A Different Construction of Gaussian Fields From

A DIFFERENT CONSTRUCTION OF GAUSSIAN FIELDS FROM

MARKOV CHAINS: DIRICHLET COVARIANCES

PERSI DIACONIS AND STEVEN N. EVANS

Abstract. We study a class of Gaussian random elds with negative corre-

lations. These elds are easy to simulate. They are de ned in a natural way

from a Markovchain that has the index space of the Gaussian eld as its

state space. In parallel with Dynkin's investigation of Gaussian elds having

covariance given by the Green's function of a Markov pro cess, wedevelop con-

nections b etween the o ccupation times of the Markovchain and the prediction

prop erties of the Gaussian eld. Our interest in such eldswas initiated by

their app earance in random matrix theory.

1. Introduction

Let X b e a nite set. Our goal is to de ne and study a rather general class of

mean zero Gaussian elds Z = fZ g with negative correlations. These elds

x x2X

may b e used for smo othing, interp olation and Bayesian prediction as in [Ylv87,

Bes74 , BG93, BGHM95, BK95 , BH99], where there are extensive further references.

The de nition b egins with a reversible Markovchain X with state space X .

Our development in the b o dy of the pap er will b e for continuous time chains,

b ecause the exp osition is somewhat cleaner in continuous time, but we will rst

explain the construction in the discrete time setting. Let P x; y b e the transi-

tion matrix of a conservative discrete time Markovchain X with state space X .

Assume for simplicity that the chain X has no holding that is, the one-step tran-

sitions of X are always to another state. Thus, P x; y 0 for all x; y 2X,

P x; y = 1 for all x 2X,and P x; x = 0 for all x 2X. Supp ose further

y 2X

that the chain X is reversible with resp ect to some probabilityvector x ;

x;y 2X

that is, xP x; y = y P y; x for all x; y 2X. The matrix given by

x; if x = y ,

x; y :=

xP x; y ; if x 6= y ,

is p ositive semi{de nite, and hence is the covariance matrix of a mean zero Gaussian

eld Z = fZ g . Note that the dep endence structure of the eld Z accords with

x x2X

the lo cal neighb ourho o d structure de ned by the transition matrix P :ifP x; y =0

that is, the chain X is unable to go from x to y in one step, then the Gaussian

random variables Z and Z are indep endent.

x y

Date : April 10, 2001.

Key words and phrases. random eld, Dirichlet form, negative correlation, simulation, Markov

pro cess, p otential theory.

Research supp orted in part byNSFgrant DMS-9504379.

Research supp orted in part byNSFgrants DMS-9703845 and DMS-0071468. 1

2 PERSI DIACONIS AND STEVEN N. EVANS

Figure 1. Signs of the Gaussian eld arising from the simple ran-

dom walk on the 50 50 discrete torus

Example 1.1. Let X b e the p oints of the n n discrete torus that is, Z Z ,

n n

where Z is the group of integers mo dulo n, and let P b e the transition matrix of

if x and y are adjacent nearest neighb our random walk on X .Thus, P x; y =

that is, x y 2f1; 0; 0; 1gand P x; y = 0 otherwise. This chain is

reversible with resp ect to the uniform distribution x= . A realisation of

the resulting eld is shown in Figure 1 for the case n = 50. Sites at whichthe

corresp onding Gaussian variable is p ositive resp. negative are coloured black

resp. grey.

For the sake of comparison, the corresp onding picture for a eld of i.i.d. Gauss-

ian random variables is shown in Figure 2. Note that the negative correlation is

apparent to the eye as a more clustered pattern. This phenomenon is an example

of the Julesz conjecture, which claims that the eye can only distinguish rst and

second order statistical features densities and correlations. A review of the litera-

ture on this conjecture and its connections to de Finetti's theorem { an early joint

interest of Bretagnolle and Dacunha-Castelle { is in [DF81].

