

(SMC) with financial aid from the Netherlands Organization for the

Advancement of Pure Research (ZWO, now the Netherlands

Organization for Scientific Research NWO).

S 7' E L LI N O E N

bij het proefschrift, "One-dependent processes" van V. de Valk, Technische Universiteit Delft, 14 juni 1988.

I. /ij M een Orlicz—functie die aan de A2-conditie voldoet, zij (X ) ° ° , een rij lianaehruiniten met (UKK)—norm.

00

Dan heeft X:=( E.fflX ), zwak—normale structuur en de Fixed Point Property. _{<n=l n'h}

M >

'1. /ij M een Orlicz—functie die niet aan de A2-conditie voldoet, maar wel aan de A2*-oonditie voldoet.

Dan lieofl, liw Heen zwak—normale structuur.

:i. '/ij M een Orlicz—functie die aan de A2*—conditie voldoet. Dan ' i ( l i w ) < 2.

'1. Zij M een Orlicz—functie die aan de A2*-conditie voldoet. Dan heeft h , . de Fixed Point Property.

•ü. Er beslaan Orlicz—functies die niet aan de A2-conditie voldoen, maar wel aan de A2*-conditie voldoen. De stellingen 2 en A impliceren nu het bestaan van ruimten die wel de Fixed Point Property maar geen zwak—normale structuur hebben. Hiermee is op oen natuurlijker manier dan in [2] een klasse van zulke ruimten geconstrueerd.

Definities bij Stelling 1 t/m 5:

Met een Orlicz—functie M correspondeert een Banachruimte hM; 00

h

M

f (

.X=1

n

' nh

^K\IP) < °°

^P > ° )

"•et norm ||(xn)™=1|| := inf { p > 0 : n| , M( \xn\/p) < 1 }.

Iti hot algemeen is de h,,-$om van een rij Banachruimten (X )°° . de ruimte

00 00

(n£l 8 Xn ' hM := < (xn)n=l : V Xn> „ I l MUK^ < " v001' a l l e ">° >■

Een Orlicz—functie M voldoet aan de A2-conditie als lim sup M(2t)/M(t) < oo.

t -» 0 Een Banachruimte X heeft (UKK)-normals

V t > 0 3 6 > 0 V ( xn) «= 1 , ||xn|| < 1 V n € IN, xn ïï x (n-oo) :

inf{i]xn-xm|j : n^m } > e => ||x|| < l-S.

Een Banachruimte X heeft zwak-normale structuur als elke zwak-compacte, convexe, uit meer dan een punt bestaande C c X een punt x € C bevat met

sup{||x-y|| :y € C ) < diam(C). Een Orlicz-functie M voldoet aan de A2*-condüie als

lim inf t.p(t)/M(t) > 1, t -»0

Voor de bewijzen van Stelling 1 t/m 5 verwijzen we naar [Ij en [3].

[1] D. van Duist & V. de Valk. "(KK)-properties, normal structure and fixed points of uonexpansive mappings in Orlicz sequence spaces", Canadian Journal of Mathematics Vol.38 (1986) pp.728-750.

[2] L.A. Karlovitz. "Existence of fixed points for uonexpansive mappings in spaces without normal structure", Pacific Journal of Mathematics Vol.66 (1976) pp.153-156. [3] V. de Valk. "Dekpunten van niet-expansieve afbeeldingen in Banachruimten en het varband met normale structuur en Kadec-Klee normen" (deel 1 en 2), doctoraal—scriptie, Mathematisch Instituut der Universiteit van Amsterdam, 1984.

6. Het vermoeden van S. Alpern ([4]);

{ Zij L,M e IN, M even, t > 0 en zij (X ) j een mengend, stationair proces, zo dal P[Xn=i]=l/M voor i=l,...,M (neZ).

Dan bestaan er gehele getallen p,q > L en een mengend, stationair proces (Y ) ■* , zo dat P[Y =i]=l/M voor i=l,...,M (neZ) en

| P[Yfl=i en Y =j] — P(X0=i en X =j] | < c voor alle i,j e {1,...,M} en alle n 6 IN en

| P[Y0 < M/2 en Y < M/2 en Y < M/2] - 1/8 | > 0.01. }

is juist in het geval dat (X ) ■* een i.i.d. rij is.

[4] S. Alpern. "Conjecture: in general a mixing transformation is not txoo-fold mixing", Annals of Probability Vol.13 (1985) pp.310-313.

7. Zij f:IR i> [0,oo) een integreerbare functie zo dat ƒ f(x)dx=l, f niet-daleud is op (-oo,0] en f niet-stijgend is op [0,a>). Zij de functie g gedefinieerd door

g(x):= ƒ f(t).f(t—x)dt. Dan is g niet-dalend op (-oo,0] en niet-stijgend op [0,oo). 8. Na de overhaaste invoering van de Wet op de Studiefinanciering heeft ook de overijlde invoering (*) van de N.W.O.—wet (**) bij de uitvoering praktische problemen

veroorzaakt, waarvoor de Minister van Onderwijs & Wetenschappen door Z.W.O. nadrukkelijk gewaarschuwd is.

Deze problemen bij de invoering van de N.W.O.—wet zijn:

slechts anderhalve maand tevoren (medio december 1987) heeft de Minister aangekondigd dal, de N.W.O.— wet per 1 februari 19S8 zal ingaan, terwijl Z.W.O. gerekend had op een invoering per I januari 1989, waardoor voor Z.W.O. het onmogelijk gemaakt is tijdig een collectieve ziektekostenverzekering te regelen voor de 1792 medewerkers die door de invoering van deze wet het ziekenfonds moesten verlaten, en waardoor Z.W.O. niet meer tijdig overgangsmaatregelen (om zo correct mogelijk met het personeel om te gaan) kon nemen.

(*) Invoering bij Staatsblad 1987, 641.

(**) Wet op de Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Staatsblad 1987, 309

9. Doordat T.U.-medewerkers bij noodweer wachten met naar huis gaan tot het weer weer opgehelderd is, bestaat er een positieve correlatie tussen noodweer en overwerken.

10. De komst van prins Johan Friso als student bij de Faculteit der Lucht—en Ruimtevaarttechniek van de T.U. Delft zal meer effect hebben op de toeloop van vrouwelijke studenten dan alle pogingen bij elkaar die tot nu toe zijn gedaan. 11. Ter verbetering van de ambtelijke stijl vermijde men de aanvoegende wijs. 12. Feminisering van de taal geschiedt niet op instigatie van de Ministers Brinkniens en Deetmens.

13. Het essentiële verschil tussen een promovendus en een pion is dat de laatstgenoemde kan promoveren zonder proefschrift.

