High-Order Correlation Functions of Binary Multi-Step Markov Chains

arXiv:physics/0610081 v1 11 Oct 2006

S. S. Apostolov, Z. A. Mayzelis

V. N.Karazin Kharkov National University, 4 Svoboda Sq.,Kharkov 61077, Ukraine

O. V. Usatenko

∗

, V. A. Yampol'skii

A. Ya. Usikov Institute for Radiophysi s and Ele troni s

Ukrainian A ademy of S ien e, 12 Proskura Street, 61085 Kharkov, Ukraine

Twoapproa hes tostudyingthe orrelation fun tionsof thebinary Markovsequen es are on-sidered. The first of them is based onthe studyof probability of o urring differentwords in thesequen e.Theotheroneusesre urren erelationsfor orrelationfun tions.Thesemethodsare appliedfortwoimportantparti ular lassesoftheMarkov hains.These lassesin ludetheMarkov hainswithpermutative onditionalprobabilityfun tionsandtheadditiveMarkov hainswiththe smallmemoryfun tions.Theex itingpropertyoftheself-similarity(dis overedinPhys.Rev.Lett. 90, 110601(2003)fortheadditiveMarkov hainwiththestep-wisememoryfun tion)isprovedto betheintrinsi propertyofanypermutativeMarkov hain.Appli abilityofthe orrelationfun tions oftheadditiveMarkov hainswiththesmallmemoryfun tionsto al ulatingthethermodynami hara teristi softhe lassi alIsingspin hainwithlong-rangeintera tionisdis ussed.

PACSnumbers:05.40.-a,02.50.Ga,87.10.+e

I. INTRODUCTION

Theproblem of long-range orrelatedrandomsymboli systems(LRCS) hasbeen under studyfor alongtime in manyareasof ontemporaryphysi s[1,2,3,4,5,6℄,biology[7,8,9,10,11,12℄,e onomi s[8,13,14℄,linguisti s[15, 16, 17, 18, 19℄, et . Among the ways to get a orre t insight into the nature of orrelations of omplex dynami systems, theuse of themulti-step Markov hains is of great importan e be ause it enables onstru ting arandom sequen e with the pres ribed orrelated properties in the mostnatural way [19, 20, 21℄. Alsothe additive Markov hainswithsmallmemoryfun tionsareofinterestforthenon-extensivethermodynami sofIsingspin hains[22,23℄. Themulti-step binary Markov hainis hara terizedby the onditional probability of the definedsymbol

a

i

(for example,

a

i

= 1

)o urring after previous symbols within thememory length

N

, i.e., after

N

-word.We definethe

L

-word

T

L

,orthewordoflength

L

,asthesetof

L

sequentialsymbols.TheMarkov hain anbeeasily onstru tedby thesequentialgenerationofsymbolsusingthepres ribed onditionalprobabilityfun tion.The onditionalprobability pointstothedire tasso iationbetweenthesymbols.Atthesametime,the orrelationfun tionsdes ribetheimpli it intera tionbetween thesymbols.Correlation fun tions ofall ordersdetermine ompletely everystatisti al property ofany randomsequen e.So, theproblemof finding the orrelationfun tions ofhigh ordersis essential.The aimof thepaperistofindthe orrelationfun tionsofdifferentordersofthebinaryMarkov hainusingtwoapproa hesand toexaminetheirproperties.

Thepaperisorganizedasfollows.These ondSe tiondes ribesthemethodof al ulatingstatisti al hara teristi s oftheMarkov hainviathe onditionalprobabilityfun tion.Herethepropertyoftheself-similarityofthe orrelated random sequen es is alsodis ussed. Thethird Se tion is devoted to amethod for finding the orrelation fun tions usingre urren erelationsforthem. Theappli ations oftheproposedgeneralalgorithms tosome on rete lassesof hainsarepresented.

II. DIRECT METHODFORCALCULATING STATISTICALCHARACTERISTICS OFTHE

MARKOVCHAIN

Inthisse tion,we des ribeadire t methodoffindingfollowing hara teristi sof the

N

-stepsMarkov hains:the probability

P (T

N

)

ofthe

N

-wordso urring,theprobability

P (T

L

)

ofo urringthewordsof anarbitrarylength

L

, andthehigh-order orrelationfun tions.