1.1. Continuous time and Dirichlet forms. As we noted ab ove, it will b e more

convenienttowork with continuous time Markovchains. To this end, let X now

b e a continuous time Markovchain on the nite state space X .Write Q for the

asso ciated in nitesimal generator and supp ose that X is reversible with resp ect to

the probability measure that is, xQx; y = y Qy; x for all x; y 2X. We

do not supp ose that X is conservative. That is, weallow Qx; y < 0, in which

Qx; y when it is in state x. case the chain is killed at rate y

GAUSSIAN FIELDS FROM MARKOV CHAINS 3

Figure 2. Signs of the i.i.d. Gaussian eld on the 50 50 discrete torus

Set L X ;:= ff : X!Rg equipp ed with the inner pro duct hf j g i :=

f xg x x. The kernel Q op erates on L by

Qf x= Qx; y f y :

Reversibilityof X is equivalent to requiring that the op erator Q is self{adjointon

L . Of course, if P is the transition matrix of a reversible, conservative, discrete

time chain with no holding as ab ove, then Q = P I is the in nitesimal generator of

a reversible, conservative, continuous time chain, namely the chain that exits state

x 2X at rate 1 and jumps to state y 6= x with probability P x; y up on exiting.

Consequently, the discrete time construction ab ove can b e subsumed under the

more general construction wearenow considering.

The usual quadratic form asso ciated with Q is the Dirichlet form:

E f; g:=hQf j g i

f x f y g x g y xQx; y =

1.1

x;y

+ f xg xx;

where

X X

x:= xQx; y = y Qy; x 0:

y y

It is clear from 1.1 that the Dirichlet form is a p ositive semi{de nite, self{adjoint

quadratic form. The two terms on the right{hand side of 1.1 are called, resp ec-

tively, the jump part and the killing part of the form. If X is conservative that

Qx; y = 0 for all x 2X and = 0. A standard is, no killing o ccurs, then y

4 PERSI DIACONIS AND STEVEN N. EVANS

reference for Dirichlet forms is [FOT94], but we nd the original pap er by Beurling

and Deny [BD58] useful and readable.

It follows from 1.1 that

1.2 x; y := xQx; y

is a p ositive semi-de nite self{adjoint matrix. Hence is the covariance of a mean

zero Gaussian eld Z = fZ g indexed by X .

x x2X

Example 1.2. Set X = Z ,theintegers mo dulo n.Take X to b e nearest neighbour

random walk with unit jump rate, so

1; if x = y;

Qx; y =

; if x y = 1;

0; otherwise.

1 1 1

,x; x= ,x; x 1 = , and x; y = 0 otherwise. When Then x=

n n 2n

n = 5 the matrix x; y app ears as the circulant

0 1

1 1

1 0 0

2 2

1 1

B C

1 0 0

B 2 2 C

1 1

B C

0 1 0

B 2 2 C

1 1

@ A

0 0 1

2 2

1 1

0 0 1

2 2

1.2. Outline of the rest of the pap er. In Section 2 wedevelop some prop erties

of this construction. We give a simple pro cedure for simulating the eld Z using

indep endent Gaussian random variables asso ciated with the \edges" x; y such

that Qx; y > 0. Generating realisations of Gaussian elds on grids or graphs with

general covariances can b e a complex enterprise. A useful review of the literature

is in [GS89].

In Section 3 we showhow the problem of using the observations fZ g to

y y 2B

predict Z for x=2 B X is intimately related to the prop erties of the o ccupation

times of the Markovchain X. In Section 4 we indicate how certain questions

f xZ sub ject to that involveminimising the variance of a linear combination

constraints can b e related to the p otential theory of X.

We conclude this Intro duction with some comments on the background and

context of this pap er.

1.3. A random matrix connection. Our interest in this construction b egan with

some results in random matrix theory.LetU b e the unitary group of n n matrices

M with MM = I . Elements of U have eigenvalues on the unit circle T in the

complex plane. The study of the distribution of these eigenvalues under Haar

measure makes up a chapter of random matrix theory see, for example, [Meh91].

2 2

Let H denote the space of functions f 2 L Tsuchthat

2 2

kf k := jf j jj j < 1;

j 2Z

by and de ne an inner pro duct on H

f hf; gi := g^ jj j:

j j

j 2Z

GAUSSIAN FIELDS FROM MARKOV CHAINS 5

is the space of functions f 2 L Tsuchthat Alternatively, H

1 f f

1.3 d d < 1;

sin

and, moreover,

1 f f g g

= d d hf; gi

sin

see Equations 1.2.18 and 1.2.21 of [FOT94].

In indep endentwork, Johansson [Joh97] and Diaconis-Shahshahani [DS94]

proved the following result whichwas extended in [DE01].