CONTENTS

INTRODUCTION

1. 1. m-dependent processes

I. 2. One—dependent Markov processes

3. 3. Renormalization theory

4. 4. m—block—factors

4. 5. A conjecture

5. 6. Two-correlations and the conjecture

5. 7. More two—correlations and applications

6. 8. Other publications on m—dependence

6. 9. Comment on the four articles

9. 10. Open problems and questions

10. 11. References

II. 12. "One—dependent processes" Summary

12. 13. "Een—afhankelijke processen" Samenvatting

13. 14. Curriculum vitae

14. 15. Acknowledgements

THE FOUR ARTICLES

1.1—1.25. The maximal and minimal 2-correlation of a class of 1—dependent

0—1 valued processes

(Report 86—47, Delft University of Technology, to appear in

Israel Journal of Mathematics)

11. 1 —II. 19. An algebraic construction of a class of one-dependent processes

(with J. Aaronson, D. Gilat and M.S. Keane)

(Report 87—65, Delft University of' Technology, submitted to the

Annals of Probability)

III.1—III.8. Extremal two—correlations of two—valued stationary one—dependent

processes

(with A. Gandolfi and M.S. Keane)

(Report 88—01, Delft University of' Technology, submitted to

Probability Theory and Related Fields)

IV.O-IV.46. A problem on 0-1 matrices.

(Report 88-03, Delft University of Technology, submitted to

Compositio Mathematica)

INTRODUCTION.

This dissertation consists of four articles, all on one—dependent processes, with connections

to combinatorics, analysis and variational problems. Before describing the articles in this

introduction, a short survey is given of the theory of m—dependence (a generalization of

one-dependence) with its applications to renormalization theory (section 3), the occurrence of'

m-dependent processes as m+1—block—factors of i.i.d.—sequences (section 4), the problem

whether or not all m—dependent processes can be obtainded as a m-t-1—block—factor (section

5), two-correlations that play an important role in this problem (section 6), and applica­

tions of the results concerning two-correlations (section 7).

In section 9 the four articles are summarized, and in section 10 some open questions concer­

ning m—dependent processes are presented.

1. 'm- dependent'/MaceAdeó

Discrete time stochastic processes (X ) -r have been studied thoroughly by probabilists.

An important class of these processes are the tndefiendent'/taaceódeó. The class of independent

processes can be considered as a part of a wider class, such as the Jt&i&cw./iaaceóóeó.

Another way of generalizing the notion of independence is by defining m-ds/iendmce

An independent process has the property that two events are independent whenever they are

separated by a time—interval with positive length, and a inr-de/wnden/fiaaced<i\\%& the

property that two events are independent whenever they are separated by a time-interval

with length more than m. To be more precise: at each (discrete) time t the future

(

n)n>t+m

P

P

(

)

< f

Although "almost everything" is known about Markov processes, not so much is known

about m-dependent processes. We give a survey in the next sections.

2. One-defterudmt'MazAau Maceóóeó

There exist processes that are both one—dependent and Markov, but that are not indepen­

dent, as the following example (see also [O'Br.]) shows.

iSccam/i/e

2

(o<p<l). Let (X ) be defined by X :=2U +U , ,. Then (X ) is a one-dependent

process (because it is a two—block-factor of an i.i.d. sequence, see section 4) and (X ) is

also Markov with state space {0,1,2,3}, and transition matrix

1-p p 0 0

0 0 1-p p

1-p p 0 0

0 0 1-p p

and finally, (X ) is clearly not an independent process.

Nevertheless, when we restrict our attention to a two—valued stationary process that is both

one—dependent and Markov, then the process is independent, as the following proposition

shows.

Let (X ) j be a 0—1 valued, stationary, one-dependent, Markov process. Then this

process is an independent process.

We write [a ,...,a ] as a short notation for the probability of the event [X =a ,...,X =a ].

Further we use the crucial observation that [a a 1 =fa a 1 holds for this process as directlv

1 i 2J L 2 i

follows from the equation

[0 0] + [0 1] = [0] = [0 0] + [1 0].

In our formulas we use the convention 0/0=0. We have

h J

= £ [ajj a-, a

] = £ [ a

a ■ a

] .[a

a

] = £ [ a . a

] .[a,, a^ =

i i i

I

k

i 1 T ^ 7

S [ a , a

1

"TaTl

With this expression we obtain

0 < E | [ a

a j ] - [a

]. 4 ^ ] y.

V l a J

E | [ a

a j ]

- 2.[a

].[a

a^ + [a

]

.[a

] j =

[ a - , ]

= [a

]

-2.[a

]

+ [a

]

= 0.

This implies

[ a

- a j ] = [a

]. v ^ ] for all a

, ^

V l a J

and this is equivalent to [a, a-] = [a.], [a.]. Combined with the Markov property, this

implies independence. a

The statement of the proposition also holds for one—dependent Markov processes assuming

more than two values, when we add the condition

fa a 1 = fa a 1 for all a , a .

1 1 2J l 2 1J 1 2

3. 9tencifrma/<&<z/ia<n Meo/m

One—dependent processes occur as limits of rescaling operations in 4ma4ma6&aét&n /Aeaau

(see e.g. [O'Br.]). Because this is an important application of one-dependence, we give some

details.

Let the process X*

' be 0—1 valued, let r be an integer greater than one, and let

4): {0,1}'' - {0,1}

be a function. We define a new process X^ ' by

X!» :- *(x(f....,X<0)

_

.

We can iterate this procedure, obtaining a sequence of processes X^

'.

When we assume X^ ' to be stationary, then it is trivial that X" ' is also stationary.

Because x j

) depends on X^°),...,x(°) and X ^ depends on X^°' , . . . , X ^ , it is easy

i

n

r

- l

-r

to see that if (X^

')°°

has a subsequence that converges (in distribution) to some limit,

then this limit is one—dependent, assuming that X^ ' satisfies the following mixing condi­

tion. O'Brien assumes that there exists a decreasing sequence ( g ( k ) ) f „ converging to zero,

such that

| P(AflB)-P(A).P(B) | <g(k)

for all events A depending on {...,—3,—2,-1} and all events B depending on {k,k+l,k+2,...}.

(P is the underlying probability-measure corresponding to the process X' '.)

4

4. m-&ac£-jtacéa/ió.

In addition to being limits of a rescaling operation, m-dependent processes can be obtained

in a simpler way: as a m+1—block—factor of an i.i.d. sequence (U ) y.

Let the process X be defined by

X

n

= «

n '

n

l ' - '

n + m >

for some function f. Obviously X is a m—dependent process.

For a m—block—factor X it is no restriction to assume that the underlying i.i.d. sequence

(U ) is identically uniformly distributed over the unit—interval.

5. iA cantecéu/ie

It is obvious to ask whether all one—dependent processes are m+1—block—factors. If the

m—dependent process is a Gaussian process, then this is true, because there is a one—one

correspondence between Gaussian stationary processes (X ) -n and autocovariance—func­

tions R . Given such a process, there exists a positive—definite function R :=E(X -X

),

and given a positive—definite function R there exists a unique Gaussian process with this

autocovariance—function. Now the notion of m—dependence means that R =0 for n>ni.