∗

Itisknownthatallstatisti alpropertiesofMarkovsequen earedetermined ompletelybythe onditional proba-bilityfun tion.Alongwiththisfun tion,itis onvenienttoemploytheprobabilities

P (T

N

)

ofthe

N

-wordso urring. All other hara teristi s of the hain an be found using these probabilities. Though the al ulation of

P (T

N

)

is quite a hallenge in the general ase, we su eed in obtaining the analyti alresults for some parti ular ases (see Subse .IIC).

The

N

-step Markov hain is asequen e ofrandom symbols

a

i

,

i ∈ Z = {. . . , −2, −1, 0, 1, 2, . . .}

.It possessesthe followingproperty: the probability ofsymbol

a

i

to have a ertain value, under the ondition that thevaluesof all previous symbolsaregiven,dependsonthevaluesof

N

previoussymbolsonly,

P (a

i

= a| . . . , a

i−2

, a

i−1

) = P (a

i

= a|a

i−N

, . . . , a

i−2

, a

i−1

).

(1) Wereferto

N

asthememory depth.

Inthispaperwe onsiderbinary hainsonly,but someresults anbegeneralizedto non-binarysequen esaswell.

A. Probabilitiesof the

N

-wordso urring

Westartexaminingthestatisti alpropertiesoftheMarkov hainswithsear hingtheprobabilitiesofthe

N

-words o urring.Thereare

2

N

different

N

-wordsin thearbitraryMarkov hainwhi his hara terizedby

2

N

probabilities ofthesewordso urring.They anbefoundusingevidentformulasoftheprobabilitytheory,

P (A) = P (A ∩ B) + P (A ∩ B),

(2)

P (A ∩ B) = P (A|B)P (B).

(3)

Here

A ∩ B

meansthat events

A

and

B

o ursimultaneouslyand event

B

isoppositeto theevent

B

.Eqs.(2), (3) yieldthefollowingequation,

P (a

1

a

2

. . . a

N

) =

X

a=0,1

P (a

N

|aa

1

. . . a

N −1

)P (aa

1

. . . a

N −1

).

(4)

Thisequation,beingwrittenforallpossiblevaluesofsymbols

a

1

, . . . , a

N

,alongwiththenormalizationrequirement,

X

ai=0,1

i=1,...,N

P (a

1

a

2

. . . a

N

) = 1,

(5)

isinfa tthesetoflinearequations.Fromthisset,one anobtainallsoughtprobabilities

P (T

N

)

.Thoughset(4),(5) ontains

2

N

_{+ 1}

equations,ithasasinglesolutionbe auseoneofEqs.(4)isalinear ombinationofothers.

B. Probabilities ofarbitrarywords o urringand orrelationfun tions

Using Eq. (2) we an al ulate the probability

P (T

L

)

with

L < N

by redu ing the

N

-words to the

L

-word

(a

1

a

2

. . . a

s

)

,

P (a

1

a

2

. . . a

L

) =

X

ai=0,1

i=L+1,...,N

P (a

1

a

2

. . . a

N

),

L < N.

(6)

This equation represents the probabilityof the

L

-wordo urring asthe average of theprobabilities to have the

N

-word ontainingthis

L

-word.Itispossibletoredu ethe

N

-wordtotheshorter

L

-wordbysummingoverdifferent setsofsymbolsattheleftandrightedgesofthe

N

-word.

Thewordoflength

L

greaterthan

N

anbepresented asthe ombinationofthewordoflength

N

and

(L − N)

symbolsfollowingafterit. A ordingtothedefinitionoftheMarkov hain, ea hofthese

(L − N)

symbols antake ona ertainvaluewiththeprobability dependingonthe pre edent

N

-word.The

(L − N)

-folduseofEq. (3)yields thefollowingequation,

P (a

1

a

2

. . . a

L

) = P (a

1

a

2

. . . a

N

)

L−N

Y

r=1

Letusdefinea orrelationfun tionofthe

s

thorder:

K

s

(i

1

, i

2

, . . . , i

s

) = (a

i

1

− ¯a

i

1

)(a

i

2

− ¯a

i

2

) . . . (a

i

s

− ¯a

i

s

),

(8) where

· · ·

isthestatisti alaverageovertheensembleof hains.We onsiderergodi hain.A ordingtotheMarkov theorem(see, e.g.,Ref.[24℄),this propertyisvalidfor thehomogenousMarkov hainsifthe onditionalprobability doesnottakeonvalues

0

and

1

.Inthis aseaveragingovertheensemble of hainsandoverthe hain oin ide.