Theorem 1.3. Choose M 2 U from Haar measure. For f in H let W M =

n f

, where ;:::; are the eigenvalues of M . Then, as n tends to f e

1 n

j =1

in nity, for any nite col lection f ;f ;:::f in H

1 2 K

K K

fW M g fZ g ;

f f

k k =1 k k =1

g is a mean zero Gaussian eld with covariance E [Z Z ]= where fZ : f 2 H

f g f

hf; gi .

The space H is an example of a Bessel{p otential function space and it coincides

1 1

2 2

with the Besov space B , the Sob olev{Leb esgue space F and the Lipschitz space

2;2 2;2

see Equations 18 and 19 in x3.5.4 and Equation 13 in x3.5.1 of [ST87].

2;2

However, for our purp oses the interesting observation is that the space H equipp ed

is nothing other than the Dirichlet space and Dirichlet with the inner pro duct h; i

form of the symmetric Cauchy pro cess on the circle see Example 1.4.2 of [FOT94].

The symmetric Cauchy pro cess on the circle is just the usual symmetric Cauchy

pro cess on the line wrapp ed around the circle. It is p ossible to carry through much

of what we do in the discrete state space setting of this pap er to Dirichlet forms

of Markov pro cesses on general state spaces, but we do not pursue that direction

here.

Note also that if wetake the complex Poisson integral of f 2 L T, namely

e + z 1

f d P f z :=

2 e z

^ ^

= f +2 f z ; jz j < 1;

0 j

j =1

then, letting m denote Leb esgue measure on the disk fz 2 C : jz j < 1g,

2 3

Z Z

dP f z

2 2 2j 1

4 5

4 rdr mdz = 2 jf j j r

j =1

=2 jf j jj j

j 2Z

6 PERSI DIACONIS AND STEVEN N. EVANS

if and only if Thus, f 2 H

dP f z

mdz < 1;

and

1 dP g z dP f z

= mdz ; f; g 2 H : hf; gi

2 dz dz

The form

1 dF z dGz

mdz

2 dz dz

is up to a constantmultiple nothing other than the Dirichlet form of Brownian

motion on the unit disk. There has b een much recentinterest in studying the

Dirichlet form of Brownian motion on such restricted domains see, for example,

[Gro99].

The ab ove connections suggest that weshouldbeableto ndaBrownian motion

or Cauchy pro cess as a limit of ob jects de ned in terms of the eigenvalues of random

unitary matrices. Wehave so far failed in this attempt.

1.4. Dynkin's isomorphism. We conclude with a brief review of Dynkin's iso-

morphism [Dyn80,Dyn83, Dyn84b, Dyn84a]. Assume that the continuous time

Markovchain X considered ab oveis transient. The Green's function Gx; y =

1 1

Q x; y y is p ositive semi-de nite and so can serveasacovariance of a

mean zero Gaussian eld Y indexed by X . Note the parallel: roughly, our basic

construction uses Q to construct a covariance while Dynkin used Q . Dynkin

related prop erties of the Gaussian eld to the underlying Markovchain. Among

other things he showed that the b est prediction of the eld at a p oint a given its

values at sites B X is a linear combination of the observed values at B with

weights the rst hitting distribution of the chain started at a when it rst hits B .

Wehaveaparallelversion in Prop osition 3.1.

Dynkin also proved the following distributional identity.Let

x 1

` 1fX = xg ds; x 2X; = x

denote the \lo cal time" pro cess for the chain X with resp ect to the measure .

Supp ose that on some probability space with exp ectation P wehave a mean zero

Gaussian eld Y = fY g with covariance G and an indep endent copyofthe

x x2X

Markovchain X. The chain X is started at x 2X and conditioned to die up on

hitting y 2X. Then, for any b ounded Borel function F : R ! R wehave

2 2

Y Y

E Y Y F Gx; y = E F + `

x y 1

2 2

Here, Y = fY g is the p ointwise square of the Gaussian eld Y and ` =

x2X 1

f` g .

x2X

In a sustained sequence of pap ers Marcus and Rosen [MR92d, MR92a, MR92c,

MR92b, MR93]have studied symmetric Markov pro cesses by using Dynkin's iso-

morphism. The isomorphism is tight enough so that re ned knowledge of Gaussian

elds e.g. continuity of sample paths can b e carried over to develop ne prop erties

of Markov pro cesses e.g., continuity of lo cal time. Sheppard [She85] gave a pro of

of the Ray{Knight theorem on the Markovianity of lo cal times of one-dimensional

GAUSSIAN FIELDS FROM MARKOV CHAINS 7

di usions that used Dynkin's result and the obvious Markovianity of the asso ciated

Gaussian eld. We do not see such depth for our construction, but nd the parallels

tantalising.