These functions correspond to the set of m+1-block-factors defined by

X := R .U + R.U , , + . . . + R. .U ,

_{n o n ! n+1 in n+m}

with (U ) an i.i.d. sequence of Gaussian variables.

Although this was conjectured for quite a long time, a one-dependent process (X ) is not

necessarily a two—block—factor if (X ) is not a Gaussian process. This has been stated yet

by Ibragimov and Linnik ([Ibr.Li.]) in 1971, but unfortunately, they did not give a counter­

example.

Other authors used this conjecture as a hypothesis. Janson ([Ja.]) studied runs of ones in

m—dependent processes. He proved his results only for m+1—block—factors and remarked

that this is sufficient under this hypothesis. Later Van den Berg ([Be.]) and O'Ciiineide

([O'Ci.]) also studied runs of ones, and they proved some of their results only for

m+1—block—factors. The results in the articles [Ja.], [Be.] and [O'Ci.] are essentially

different from those of this thesis.

In 1987 Aaronson and Gilat ([A.G.J) found a one-parameter-family of counterexamples.

Later, in collaboration with Keane and De Valk ([A.G.K.V.], the second article of this

thesis), they found a two—parameter—family. These counterexamples are all 0—1 valued

one—dependent processes where a run of three ones has probability zero. We summarize the

facts in a diagram (the definition of f(n)—dependence will be given in section 8).

Markov processes |independent processes |

two-block-factors

one-dependent processes

m-dependent processes

f(n)-dependent processes

6. J^Wt-■cabZe/aÜomd <xnd'me can^ectu/le

Although the conjecture does not hold generally, it is true under certain extremal conditions

on 0—1 valued one—dependent processes.

Fix an a in the unit—interval and consider the subclass of 0—1 valued one—dependent

processes with probability of a one equal to a. In [G.K.V.] (the third article of this thesis) is

proved that in this subclass the probability of a run of two ones (a ht<3r-<x>M£/aüo^ has

,3/2 3/2

maximal value a '" (if a>l/2) and 2a— 1 + (I—a) ' (if a<l/2). This supremum is

attained uniquely if a is not equal to 1/2, and for a = l / 2 there exist exactly two processes

with maximal two-correlation. The processes with maximal two-correlation are all two—

block-factors.

Further, a 0—1 valued one-dependent process with minimal two—correlation (for fixed a) is

necessarily a two-block-factor if o£(l/4,3/4) ([G.K.V.]).

7. Ma/u ^ma—caMe/auoMA ana aAAÜcauanó

The maximal two-correlation of two—block—factors (translated to our terminology) was

computed by Katz ([Ka.]) and later by Finke ([Fi.]), who interpreted Katz' mathematical

objects as two-correlations in stochastic processes. The minimal two—correlation of two—

block—factors is computed in [V.I.] (the first article in this thesis). These two—correlations

6

have applications to matrix—theory and graph—theory, when we restrict our attention to 0—1

valued, one-dependent processes that are two-block—factors of an independent sequence of

random variables, uniformly distributed over a finite number of values.

The problem of the maximal or minimal value of a two-correlation in this discretized selling

is equivalent to the problem of finding the maximal or minimal number of paths of length

two in a directed graph (as was remarked in [Fi.]) with a fixed number of edges and vertices.

This problem is also equivalent to finding the maximal or minimal value of | | M

1 | over

the class of 0—1 valued NxN matrices M with K ones (for fixed N and K). This problem is

solved in [V.2.] (the fourth article in this thesis).

Although the following articles concern different problems than those dealt within this

thesis, they are mentioned to give a survey of the field of m—dependence.

Hoeffding and Robbins ([Ho.Ro.]) have studied f(n)-dependent processes, i.e. processes

(X

)™

such that

{X ,...,X, } is independent of {X, ,...,X }

-whenever k — k > f(n) , for some function 1'.

When f is constant, then we have m-dependence. They proved central limit theorems for

these processes.

Later Diananda ([Di.J) proved central limit theorems for m—dependent processes.

Haiman ([Ha.]) wrote a paper on extreme value theory for m—dependent processes.

9. 'ê'ammerU'-cm m€ Jau/t ti/lÜckd

I. [V.I.] "The maximal and minimal 2-correlation of a class of J-dependent 0-1 valued

processes."

In this article 0-1 valued two-block-factors (X ) of an independent sequence (U ) of

random variables that are uniformly distributed over the unit—interval, are considered.

Because such two—block—factors are completely determined by the indicator—function of a

subset A of the unit—square, defining

these processes are also called wcücaéat-feiaceAAei The probability of a one is equal to the

Lebesgue—measure of A, and the probability of a run of two ones (a two-correlation) is eciual

to

I

:= / H

(x).V

(x) dx

where H. and V , are the horizontal and vertical sections of A.

The computation of the least possible two—correlation (for fixed probability of a one) over

the class of 0-1 valued two-block-factors turns out to be a variational problem, equivalent

to computing the minimal value of I» for fixed Lebesgue—measure of A.

This problem gives rise to some questions (see also section 10), some of which are solved in

[G.K.V.]. The papers [G.K.V.] and [V.2.] can be considered as continuations of this article.

II. [A.G.K.V.] "An algebraic construction of a class of one-dependent processes." (with J.

Aaronson, D. Gilat and M.S. Keane)

In this article a rather old conjecture is disproved. The authors construct in an algebraic

way a continuum number of 0—1 valued stationary one—dependent processes that are not

two—block—factors of an i.i.d. sequence, and in this way they disprove the conjecture that

each 0—1 valued one—dependent process is a two—block—factor.

All these counterexamples have the property that a run of three ones has probability zero.

The class of counterexamples is parametrized by a (the probability of a one) and /? (the

probability of a run of two ones; a two—correlation). These parameters (together with the

fact that a run of three ones has probability zero and the property of one—dependence)

uniquely determine the measure of all cylinder—sets. To determine for which values of' the

parameters a process exists, it is enough to check whether the measures of all cylinder-sets

are non-negative. This turns out to be equivalent to the problem whether the orbit of (1,1)

under successive applications of certain mappings y>„ and ipi : IR

-» R

in any order always

remains in the unit-square. It is known for which values of a and fi a two-block-1'actor

exists (by methods as in [V.l.]), and it turns out that there exists a two—parameter-family

of counterexamples to the conjecture.

8

III. [G.K.V.] "Extremal two -correlations of two-valued stationary one-dependent

processes." (with A. Gandolfi and M.S. Keane)

This article can be considered as a continuation of [V.1.].

The authors compute the maximal value of a two-correlation (probability of a run of two

ones) over the class of 0—1 valued, stationary, one—dependent processes. This is a simplifica­

tion and generalization of [Ka.], where the maximal two-correlation over the class of

two—block—factors was computed. The authors prove that this supremum is uniquely

attained when the fixed probability of a one is not equal to 1/2, and that there exist exactly

two processes with maximal two—correlation when the fixed probability of a, one is equal to

1/2. The processes with maximal two-correlation are all two—block—factors.