Formally,fun tion

K

s

dependson

s

arguments(

s

differentindexesofthesymbols),butwedo onsiderhomogenous Markov hains.Therefore, orrelationfun tion

K

s

dependson

(s−1)

arguments,i.e.,thedistan esbetweentheindexes,

r

1

= i

2

− i

1

, r

2

= i

3

− i

2

, . . . , r

s−1

= i

s

− i

s−1

.Itsdefinitionis writtenas

K

s

(r

1

, r

2

, . . . , r

s−1

) = (a

0

− ¯a)(a

r

1

− ¯a)(a

r

1

+r

2

− ¯a) . . . (a

r

1

+...+r

s−1

− ¯a).

(9) Here

¯

a

istheaveragenumberofunitiesin thesequen eand notation

· · ·

isthestatisti alaverageoverthe hain,

f (a

r

1

, . . . , a

r

1

+...+r

s

) = lim

M→∞

1

2M + 1

M

X

i=−M

f (a

i+r

1

, . . . , a

i+r

1

+...+r

s

).

(10)

Introdu ingthenotation,

R

0

= 0,

R

k

=

k

X

i=1

r

i

,

d

i

= a

i

− ¯a,

(11)

werewritedefinition(9)ofthe orrelationfun tion,

K

s

(r

1

, r

2

, . . . , r

s−1

) =

X

aR

i

=0,1

i=0,...,s−1

d

R

0

d

R

1

. . . d

R

s−1

P (a

R

0

a

R

1

. . . a

R

s−1

).

(12)

Nowwe an omplement theset ofsymbols

a

R

0

, a

R

1

, . . . , a

R

s−1

with thesymbolsbetween them. Finallywe have thefollowingformula,

K

s

(r

1

, r

2

, . . . , r

s−1

) =

X

ai=0,1

i=0,...,Rs

−1

d

R

0

d

R

1

. . . d

R

s−1

P (a

0

a

1

. . . a

R

s−1

).

(13) Hereprobabilities

P (a

0

a

1

. . . a

R

s−1

)

should be al ulatedusingEqs.(6) or(7).

C. PermutativeMarkov hains

Solvinglinearsystem(4),(5)forthegeneral aseisa hallengingproblem.Herewedemonstratetheappli ationof thedes ribed methodto a ertain lass ofthe Markov hains. Weassumethat the onditional probability fun tion of the hainunder onsiderationis independent of the order of symbols in the previous

N

-word.We refer to su h sequen esasthepermutative Markov hains.

1. Probabilitiesofword o urringinthepermutativeMarkov hain

It is onvenient to introdu e a new abbreviated notation for the onditional probability fun tion

P (a

N +1

|a

1

a

2

. . . a

N

) = p

k

(a

N +1

)

. Here

k = a

1

+ a

2

+ . . . + a

N

is the number of unities in

N

-word

(a

1

a

2

. . . a

N

)

. Besides,we define

p

k

(1)

as

p

k

.

Nowwe seekthesolutionofsystem(4),(5)intheform

b

N

(k) = b

N

(a

1

+ a

2

+ . . . + a

N

) = P (a

1

a

2

. . . a

N

).

(14)

Inotherwords,theprobabilityofthe

N

-wordo urringdependsonthenumberofunitiesinthis

N

-wordonly.Then Eq.(4) anberewrittenasthefollowingre urren erelation,

Thesolutionofthisequationis

b

N

(k) = b

N

(0)

k

Y

r=1

p

r−1

1 − p

r

.

(16)

Hereprobability

b

N

(0)

anbeobtainedfromEq.(5) ,

b

N

(0) =



X

N

k=0

C

k

N

k

Y

r=1

p

r−1

1 − p

r



−1

,

(17)

C

n

k

=

n!

k!(n − k)!