Dynkin's construction has b een used in statistical applications by Ylvisaker

[Ylv87]. He used the Gaussian elds as Bayesian priors for prediction and de-

sign problems. Dynkin's elds only have p ositive correlations while the elds we

construct have negative correlations; using indep endent sums of b oth constructions

may prove useful.

The relationship b etween a Markovchain and the Gaussian eld with covariance

given by the asso ciated Green's function was discovered indep endently byseveral

p eople. In physics, there is work of Symanzik [Sym69] followed bywork of Brydges

et al. [BFS82]. In statistics, Ylvisaker [Ylv87]gives references to Hammersley's

[Ham67]work on harnesses as followed up by Williams [Wil73], Kingman [Kin85,

Kin86] and Dozzi [Doz81, Doz89]. Variants of Dynkin's isomorphism have b een

established by Eisenbaum [Eis95], as well as by Marcus and Rosen in the pap ers

cited ab ove. Markovchain representations of elds other than Gaussian ones have

also b een studied: a recent pap er with an extensive bibliography is [Bol01].

Here are two lesser known alternative app earances of this connection. Bhat-

tacharya[Bha82] establishes general results that sp ecialise in our nite setting to

the following. Supp ose that the chain X is ergo dic. For f in the range of Q,

f X ds converges in distribution as T !1 to a Gaussian eld with co-

variance G

In a more applied context, various authors see, for example, [BM74, Bor79,

Bor01] have considered optimal estimates of heightinsurveying problems. There

are n p oints and estimates of height di erences are available for some pairs. Forming

an undirected graph with the pairs as edges assumed connected, they nd the b est

linear unbiased estimates of the true heights h . Assuming one true height, sayat

site z ,isknown, they showthat

b b

Covh ; h = G x; y

x y z

q y

with G x; y the exp ected numb er of times y is hit starting at x by a dis-

crete time reversible Markovchain constructed from edge weights 1= x; y .

Here x; y isthevariance of the x; y th height di erence measurementand

q y = 1= y; x. The walk is killed when it hits z . If the measurement

errors are assumed Gaussian, then h is a Gaussian eld with covariance given by

the Green's function: Known asymptotics of G x; y in planar grids can then b e

used to understand howthecovariances fall o with separation.

2. Finite State Spaces

The following result gives a representation of the eld Z in terms of indep endent

Gaussian random variables and hence furnishes a simple way to simulate sucha

eld.

Prop osition 2.1. Let X beareversible Markov chain with nite state space X .

Form a graph with vertex set X by placing an undirectededge from x to y if

Qx; y > 0.Choose an orientation for each edge e of the graph, that is, a func-

tion from the edge set to f1g. This orientation may be chosen arbitrarily but,

of course, x; y =y; x. Associate a mean zero, variance x; y , Gaussian

8 PERSI DIACONIS AND STEVEN N. EVANS

random variable W x; y to each edge and a mean zero, variance x, Gaussian

random variable W x to each vertex, with al l of these random variables being in-

dependent. Set

Z = x; y W x; y +W x;

y :Qx;y >0

then the Gaussian eld fZ g has covariance .

x x2X

Proof. The random variable Z has mean zero and variance

X X X

x; y +x= xQx; y xQx; y

y 6=x

y :Qx;y >0

= xQx; x= x; x:

Further, for x 6= y ,

E [Z Z ]= x; ay; bE [W x; aW y; b]:

x y

a;b:Qx;aQy;b>0

The sum is zero unless Qx; y > 0, and then it contains the single term

x; y y; xx; y =x; y ;

as desired.