Further, the minimal two—correlation over the class of 0—1 valued, stationary, one—depen­

dent processes is computed in the case that the fixed probability of a one is < 1/3 or > 2/3.

The computed lower bound is the same as the minimal two-correlation over the class of

two-block-factors ([V.1.]). In the case that the fixed probability of a one is < 1/4 or > 3/4,

it is proved that the infimum over the class of one-dependent processes is uniquely attained,

and the corresponding processes are all two-block-factors.

The upper— and lower—bounds for the two-correlation are computed by showing that the

measure of some cylinder—sets becomes negative when we assume that the two—correlation

has a value greater than the upperbound c.q. smaller than the lowerbound. So the computa­

tion is probabilistic, in contrast to the analytic and combinatoric computation in [V.1.].

IV. [V.2.] "A problem on 0-1 matrices."

In terms of matrices, the maximal and minimal value of | | M

| | is computed over the class

of 0—1 valued NxN matrices M with K entries equal to one (for fixed N and K.). In terms of

one—dependent processes, the maximal and minimal value of the two—correlation over the

class of 0-1 valued two—block-factors of the N—shift (for fixed N and fixed probability of a

one) is computed. This article can be considered as a discretized version of [V.1.]. In terms

of graphs, this corresponds to the maximal and minimal number of different paths of length

two in a directed graph with N vertices and K edges (for fixed N and K). The solution is

found by means of analysis and combinatorics.

10. 0/wn ftaa&etnA and'-pu&i&amd

In this section we will give a contribution to the perpetuation of mathematics by a list of

open problems, to which this thesis gives rise.

On [V.l]. and [G.K.V.].

(1) Is the value of the minimal two-correlation (for fixed probability of a one) over the

class of 0—1 valued two—block—factors (as in [V.l.]) equal to the value of the minimal

two-correlation over the class of 0—1 valued one—dependent processes ? In [G.K.V.] this

problem is solved in the case that the fixed probability of a one is < 1/3 or > 2/3. It seems

that this problem becomes more and more complicated when the fixed probability of a one

tends to 1/2.

(2) If the answer to question (1) is yes, are the one-dependent processes with minimal

two-correlation all two-block-factors ? In [G.K.V.] this problem is solved in the case that

the fixed probability of a one is < 1/4 or > 3/4. Just as question (1), it seems that this

problem becomes more and more complicated when the fixed probability of a one approaches

1/2. In particularly we do not know whether the minimal two—correlation is equal to 1/6

when the fixed probability of a one is equal to 1/2 (question (1)), and if the answer to this

question is yes, we do not know whether this minimum is uniquely attained in the following

process (X ) j (question (2)). Let (U ) y be an i.i.d. sequence of random variables,

uniformly distributed over the unit—interval. Let X :=0 if U < U , , and X :=1 if

n n n+1 n

U > U ,. This problem seems to be interesting in the theory of order-statistics.

(3) Can the computation of the minimal two—correlation in [V.l] be simplified, just as the

computation of the maximal two-correlation in [Ka.] is simplified (and generalized) in

[G.K.V.] ?

(4) The computation in [V.l.] is not probabilistic but analytic and combinatoric. Can the

computation in [V.L] be "probabilized", just as [Ka.] is probabilized by [G.K.V.] ?

(5) What extremal conditions on n-correlations (the probability of a run of n ones) are

needed to assure that m-dependent processes are always m+1-block-factors ?

On [A.G.K.V.].

(6) The counterexamples of one-dependent processes that are not two—block—factors, are

constructed in an algebraic way. Can they be constructed in a probabilistic way, such that

the structure becomes more natural and clear (can the counterexamples be probabilized) ?

(7) Are these counterexamples m—block—factors of i.i.d. sequences for some m greater than

two ?

10

(8) Do there exist counterexamples, not having the property that a run of three ones has

probability zero, or even having the property that each cylinder—set has positive measure ?

(9) Can these counterexamples be described as limits of a rescaling-operation (see [O'Br.])

of a mixing process ?

(10) For which values of the parameters a and /? do there exist processes in the "unex­

plored area" ? It seems that this problem becomes more and more complicated when (a,/?)

approaches (1/3,1/27).

(11) Do there exist m—dependent processes (for some m larger than one) that are not

m+1—block—factors, and that are not m-1-dependent ?

(12) Are the counterexamples in the article functions of Markov processes, or even func­

tions of m-dependent Markov processes ?

On [V.2].

(13) Can the computation be more straightforward ? There exist values of N and K such

that int{N

.Max(K/N

)} > Max(N,K) and other values such that 1 + int{N

.Min(K/N

)}

< Min(N,K) (int(x) is the integer-part of x), and therefore it is not possible to prove the

maximality or minimality of some matrix M by stating that I». (an integer) is in this case

the best integer—approximation (the entier) to N

.Max(K/N

) (in the maximum case), c.q.

the best integer—approximation (one + the entier) to N

.Min(K/N

) (in the minimum

case). Note that always: N

.Max(K/N

) > Max(N,K) and N

.Min(K/N

) < Min(N,K).

11. SL^eïencai

[A.G.] J. Aaronson and D. Gilat. "On the structure o f stationary one dependent processes",

School of Mathematical Sciences, Tel Aviv University, Israel, 1987.

[A.G.K.V.] J. Aaronson, D. Gilat, M.S. Keane and V. de Valk. "An algebraic construction

of a class of one-dependent processes", Faculty of Technical Mathematics and Informatics,

Delft University of Technology, Report 87-65, submitted to the Annals of Probability, also

in this thesis, 1987.

[Be.] J. van den Berg. "On some results by S. Janson concerning runs in in-dependent

sequences", preprint, 1986.

[Di.] P.H. Diananda. "Some probability limit theorems with statistical applications", Proc.

Cambridge Phil. Soc. Vol. 49 (1953) pp.239-246.

[Fi.] L. Finke. "Two maximization problems", a paper submitted to Oregon State University

in partial fulfillment of the requirements for the degree of Master of Arts. 1982.

two-valued stationary one-dependent processes", Faculty of Technical Mathematics and

Informatics, Delft University of Technology, Report 88-01, submitted to Probability Theory

and Related Fields, also in this thesis, 1988.

[Ha.] M.G. Haiman. " Valeurs extremales de suites stationnaires de variables aleatoires

m-dependantes", Ann. Inst. H. Poincare Sec. B Vol. 17 (1981) pp. 309-330.

[H.L.P.] G.H. Hardy, J.E. Littlewood and G. Polya. "Inequalities", Cambridge University

Press, 1934.

[Ho.Ro.] W. Hoeffding and H. Robbins. " The central limit theorem for dependent random

variables", Duke Math. Journal Vol. 15 (1948) pp. 773-780.