=

Γ(n + 1)

Γ(k + 1)Γ(n − k + 1)

.

Theprobability

P (a

1

a

2

. . . a

N

)

,Eq.(14) ,does notdepend ontheorderofsymbolsin

N

-word

(a

1

a

2

. . . a

N

)

. From Eq. (6) and the above statement it followsthat theprobability

P (T

L

)

of a short word (with

L < N

)o urring is likewiseindependentoftheorderofsymbolsinthisword.Denotingthisprobabilityby

b

L

(k) = b

L

(a

1

+a

2

+. . .+a

L

) =

P (a

1

a

2

. . . a

L

)

wearrivedatthefollowingformula,

b

L

(k) =

N −L

X

m=0

C

N −L

m

b

N

(k + m).

(18)

Theprobabilityofthelong

L

-word(with

L > N

)o urringdoesdependontheorderofthesymbolsinthisword. UsingEq.(7)we have

P (a

1

a

2

. . . a

L

) = b

N

(q

0

)

L−N

Y

r=1

p

q

r−1

(a

N +r

).

(19)

Here

q

i

= a

i+1

+ a

i+2

+ . . . + a

i+N

. Equations(16)-(18)are thegeneralizationof resultsearlier obtainedin [21,25℄ fortheadditivebinaryMarkov hainwiththestep-wisememoryfun tion.

2. Correlationfun tions

Herewepresentanalyti alresultsfrom the al ulationofthe orrelationfun tionsofarguments

r

1

, . . . , r

s−1

satis-fyingthe ondition

r

1

+ . . . + r

s−1

< N

.Forthispurposeweexpressthefra tionofunitiesinthe hainusingEq.(18),

a = b

1

(1) =

N

X

k=1

b

N

(k)C

_{N −1}

k−1

.

(20)

Forthepermutative Markov hainsunder onsideration,the orrelationfun tion ofarguments

r

1

, . . . , r

s−1

depends ontheirnumber

s

only.Equation(13)yields

K

s

(r

1

, r

2

, . . . , r

s−1

) = K

s

=

N

X

k=0

b

N

(k)S

N

(k, s, a),

(21)

S

N

(k, s, ¯

a) =

min{s,k}

X

j=0

(−¯a)

s−j

C

_{N −j}

k−j

C

s

j

.

(22)

Inparti ular,thebinary orrelationfun tion

K

2

(r)

ofanypermutative hainis onstantforthevaluesofarguments lessthanthememorydepth:

K

2

(r) = K

2

,

r < N

.

IfweapplyallderivedformulastotheadditiveMarkov hainwiththestep-wisememoryfun tion,

p

k

=

1

2

− ν + µ

 2k

N

− 1



,

(23)

wegetthe orrelationfun tionsoforder

s

,

K

s

=

Γ(n

1

+ n

2

)

Γ(n

1

)

s

X

k=0

(−¯a)

s−k

C

s

k

Γ(n

1

+ k)

Γ(n

1

+ n

2

+ k)

.

(24) Here

¯

a =

n

1

n

1

+ n

2

,

n

1

=

N (1 − 2(µ + ν))

4µ

,

n

2

=

N (1 − 2(µ − ν))

4µ

.

(25)

For

s = 2

we re overtheresultspreviouslyobtainedin[21, 25℄forthebinary orrelationfun tion

K

2

.

D. Self-similarity ofthe permutativeMarkov hain

In this subse tion we point out an interesting property of the permutative Markov hains, namely, their self-similarity.Thedis ussionofthisissueforthestep-likememoryfun tionispresentedin paper[25℄.

Letusredu e the

N

-stepMarkovsequen eby regularly(orrandomly)removingsomesymbolsand introdu ethe de imationparameter

λ = N

∗

_{/N, 1/N < λ < 1}

,whi h representsthefra tionofsymbols keptinthe hain.

Due to approa hmentof symbols in thesequen e afterthe de imation pro edure,the binary orrelation fun tion

K

2

(r)

transformsinto

K

∗

2

(r) =







(1 − {r/λ})K

2

([r/λ]) + {r/λ}K

2

([r/λ] + 1),

regularde imation,

∞

P

l=0

K

2

(r + l)C

l+r−1

l

(1 − λ)

l

λ

r

,

randomde imation, (26)

where

[x]

isthemaximalintegernumberlessthan

x

and

{x} = x − [x]

.