Remark 2.2. i The construction is not limited to Gaussian variables. It gives a

2nd order eld with the prescrib ed covariance for other uncorrelated choices of

W x; y and W x with the variances set out in Prop osition 2.1. Vertices with

no edge b etween them are uncorrelated.

ii If = 0 that is, there is no killing, then Z =0.

x2X

iii For simple random walk on Z Example 1.2, cho ose a clo ckwise orientation

and let Z = W W indices mo d n with W indep endent N 0;

j j j 1 j

variables.

iv Conversely,if=x; y is a covariance matrix with p ositive diagonal

x;y 2X

entries, non-p ositive o -diagonal entries, and non-negativerow sums, then

can b e realized as a matrix arising from the Markovchain construction in

many di erentways.Justtake to b e an arbitrary probability measure on

the nite set X with x > 0 for all x and put Qx; y = x x; y .

v The representation of Prop osition 2.1 can b e thought of as a factorisation

=AA ,where A is a matrix that has a row for eachelementofX and a

column corresp onding to each of the random variables W x; y andW x. This

factorisation should b e compared to the Karhunen{Lo eve decomp osition =

1 1

0 0

2 2

D D =D where the columns of are the normalised eigenvectors

of corresp onding to non{zero eigenvalues and D = diag ;:::; , k =

1 k

rank , is the diagonal matrix that has these eigenvalues down the diagonal.

Of course, the Karhunen{Lo eve decomp osition leads to another representation

of the eld Z, namely

Z = V ; x 2X;

x x

are indep endent mean zero Gaussian random variables, with ;:::;V where V

k 1

V having variance .

GAUSSIAN FIELDS FROM MARKOV CHAINS 9

For the n n discrete torus eld in Example 1.1, rank n whereas the

construction of Prop osition 2.1 requires 4n indep endent Gaussian random

variables. However, the computation of a particular Z requires only 4 of

these variables, whereas the Karhunen{Lo eve expansion requires the use of

2 2

all rank n variables. Thus simulation the entire eld Z requires 4n

additions using our representation and the order of n multiplications and

additions for the Karhunen{Lo eve representation. In this particular example,

the fast Fourier transform can b e used to cut the latter numb er of op erations

down to the order of n log n, but this improvementisnotavailable for general

chains that lack such group structure.

vi Most constructions of Gaussian elds lead to p ositive correlations for near

neighb ours. Of course, this is often scienti cally natural. However, elds in

which all sites are negatively correlated could arise in settings where growth

in one region deletes supplies from other regions. In situations likeoursin

which all sites are negatively correlated, there are constraints on the strength

of the correlation. This is related to the well{known fact that n exchangeable

. random variables have correlations at least

More generally,let G =X ;E b e an undirected graph with vertex set X

and edge set E . Suup ose that the automorphism group G of G is such that

0 0 00 00

given twoedges fx ;y g and fx ;y g there exists an element g of G such that

0 00 0 00

gx = x and gy = y .LetY = fY g b e a mean zero Gaussian eld

x x2X

that is invariant under the action of G. By renormalising if necessary,wemay

supp ose that the common value of E [Y ], x 2X,isd=2jE j, where d is the

common degree of the vertices of G . Supp ose the E [Y Y ] 0forallx; y 2X.

x y

Write for the common value of E [Y Y ]whenfx; y g is an edge. Wehave

x y

2 3

4 5

0 E Y

x2X

=1+ E [ Y Y ]

x y

x6=y

E [ Y Y ] =1+2jE j +2

x y

fx;y g2= E

1+2jE j:

Thus 1=2jE j with equality if and only if E [Y Y ]=0forallfx; y g 2= E .

x y

This extremal case when all the non-edge covariances are zero corresp onds to

our Markovchain construction with

; x 2X; x=

2jE j

and

1; if x = y;

Qx; y =

; if x 6= y:

That is, the chain X exits from any state at rate 1, and when it exits it jumps

to each of the d neighb ouring states with equal probability. Examples 1.1 and

1.2 t into this framework. The exchangeable case also ts into this framework,

with the graph G b eing the complete graph.