[Ibr.Li.] I.A. Ibragimov and Y.V. Linnik. "Independent and stationary sequences of random

variables", Wolters Noordhoff, Groningen, 1971.

[Ja.] S. Janson. "Runs in m-dependent sequences", Ann. Prob. Vol. 12 (1984) pp. 805—818.

[Ka.] M. Katz. "Rearrangements of (0-1) matrices", Israel Journal of Math. Vol. 9 (1971)

pp. 53-72.

[Kh.] A. Khintchine. "Ubereine Ungleichung", Mat. Sb. Vol. 39 (1932) pp. 35-39.

[Lo.] G.G. Lorentz. "A problem of plane m,easure", Amer. Journal Math. Vol. 71 (1949) pp.

417-426.

[Lux.] W.A.J. Luxemburg. "On an Inequality of A. Khintchine for Zero-One Matrices",

Journal of Combinatorial Theory Vol. 12 (1972) pp. 289-296.

[O'Br.j G.L. O'Brien. "Scaling transformations for {0,1}-valued sequences", Zeit.

Wa.hr

.

Vol. 53 (1980) pp. 35-19.

[O'Ci.] C.A. O'Cinneide. " Some properties of one-dependent sequences", preprint, 1987.

[V.I.] V. de Valk. " The maximal and minimal 2-correlation of a class of 1-dependent 0-1

valued processes", Faculty of Technical Mathematics and Informatics, Delft University of

Technology, Report 86-47, accepted by the Israel Journal of Math., also in this thesis, 1986.

[V.2.] V. de Valk. "A problem on 0-1 matrices", Faculty of Technical Mathematics and

Informatics, Delft University of Technology, Report 88-03, submitted to Compositio

Mathematica, also in this thesis, 1988.

12. " One-défiencfent'fiüaceóóeX tfummafru

This thesis consists of four articles on one—dependent processes. Therefore, the subject is in

the first place probability theory, although the methods and applications not only appear in

probability theory, but also in analysis, variational—problems, matrix theory and combinato­

rics.

12

One—dependent processes are stationary, discrete-time processes (X ) j with the property

that at each time t the future (X ) is independent of the past (X ) . Such processes

ii 11-^* L il n ^. ij

can be constructed as a two-block-factor of an i.i.d. sequence (U ) j by defining

X

=f(U ,U , ,) for some function f. Although it was conjectured for quite a long time

that each one—dependent process is a two—block—factor, in the second article of this thesis a

continuum number of counterexamples are constructed of 0—1 valued one-dependent

processes that are not two-block-factors.

In the third article of this thesis is proved that under certain extremal conditions on the

two—correlations (the probability of a run of two ones) a 0—1 valued one-dependent process

is a two—block—factor. The maximal value of a two-correlation over the class of 0—1 valued

one—dependent processes (for fixed probability of a one) is computed and it turns out that

the processes where this maximum is attained, are all two—block—factors. If the fixed

probability of a one is not equal to 1/2, this maximum is uniquely attained and there exist

exactly two processes with maximal two-correlation in the case that the fixed probability of

a one is equal to 1/2. Further, partial results are proved on minimal two-correlations. The

third article of this thesis is also a simplification and a generalization of [Ka.], where the

maximal two-correlation over the class of 0—1 valued two—block—factors is computed.

In the first article of this thesis the minimal two—correlation over the class of 0—1 valued

two-block-factors is computed (for fixed probability of a one).

In the fourth article of this thesis the maximal and minimal value of 11 M"| | is computed

over the class of 0-1 valued NxN matrices M with K ones (for fixed N and K). In terms of

two-correlations, this corresponds to the maximal and minimal value of the two—correlation

over the class of 0-1 valued two-block-factors of an i.i.d. sequence of random variables that

are all uniformly distributed over N values, (for fixed N and fixed probability of a one).

13. "'S'en-oM&nAelc-fle/vtaceóóeiï dftvmerwaééi/na

Dit proefschrift bestaat uit vier artikelen over een-afhankelijke processen. Het gaat derhal­

ve in de eerste plaats over kansrekening, ofschoon de gebruikte methoden en toepassingen

niet alleen uit de waarschijnlijkheidsrekening komen, maar ook uit de analyse, de variatie­

rekening, de matrix—theorie en de combinatoriek. Een—afhankelijke processen zijn, statio­

naire, discrete tijd processen (X ) ■» met de eigenschap dat op elk tijdstip t de toekomst

(X ) onafhankelijk is van het verleden (X ) Zulke processen kunnen geconstrueerd

worden als twee—blok-factor van een onafhankelijke rij gelijkverdeelde stochasten (U ) -* ,

door X :=f(U ,U .) te definiëren voor een functie f. Hoewel men vrij lang heeft vermoed

dat elk een-afhankelijk proces een twee—blok-factor is, wordt in het tweede artikel van dit

proefschrift een continu aantal tegenvoorbeelden geconstrueerd van 0—1 waardige een—afhan­

kelijke processen die geen twee—blok—factor zijn.

In het derde artikel van dit proefschrift wordt bewezen dat onder zekere extremale condities

op de twee—correlaties (de kans op twee opeenvolgende enen) een 0-1 waardig een-afhanke­

lijk proces een twee-blok—factor is. De maximale waarde van een twee-correlatie over de

klasse van 0-1 waardige een-afhankelijke processen (bij vaste kans op een 1) wordt berekend

en de processen waar dit maximum wordt aangenomen blijken allen twee—blok—factoren te

zijn. Als de kans op een 1 ongelijk is aan 1/2, wordt dit maximum uniek aangenomen, en er

bestaan precies twee processen met maximale twee-correlatie in het geval dat de kans op

een 1 gelijk is aan 1/2. Verder worden gedeeltelijke resultaten over minimale twee—correla­

ties bewezen. Het derde artikel van dit proefschrift is tevens een vereenvoudiging en een

generalisatie van [Ka.], waarin de maximale twee—correlatie over de klasse van 0—1 waardige

twee-blok—factoren wordt berekend.

In het eerste artikel van dit proefschrift wordt de minimale twee—correlatie over de klasse

van 0—1 waardige twee—blok—factoren berekend (bij vaste kans op een 1).

In het vierde artikel van dit proefschrift wordt de maximale en minimale waarde van

| |M | | berekend over de klasse van 0-1 waardige NxN matrices M met K enen (bij vaste N

en K). In termen van twee-correlaties correspondeert dit met de maximale en minimale

waarde van de twee-correlatie over de klasse van 0—1 waardige twee—blok—factoren van een

rij onafhankelijke stochasten die elk uniform verdeeld zijn over N waarden (bij vaste N en

vaste kans op een 1).

14. 'êafoiicuAim ttiéae

Op verzoek van de universiteit volgt hier een kort curriculum vitae.