The orrelation fun tion

K

2

(r)

of the permutative Markov hain is equalto onstant value

K

2

with arguments

r 6 N − 1

.A ordingto Eq.(26)(regularde imation),thevaluesofthe orrelationfun tion

K

∗

2

(r)

at

r 6 λ(N − 1)

takeon thesamevalue

K

2

.In the aseofthe randomde imation wehave onlyan exponentiallysmall (at

N ≫ 1

) differen ebetweenthe orrelationfun tion

K

∗

2

(r)

andvalue

K

2

,

|K

2

∗

(r) − K

2

| 6

√

A

¯

λN

exp − a(¯λ, λ)N

,

r = ¯

λN,

¯

λ < λ.

(27)

Herethenewfun tion

a(¯

λ, λ)

and onstant

A

areintrodu ed,

a(¯

λ, λ) = ¯

λ ln

 ¯

λ

λ



+ (1 − ¯λ) ln

 1 − ¯λ

1 − λ



> 0,

A =

√

e

π

|K

2

| + max

r>N

|K

2

(r)|

.

(28)

Thus, the orrelation fun tion of the de imated sequen e is onstant (or asymptoti ally onstant) at the region

r < N

∗

.Thispropertyisreferredtoastheself-similarity.

Theself-similarity is the intrinsi propertyof the permutative Markov hains.It is possible to showthat, in the general ase, the onditional probability fun tion of the de imated

N

-step Markovsequen e is of infinite memory depth. This is be auseits onditional probabilityis blurred by the de imation pro edureand be omes dependent onallprevioussymbols.

So,the self-similarity istheproperty of thebinary orrelationfun tion onlyand inheresin allrandomsequen es witha onstantbinary orrelationfun tion

K

2

(r)

at

r < N

,

K

2

(r) = K

2

, 1 6 r < N ⇒

Self-similarity. (29)

III. CORRELATION FUNCTIONS ANDCHARACTERISTICEQUATIONS

InthisSe tion,wedealwiththere urren erelationstofindthe orrelationfun tions.InSubse tionAweshowthat theserelationsyieldtheexpli itexpressionforthe orrelationfun tionsviatherootsofthe hara teristi equations. Subse tion B is devoted to appli ations of this method. The last Subse tion ontains some generalizations of the above-mentionedre urren erelations.

The pro edure of finding the orrelation fun tions

K

s

is based on the mathemati al indu tion method. In other words,we supposethat all orrelationfun tions

K

l

oftheorderslessthan

s

arefound.Forthe onvenien esake,we admitthat

K

0

= 1

and

K

1

= 0

.

A. High-order orrelation fun tionsofthe additive Markov hain

Considerthe

N

-stepMarkov hainwiththeadditive onditionalprobabilityfun tion,

P (a

i

= 1|a

i−N

. . . a

i−1

) = ¯

a +

N

X

r=1

F (r)(a

i−r

− ¯a).

(30)

Herefun tion

F (r)

,

r = 1, . . . , N

,isreferredtoasthememory fun tion and

¯

a

isthefra tionof unitiesintheabove sequen e(fordetailsseeRef. [26℄).

Letusfindthere urren erelationsforthe orrelationfun tionsof

N

-stepadditiveMarkov hain.Forthispurpose, wefirst al ulateexpli itlytheaverageoversymbol

a

r

1

+...+r

s−1

inEq.(9).UsingthenotationofSe .IIBandEq.(30), and takingintoa ountequation

P (a

R

s−1

= 1|·) + P (a

R

s−1

= 0|·) = 1

,we anrewriteEq. (9)forarbitrary

r

i

> 0

,

i = 1, . . . , s − 1

,intheform,

(a

R

0

− ¯a) . . . (a

R

s−1

− ¯a) =

= (a

0

− ¯a) . . . (a

R

s−2

− ¯a)((1 − ¯a)P (a

R

s−1

= 1|T

N,R

s−1

) − ¯aP (a

R

s−1

= 0|T

N,R

s−1

)) =

(31)

= (a

0

− ¯a) . . . (a

R

s−2

− ¯a)

N

X

r=1

F (r)(a

R

s−1

−r

− ¯a).