10 PERSI DIACONIS AND STEVEN N. EVANS

3. Prediction and conditional distributions

The following result relates the dep endence structure of the Gaussian eld Z =

fZ g to the prop erties of the o ccupation times of the original Markovchain X.

x x2X

For B a prop er subset of X , supp ose the eld is observed at x 2 B and wewantto

predict it at y=2 B . The mean square optimal prediction is a linear combination of

the observed values Z = fZ g . In order to describ e the asso ciated weights in

B x x2B

terms of quantities for the chain X,let

1fX 2 C g ds; t 0;C X; L C :=

s t

denote the occupation time eld for the chain X.Write

:= inf fs 0: L C = tg

C C

and let X be the chain X time{changed according to L C : that is, X is a

Markovchain with state space C such that the lawofX starting at c 2 C is the

same as that of fX C : t 0g starting at c. Denote by

S := inf ft 0:X 6= X g

t 0

the rst time that the chain X leaves its initial state and by

R := inf ft S : X 2 D g; D X;

D t

the rst time after leaving its initial state that the chain X enters the subset of

states D .

Prop osition 3.1. Let Z = fZ g bea mean zero Gaussian eld with covariance

x x2X

given by 1.2. For a proper subset B X and A = XnB ,theconditional

distribution of Z = fZ : x 2 Ag given Z = fZ : x 2 B g is Gaussian with mean

A x B x

E [Z j Z ]=M Z ,where

A B B

aQa; a

E [L fbg;X 2 B ] ; M a; b=

a R S

and covariance given by

0 A 0 00 0 00

a Q a ;a ; a ;a 2 A;

A A

where Q is the in nitesimal generator of the time{changed chain X .

Proof. Classical theory gives that the conditional distribution of Z given Z is

A B

Gaussian with mean

E [Z j Z ]= Z :

A B AB B

Now=Q where := diag x, and thus

1 1 1

: = Q Q

AA AB AB

BB BB BB

By direct expansion,

00 0 00

0 00 0

= E [L fb g] ; b ;b 2 B: Q

b b b T

Moreover,

Q = Qa; aP [X = b] ; a 2 A; b 2 B:

a;b a S

Therefore,

aQa; a

fbg;X 2 B ] : E [L =

AB S a R ab

GAUSSIAN FIELDS FROM MARKOV CHAINS 11

Classical theory also gives that the covariance of the conditional distribution of

Z given Z is

A B

1 1

= Q Q Q Q ;

AA AB BA AA AA AA AB BA

BB BB

and it is straightforward to see that

A 1

Q = Q + Q Q Q :

AA AA AB BA

The following result is immediate from Prop osition 3.1.

Corollary 3.2. In the notation of Proposition 3.1, construct a graph with vertex

set X by placing an undirected edge between two vertices, x 6= y if x; y < 0.

Fixapoint a 2 A. Say that a point b 2 B is shielded from a if every path from

a to b passes through a point of Anfag. Write B for the set of points in B that

are shieldedfrom a. Then Z is conditional ly independent of Z given Z

a B

B nB

equivalently, Z is conditional ly independent of Z given Z . Moreover, if

a B

B nB

B nB 6 B B ,thenZ is not conditional ly independent of Z given Z

a a B

Remark 3.3. i For large state spaces, Ylvisaker [Ylv87] suggested using simu-

lation of the Markovchain as an aid to computing regression co ecients via

Dynkin's construction. Prop osition 3.1 can b e used similarly.

ii Note from the assumption of reversibilitythat

Qb; a a

= ; a 2 A; b 2 B;

b Qa; b

if the numerator and denominator on the right{hand side are p ositive. In this

case

Qb; aQa; a

M a; b= E [L fbg;X 2 B ] ;

a R S

Qa; b

In any case, only needs to b e determined up to a constant.

iii The co ecients M a; b are always non-p ositive.

iv Note the parallel with the form of the co ecients in Dynkin's construction

describ ed in the Intro duction.

Example 3.4. Consider our running example of simple random walk on Z Ex-

ample 1.2. Cho ose a partition A; B of Z into two non-empty subsets. Fix a

p oint a A.Ifwehave a +1;a +2;:::;a + r 2 B and a + r +1 2 A, then set

B = fa +1;a+2;:::;a + r g. Similarly,ifwehave a 1;a 2;:::;a ` 2 B and

a ` +1 2 A, then set B = fa 1;a 2;:::;a `g. Of course, B or B may

b e empty. Then, in the notation of Corollary 3.2, B = B nB [ B .