De auteur is geboren op 11 december 1959 te Amsterdam. Na het examen Gymnasium—B

aan het Sint—Nicolaaslyceum te Amsterdam in 1978 ging hij wiskunde studeren aan de

Universiteit van Amsterdam. In september 1980 werd hij (0.2) student—assistent en na het

(cum laude) afleggen van het kandidaats—examen in december 1980, werd hij (0,4) kandi­

daat-assistent bij Prof.dr. D. van Duist, bij wie hij in augustus 1984 (cum laude) het

doctoraal-examen met specialisatie functionaal—analyse aflegde. Zijn doctoraal—scriptie was

getiteld "Dekpunten van niet—expansieve afbeeldingen in Banachruimten en het verband

14

met normale structuur en Kadee—Klee—normen". In september 1984 trad hij bij de Neder­

landse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O., thans de Nederlandse

Organisatie voor Wetenschappelijk Onderzoek, N.W.O.) in dienst als onderzoekmedewerkei'

op het project "Coderingsproblemen in de Ergodentheorie". Dit onderzoek, dat o.a. is

uitgemond in dit proefschrift, voerde hij uit bij de vakgroep Statistiek, Stochastiek en

Operationele Analyse (S.S.O.R.) van de Faculteit der Technische Wiskunde en Informatica

aan de Technische Universiteit Delft. Tijdens zijn onderzoek heeft hij samengewerkt met

Prof.dr. M.S. Keane (projectleider), drs. A. Gandolfi (C.N.R., Italië), dr. J. Aaronson en dr.

D. Gilat (beide van de Tel Aviv University, Israel).

15. ^c&no4a/ed!i3e77ienli

Since September 1984 my research was supported by the Netherlands Foundation for

Mathematics (S.M.C.) with financial aid from the Netherlands Organization for the Advan­

cement of Pure Research (Z.W.O., now the Netherlands Organization for Scientific Re­

search, N.W.O.). I thank these organizations for their support. Further I thank the mem­

bers of the Vakgroep Statistics, Stochastics and Operations Research (S.S.O.R.) of the

Faculty of Technical Mathematics and Informatics of the Delft University of Technology for

their collegiality and the university for its hospitality.

THE MAXIMAL AND MINIMAL

2-CORRELATION OF A CLASS OF

1-DEPENDENT 0-1 VALUED PROCESSES

BY

V. DE VALK*

Department of Mathematics & Informatics,

Delft University of Technology. P.O. Box 356, 2600 AJ Delft, The Netherlands

ABSTRACT

We compute the maximal and minimal value of P[XN — Xs+X — 1] for fixed P[XS = 1], where (Xs)Nez is a 0-1 valued 1-dependent process obtained by a

coding of an i.i.d.-sequence of uniformly [0.1] distributed random variables with a subset of the unit square.

1. Introduction

A stationary, 0-1 valued, stochastic process (X

)

is [-dependent if

P[X^y = /_.v, . ■ • , X_ i = /_[, X\ = l\, . . . , A.v = ',vj

= p[x.

= /_,v, ...,*_, = /_,] -P[x

= / „ . . . ;x

= i

]

for all N i 1 and for all z'_

,..., i-

i

,..., /,

E {0, 1}.

For quite a long time it seemed to be folklore to conjecture that each

1-dependent process is an indicator process (we will define that), but recently

Aaronson and Gilat ([AG]) found a counterexample of a 1-dependent process

that is not an indicator process. A paper by Aaronson, Gilat. Keane and De

Valk on a two-parameter family of such counterexamples is in

preparation-Let Jbe the unit interval, J

the unit square, let A and n be Lebesgue measure

on J and J

resp. and let A be the collection of \x-measurable sets in J

.

f This research was supported by the Netherlands Foundation for Mathematis (S.M.C.) with

financial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO). Received November 5. 1986 and in revised form August 16, 1987

2 V. DE VALK Isr. J. Math.

Let (U

)

be an i.i.d. sequence of random variables uniformly distributed

over J. Define for each A E A the corresponding indicator process (X

)

:

[0, if(U

,U

)£A,

[l, if (U

,U

)EA.

It is easy to see that each indicator process is a 1-dependent process and that

P[X

= \]=n(A).

From now on we reserve a for the Lebesgue measure of A (thus a = n(A) is the

probability of a one).

In 1971 Katz [Ka] computed (translated to our terminology) the maximal

value of a 2-correlation P[X

= X

+, = 1 ] over the class of indicator processes

for fixed a.

Finke [F] (1982) was the first to interpret Katz's mathematical objects as

correlations in stochastic processes.

Recently Gandolfi, Keane and De Valk [GKV] (in preparation) proved a

more general result about the maximal value of a 2-correlation over the class of

1-dependent processes. They computed that the 2-correlation (for fixed prob­

ability of a one) has the same upper bound over the class of 1-dependent

processes as over the class of indicator processes.

Further, they proved that there exists a unique 1-dependent process with

this 2-correlation if the probability of a one is not {. If the probability of a one is

j , there exist exactly two 1-dependent processes with this 2-correlation (and

both are indicator processes). So, the conjecture mentioned in the beginning of

this section does not hold in general, but is true for these extremal cases.

In this paper we will compute the minimal 2-correlation for all indicator

processes. For a £ (3,3) we have been able to compute the minimal 2-correla­

tion for 1-dependent processes, finding the same lower bound ([GKV]).

For a £ ( i i ) we know that there exists a unique process with this

2-correlation ([GKV]).

2. Basic properties

For A E A we define the horizontal and vertical sections H

and V

:

H

{y):=>X{xEJ:{x,y)EA), yEJ,

and we define I

:

I

:= V H

{x)V

{x)dX(x).

Jo

LEMMA

1. The 2-correlation P[X

= X

= 1] of an indicator process is

equal to I

.

PROOF.

Directly from the definitions,

P[X

= X

= l] = P[(U

U

,)EA,{U

, U

)EA]

= rP[(U„, U

)EA,{U

U„

)EA 1 U

~x]dX(x)

Jo

= [

H

(x)V

(x)dk{x)

Jo

=/..

□

We define the maximal and minimal 2-correlations of an indicator process

by

Max(a) := sup{/

:A EA,fi{A) = a),

Min(a): = inf{I

:AEA,/i(A) = a}, a £ 7 .

Before we describe the sets for which these extremal values are attained, we

state three simple lemmas.

Let A

: = J

\A be the complement of A.

LEMMA

2 (Complement Lemma). For A EA with fi(A) = a we have

I

= I

C + 2a - 1

and therefore (for aEJ)

Min(a) = M i n ( l - a ) + 2 a - 1 and Max(a) = Max(l - a) + 2a - 1.

We have H

<(x) = I - H

(x) and V

<(x) = 1 - V

(x) which

4 V. DE VALK Isr. J. Math.

ƒ,,= f* (l - HAxW - V

(x))dk(x)

Jo

= ƒ ' {1 - HAx) - V

(x) + H

(x)V

{x))dX{x)

= 1 - la + I

. D

Note that the supremum (infimum) is attained in A for a iff the supremum

(infimum) is attained in A

for 1 — a, so that we may assume a è i

We call the sets {(x,x)EJ

:xEJ), {{x, 1 -x)EJ

:xEJ) the diagonal

resp. the cross diagonal.