Here

T

N,R

s−1

isthesetofsymbols

(a

R

s−1

−N

, . . . , a

R

s−1

−1

)

.Inthatway,weobtainthefundamentalre urren erelation onne tingthe orrelationfun tions ofdifferentorders

s

,

K

s

(r

1

, . . . , r

s−1

) =

N

X

r=1

F (r)K

s

(r

1

, . . . , r

s−1

− r).

(32)

Intheparti ular ase

s = 2

,this equation,

K

2

(r

1

) =

N

X

r=1

F (r)K

2

(r

1

− r),

(33)

was obtained and dis ussed in [26℄. Re urren e relation (32) is orre t for

r

i

> 0

,

i = 1, . . . , s − 1

. Provided that

r

s−1

6

N

,thelastargument ofthe orrelationfun tionin theright-handsideofEq.(32)isnegativeorzeroandone shouldinterpretitinthefollowingmanner,whi hisreferredtoas ollating.Ifthe orrelationfun tionhasnegative arguments,we mustreorganizeita ordingtodefinition(9).Forexample,

K

4

(2, 2, −3) =

(a

0

− ¯a)(a

2

− ¯a)(a

4

− ¯a)(a

1

− ¯a)



=

=

(a

0

− ¯a)(a

1

− ¯a)(a

2

− ¯a)(a

4

− ¯a)



= K

4

(1, 1, 2).

(34)

Ifthe orrelationfun tion haszeroarguments(indexes

i

and

k

oftwomultipliers

(a

i

− ¯a)

and

(a

k

− ¯a)

oin ide)one shouldemploythepropertyofthebinary hain:

a

2

i

= a

i

,

a

i

= {0, 1}

.Thus, wehave

(a

i

− ¯a)

2

= (1 − 2¯a)(a

i

− ¯a) + ¯a(1 − ¯a).

With this property we an write the useful relations for the orrelation fun tion ontaining zero arguments in differentpositionsamongallargumentsof

K

s

,

K

s

(0, r

2

, . . . , r

s−1

) = (1 − 2¯a)K

s−1

(r

2

, . . . , r

s−1

) + ¯

a(1 − ¯a)K

s−2

(r

3

, . . . , r

s−1

),

K

s

(r

1

, . . . , r

k−1

, 0, r

k+1

, . . . , r

s−1

) = (1 − 2¯a)K

s−1

(r

1

, . . . , r

k−1

, r

k+1

, . . . , r

s−1

)+

+¯

a(1 − ¯a)K

s−2

(r

1

, . . . , r

k−1

+ r

k+1

, . . . , r

s

),

k 6= 1, s,

K

s

(r

1

, . . . , r

s−2

, 0) = (1 − 2¯a)K

s−1

(r

1

, . . . , r

s−2

) + ¯

a(1 − ¯a)K

s−2

(r

1

, . . . , r

s−3

).

ThegeneralsolutionofEq.(32) an berepresentedasalinear ombination,

K

s

(r

1

, . . . , r

s−1

) =

N

X

j=1

L

j

(r

1

, . . . , r

s−2

)ξ

j

r

s−1

, r

i

> 0 (i = 1, . . . , s − 1),

(36)

ofpowersoftheroots

ξ

j

,

j = 1, . . . , N

, ofthe hara teristi equation,

ξ

N

₋

N

X

j=1

F (j)ξ

N −j

_{= 0.}

(37)

Newfun tions

L

j

(r

1

, . . . , r

s−2

)

, i.e. oeffi ients oflinearform Eq.(36) , intheirturn, should bedefined. All one has todoistosubstituteEq.(36)into ollatedEq.(32)for

0 < r

s−1

< N

.Finally,thispro edureyieldsthere urren e relationsforthesoughtfun tions

L

j

.