a +

More generally, standard probability calculations can b e used to calculate the

matrix M of Prop osition 3.1 for various con gurations. For example, supp ose that

a = 0 and B = f1; 2;:::;rg. The probability that the random walk gets to

1 b r b efore returning to a or hitting another p ointof A is, by the classical

1 1

. Moreover, the probability that the walk returns to b Gambler's Ruin problem,

2 b

b efore hitting A is, again byGambler's Ruin,

1 1 1 1

1 + 1 ;

2 r +1 b 2 b

and so the distribution of the numb er of visits to b given that b is reached at all is

the same as the numb er of trials up to and including the rst success in Bernoulli

12 PERSI DIACONIS AND STEVEN N. EVANS

trials with this return probability as the failure probability. It follows after a little

algebra that

: M a; b= 1

r +1

4. Minimizing variances subject to constraints

In this section we will study the problem of minimizing the variance

3 2

5 4

E f xZ

f xZ under certain constraints on the co ecients. of a linear combination

Here Z = fZ g is a mean zero Gaussian eld with covariance that has

x x2X

p ositive diagonal entries, non-p ositive o -diagonal entries, and non-negativerow

sums. We will also assume that is irreducible in the sense that for x 6= y we

can nd a sequence x = z 6= z 6= ::: 6= z = y such that z ;z 6= 0 for

0 1 k i i+1

0 i k 1.

Prop osition 4.1. Suppose that has at least one row sum positive. Given a

proper subset B X,consider the problem of minimizing E [ f xZ ] subject

to f x=1, x 2 B . The minimum is achievedby

f x:=P fT < 1g

B x B

where T =infft 0:X 2 B g for X a Markov chain with in nitesimal generator

B t

Qx; y = x x; y for any probability vector with positive entries.

Proof. This follows immediately from Theorem 4.3.3 of [FOT94] once we note that

the condition that at least one row sum of is p ositive is equivalenttothechain

X b eing transient.

Prop osition 4.2. Suppose that has al l row sums zero. Given a proper subset

B X and a probability vector with positive entries, consider the problem of

P P

minimizing E [ f xZ ] subject to f x=1, x 2 B ,and f x x=0.

x x

The minimum is given by

E [T ] ;

where T = inf ft 0:X 2 B g for X the Markov chain with in nitesimal

B t

generator Qx; y = x x; y .Moreover, the minimum is achievedby

E [T ]+E [T ]

x B B

f x= :

E [T ]

Proof. We rst recall some standard facts. For > 0, let E denote the inner

pro duct E + h j i.Write Cap for the corresp onding capacity. Then

4.1 Cap B = inf fE f; f :f =1g = E p ;p ;

B B

x:=P [exp T ] see Theorem 4.2.5 of [FOT94] for the case = where p

1, the pro of for general involves just obvious changes. By Theorem 4.3.1 of

[FOT94], p and 1 p are orthogonal with resp ect to E and so

B B

j 1i = Cap B + E [exp T ]; =Cap B + hp ; 1 p 0=E p

B B B

GAUSSIAN FIELDS FROM MARKOV CHAINS 13

where wehave used the fact that E 1; 1 = 0. Thus

4.2 c := p x x=E [exp T ] = Cap B :

B B

Wehave

c = inf fE f; f :f =1 on B g

= inf fE f; f :f =1 on B; f x x=c g:

Thus, if weput

p x c

B B

; f x:=

1 c

then

4.3 inf fE f; f :f =1 on B; f x x=0g = E f ;f =

B B

1 c

after a little algebra. By the assumption of irreducibility, P fT = 1g =0,andso

s 1

= lim E e ds = E [T ] 4.4 lim 1 c

0 0

and the last term in 4.3 converges to E [T ] as 0.

Note, by the same argument that gave 4.4, that

lim 1 p x = E [T ]:

x B

Therefore, in order to establish the claim of the prop osition, it suces to observe

that

lim inf fE f; f :f =1 on B; f x x=0g

=inffE f; f :f =1 on B; f x x=0g

and that

= E f ;f : ;f lim E f

B B

Acknowledgements. We thank Marc Coram for pro ducing the graphics and Don

Ylvisaker for a numb er of helpful conversations.

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E-mail address : [email protected]

Department of Mathematics, Stanford University, Building 380, MC 2125, Stanford,

CA 94305, U.S.A.

E-mail address : [email protected]

Department of Statistics 3860, University of California at Berkeley, 367 Evans Hall, Berkeley, CA 94720-3860, U.S.A