Let R

, resp. R

be reflection w.r.t. these diagonals. We call a transformation

(rxr):/

2

->;

2

a product isomorphism if T:J-~J is measurable, measure preserving and

almost everywhere 1-1.

LEMMA

3 (Reflection and Invariance Lemma). For A EA and for a product

isomorphism T XT we have

'A ~ 'RjA

'R<A ~

Arxrvt-PROOF.

We have H^ = V

, H

Jx) = V

{1 -x) and H

(x) =

H

{T~

x) (and similar formulas for V

) which imply the statement. D

We will identify two sets A and B if ^(AAB) = 0, and we introduce the

habitual metric d:

d{A,B): = fi{A/\B), A,BEA.

LEMMA

4 (Continuity Lemma). For A, B E A we have

\lA-h\e2MAAB

)

and therefore (for a, /? E J)

| M a x ( « ) - M a x ( £ ) | S 2 | a - y ? | and |Min(a)-Min(/?)| < 2 | a - / ? | .

The first inequality follows from

\IA-IB\ = \JH

(V

- V

)

+ V

(H

- H

)dk

^j\V

~V

\dX + J\H

-H

\dk

The second inequality follows by choosing for a > /? a set A with measure a

such that I

is close to Max(a), and a subset B of A with measure /?. Then

M(AAB) = a — P, and application of the first inequality yields the second

inequality.

The third inequality follows analogously. D

3. The sets where the maximal and minimal 2-correlations are attained

We define the following sets for 0 ^ a ^ $:

Ar := ([0, 1 - V\-a] X [0, 1]) U ([1 - V\ - a, 1] X [0, 1 - V\ - a]).

For a < i, let

/N+l

l+Vl-2a

where

N__

N+l

1

A ^ i n t

1 -la

is such that

1

1

1 1

2 2N 2 2(N+l)

Now let

Af

:= {{x,y)EJ

:y^s-int(x/s)}

or equivalently

A™

:= Ü ([«,(/ + \)s] X[0,is])U([Ns, l ] X [0,M]).

Finally we define

A™:={(x,y)EJ

:y^x}.

We call A™

a staircase set.

Straightforward computations show that both <4

and A™" have measure a

(see Figs. 1-11).

We call a set A with (i{A) = a < \ a disturbed staircase set if ^ is not a

staircase set and if there exists a set 5 such that (yVand s as above)

6 V. DE VALK Isr. J. Math. 1 l - V \ - a töx-x'xvxjxvxjxjxy'y O 1 - A / 1 - O Fig. 1. /l0m" (0 2 a S J). \ / a 0 Va 1 Fig. 2. ^ U r « )c( l 2 « S l ) . 0 s 1 Fig. 3. .4™" (0 S a < i). O s 25 1 Fig. 4. ^ami" (1S « < }). 35 25 0 5 25 35 i 0 5 25 MS 1 Fig. 5. Af" (i S « < Ö- F'8- 6- -4»"1'" (i _ 1/<2A') S a < i - 1/(2(JV + i)))

1 A's 0 5 Ns 1

Fig. 8. RA^Y(l+V{2(N+ !))<«< i + 1/2JV).

Fig. 9. RAAr

Y(l<a£]).

s 2s 1

Fig. 10. RMr.Y(i<«^i).

0 i 1

Fig. II. RAAr.r(i<a£-\).

8

with O ( = x

) < x , < • • ■ <x,v_i <x

= 1, and for some ƒ„(

(0,1 N-l)

Xi+l-Xi

l-(N-l)s, if/ = /

U, i f / ^ / o , / e ( 0 , ï j v - 1}

and, for some 0 < y < x

, - x

, B is a subset of

[x,

+ 7,.r,

]X[.r,

,x,

+ 7].

See Figs. 12-15.

-li + i -f.

** + >'

Fig. 12. Two disturbed staircase sets and the complements of the reflected (w.r.t. the diagonal)sets of two disturbed staircase sets. B

/

7_

£

^

/

4. Results

[ 2 a - 1 +(1 - a )

'

, 0 < a S i

Max(a) = \

U

, i ^ a < l .

PROPOSITION

1. This supremum is attained in the sets A™

for 0 ' ^ a é \

and in (A^y for J < a < 1.

Conversely, each set A with measure a and I

= Max(a) is product

isomor-phic to one of the above-mentioned sets.

For the proof of Theorem 1 we refer to Katz [Ka] or Finke [F] or Gandolfi,

Keane and de Valk [GKV].

Proposition 1 is proved in [GKV].

UN — l)N

Min(a) =

6(iV+ l)

5, if a = i,

2 a - 1 + M i n ( l - a ) , if\<a£\,

with

«■"'(n;) « - i - V -

2

^ )

-REMARK.

For i - 1/(2JV) ^ a < j - 1/(2(JV + 1)) we have \/N è S > 0, so

S -* 0 if a — \. Note that if 1/(1 — 2o) is an integer we have

Min(a) = Min - ± — = =

\2 IN! 6N

3

and in these points the function Min has a left derivative which is smaller than

the right derivative. For the function Max this phenomenon only occurs at

a = i. (See Fig. 16.)

10 V. DE VALK Isr.J. Math. 1"" 0.950.9 - 0.850.8 - 0.750.7 - 0.650.6 - 0.550.5 - 0.450.4 0.35 0.3 0.25 -0.2 . 0.15 . 0.1 -0.05 . 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a — Fig. 16. The functions Max and Min.

PROPOSITION

2. The infimum is attained in the staircase sets A™

for

0 | a | j , ( ^ ™i

)

for 5 s a ^ 1, and it is also attained in the disturbed staircase

sets for a < ^ and in the complements of these for a > ^.

Conversely, when 1/(1 — 2a) is an integer or a = i, if the infimum is attained

in some set A EA with measure a, then A is product isomorphic to a staircase

set (a 2 i), or to the complement of a staircase set (a > i).

When a ¥= i and 1/(1 — 2a) is not an integer, if the infimum is attained in

some set A EA with measure a, then A is product isomorphic to a staircase set

or to a disturbed staircase set (a < \) or to the complement of one of these sets

5. Proof of Theorem 2

Let a > 0 be fixed. In six steps we will, by various rearrangement procedures,

gradually diminish the size of the collection of sets A for which /,, = Min(a),

until we reach the staircase sets, so proving the statement of Theorem 2.

Step 1. Standardization

By the continuity lemma we may approximate a set A (n(A) = a) by a finite

union of squares of the form [x, x + S) X [ v, y + 6) with x, y E.J, where 8 > 0

is the reciprocal of an integer.

Then H

and V

are constant on intervals. We rearrange /with a transforma­

tion T (a permutation of intervals) such that H

is non-increasing (see

Figs. 17 and 18). We use the notation x := T X T.

The Invariance Lemma implies that I

= I

.

We say that a set is in standard form if it is the set under (the graph of) a

non-decreasing function. We will obtain from rA a set A' in standard form with

l

, ^ I

. This is accomplished by moving squares horizontally to the right.

If tA is not in standard form, then there exist squares S

and S

such that

S

:= [x,,x, + <5)X [y,y + S) is a subset of rA,

S

■ = [x

, x

+ 5) X [y, y + ö) is disjoint with xA,

for some x

<x

. Define the set axA (Fig. 19):

x, + S Xi + ö -v, +S x2 + 6

Fig. 17. Fig. 18. Fig. 19.

oxA: = {xA\S

)US

,

then n(oxA) = pi(xA) and we will prove that I„

5i I

■

12

Therefore

V,

(x)-ö, if* e [*„*, + <$),

V«A(x)=\ V<A(x) + ö, if x £ [x

, x

+ ö),

V

(x), else.

UA - Iau = SH(x)dx - ÖH(x)dx

= S

{H(

) - H(x

)}

Note that we have equality iff H is constant on [x,, x

+ S). The set A' (in

standard form) is obtained from A by applying x (once) and a finite number of

shifts of the type a. Using these facts we obtain the next claim, in which we

introduce the notation f

and A

(to stress the correspondence between a

non-decreasing function/and a set ,4 in standard form that is the set under/).

CLAIM

1 (Standardization).

Min(a) = inf{I

: fi(A) = a, A = A

EA in standard form,

f

finite valued} (aEJ).

REMARK.

From Helly's selection principle ([Luk] par. 3.5) it follows that

the infimum is actually attained in some set in standard form.

Step 2. (Under the diagonal)

Because of the Complement Lemma we assume that a < \. It is easy to see

that Min(a) = 0 for a ^ J; take e.g. A = [1 - Va, 1] X [0, Va].

Therefore we assume further in this proof that | < a < {.

Take a set A in standard form with measure a and such that f

is finite

valued. Assume that A does not lie under the diagonal (a set A lies' under the

diagonal if A is a subset of A™™). We will transform A to a set lying under the

diagonal such that I

does not increase.

Let A be a union of 5 X S squares. We choose

S, : = [,Y„ x

+ S) X [y

y, + 5) subset of A

and

■S:

= [*2, x

+ S) X [^2, v

+ 8) disjoint with A

from ö to \S we may assume that there exist such squares entirely above or

under the diagonal), and such that

(these conditions guarantee that the transformed set will be in standard form).

Let g be such that

A

= (A

\S

)uS

.

We will prove that I

< I

.

We say that a rectangle [,Y', X") X [y', y") (disjoint with the diagonal and a

subset of a set A in standard form) interferes with the horizontal sections H

{x)

with x' S x <x" and with the vertical sections V

(y) with y' ^ y <y".

We introduce this definition because the removal of this rectangle from A

decreases I

by the amount (as follows from the computation in this step)

{y" -y')-{x" -x')- ■

f' H

(x)dx P V

(y)dy

(x" - x') {y" - y')

i.e., the change in I

equals the area of the rectangle times the average value of

the sections with which the rectangle interferes.

The intuitive idea behind the inequality I

< I

is the fact that the square 5,

interferes with the sections marked with a — sign and the square S

interferes

with the sections marked with a + sign (in Fig. 20); the first total is larger than

the second. We have

h, ~ I

s, = r

H

(x)ödx + n

V

(y)ödy

^S-(l -x

)-S+S-(y

+S)-S

and analogously

/ « - / , , = - H

(x)Sdx-\ V

{y)8dy

^ -<J-(1 -x

-ö)-S-Sy

ó

which implies

I

- /,, i S

{ y, - x, +x

-y

+ 26}* 4S> > 0

since x, + S ^ y

and x

^ y

+ 5.

14

-s,

1 g

y/#Z

Éfcfii

:S:

/ \

/

1

:•*:

:|*:

1

1

-'S

f*

x^:'*:

$

Vit-:

•$&

*#:

ë £ 1—i

CLAIM

2 (Under the diagonal). For £ < o < { we have

Min(a) = inf{/

: A E A, n{A) = a, ^ in Standard form,

,4 under the diagonal, f

finite valued).

LEMMA

5 (Windowing). Lei f

:J~J be a non-decreasing function such

that f

(a) = a,f

(b) = b for some O^a <b£ I. Define

A

: = A n([a,b]X[a,b})

and let H

and V" be the corresponding sections on [a, b] (Fig. 21),

H

:= H

- (I - b) and V»:=V

-a.

Fig. 21.

then

l

- = f H

(x)V

(x)dx,

h = Lr + f H

(x)V

(x)dx + (l-b+ a)a

+ (b-a)(\~ b)a.

J [0.a|ul6,lJ PROOF.

We have

I

- f H

(x) V,{x)dx = f (H

(x) +l-b)( V»{x) + a)dx

= f H

(x)V

(x)dx + (l-b) f V"(x)dx

+ a f H

(x)dx + (b-a){\- b)a

= I

- + (l-b + a)a

+ (b-a)(l-b)a. D

COROLLARY.

Assume f

is as in this lemma, and we changef

on {a, b) area

preservingly tof

(i.e.n(A) =[i(B)). ThenI

will change by the same amount as

Step 3. (Moving to the diagonal)

Let A be a set in standard form, lying under the diagonal, such that for some

positive integer N

16

Af i - 1

with O = x

< x, < • • • < A-.v +1 = 1.

Let d-,: = x, - x, _, and c,: = y, - y, _, (/ = 1 , . . . , TV + 1). Assume that

L ' ( / J : = c a r d { / : y , < . v , } > 0 ,

then we will prove the existence of a set B in standard form, lying under the

diagonal, with the same measure as A, and with a finite valued function^ such

that / „ < ƒ , and V{f

)<U{f

).

We first give an intuitive sketch of our procedure (cf. Figs. 22 and 23). Let /'

be the first index such that y, < x,. We will change f

on [A-,_ ,, .r,

,). Because of

the Windowing Lemma we may restrict our attention to the square [A,_ ,, 1] X

[*/-., 1].

We transform the rectangle [A-,, A,

,) X [y,_

y,) (with area ^, + ,-c,) such

that U{f

) reduces by one. We change it to a rectangle with height c, + c

.

This is possible if (Case I, Figs, 22 and 23)

y.--i

X j _ , Xj .*,■ + , Xi+i-a

Fig. 23. Case I. After the transformation.

-v,-,r

Fig. 24. Case II. Before the transformation.

Otherwise (Case II, Figs. 24 and 25) we transform it to a rectangle that has one

corner on the diagonal and lies as far as possible to the right. The rectangle with

area d

-c

interferes before the transformation with the set of horizontal