Thegeneralsolutionofthisre urren erelationisredu edtotheform,

L

i

(r

1

, . . . , r

s−2

) =

N (N −1)/2

X

k=1

M

ik

(r

1

, . . . , r

s−3

)η

k

r

s−2

,

(38)

where

η

k

,

k = 1, . . . , N (N − 1)/2

,aretherootsofthenew hara teristi equation

det Υ(η) = 0,

(39)

Υ

ij

(η) = (η/ξ

j

)

i−1

− ξ

j

i−1

+ δ

1i

,

i, j = 1, . . . , N.

(40) Fromthe ollating pro edurewe anfindthenextre urren erelationsforfun tions

M

ij

et .Using thisalgorithm we anfind,in prin iple,the orrelationfun tionsofallorders.

B. Appli ation ofthe algorithm

All results obtainedin this subse tionare orre tfor the additiveMarkov hain, but someof them arevalid for thenon-additive hainaswell.

Thefirstsimpleresultisthatthe orrelationfun tionsofalloddordersarezerofortheadditiveMarkov hainwith

¯

a = 1/2

.ThisresultsfromEq. (35):fun tion

K

2m+1

(·)

isexpressedonlyin termof

K

2m−1

(·)

and

K

1

= 0

.

The orrelationfun tion ofthese ondorder

K

2

(·)

(the binary orrelationfun tion) foranadditiveMarkov hain anbefoundfrom Eq.(32):

K

2

(r) =

N

X

j=1

L

j

ξ

j

r

,

r > −N,

(41)

where

ξ

j

,

j = 1, . . . , N

, are therootsof the hara teristi equation (37) .Coeffi ients

L

j

should beobtainedby the ollating:

K

2

(0) = ¯

a(1 − ¯a);

K

2

(−r) = K

2

(r),

0 < r < N.

(42)

Paper[21℄ ontainsananalysisofthisequationsinthe aseoftheadditiveMarkov hainwiththestep-wisememory fun tion.Belowwe presenttheresultsof al ulationforthe orrelationfun tion ofaweakly orrelated hain,

|P (a

i

= 1|a

i−N

. . . a

i−1

) − ¯a| ≪ 1.

(43)

This aseisveryimportantfromthephysi alpointofviewbe ausethestatisti alpropertiesoftheequilibrium long-range orrelatedIsing hains anberepresentedand onsideredastheMarkov hains.Spe ifi ally,theMarkov hain withthesmallmemoryfun tionisstatisti allyequivalent totheweakly orrelatedIsing hain(seeRefs.[22, 23℄).

Forthe orrelationfun tion ofevenorderof theunbiased (

a = 1/2

¯

)additive hainwehave

K

2

(r) ≈

1

4

F (r),

0 < r < N,

K(0) =

1

4

,

(44)

K

2s

(r

1

, . . . , r

2s−1

) =

s

Y

j=1

K

2

(r

2j−1

),

r

j

> 0, j = 1, . . . , 2s − 1.

(45)

Thelatter equation is orre t forthe non-additivesequen e aswell. Inthe ase ofone-step Markov hain,

N = 1

, thisequationis exa t.Butif

N > 1

oneshould interpretitastheasymptoti alequality.At

N = 2

,theexa t result forthe orrelationfun tion

K

4

(·)

oftheadditive hainis,

K

4

(r

1

, r

2

, r

3

) = K

2

(r

1

)K

2

(r

3

) + (−ξ

1

ξ

2

)

r

2

 1

4

K

2

(r

1

+ r

3

) − K

2

(r

1

)K

2

(r

3

)



.

(46)

Here

ξ

1

and

ξ

2

aretherootsofthe hara teristi equation,

ξ

2

− F (1)ξ − F (2) = 0.

(47)

Andfinally,Eq.(32)allowsustoexpressthememoryfun tionintermsthepres ribed orrelationfun tions

K

s

.In Ref. [26℄,themethodof onstru tingthe orrelatedbinarysequen e withpres ribedbinary orrelationfun tion

K

2

wasearlierdis ussed.

C. Generalized algorithm offindingthe orrelationfun tions

Herewe generalizethealgorithmproposedin Subse .IIIAfortheadditiveMarkov hainsto thearbitrarybinary multi-stepMarkov hains.

The onditionalprobabilityfun tionofthe

N

-stepbinaryMarkov hain anbewrittenas

P (a

i

= 1|a

i−N

. . . a

i−1

) =

X

lj =0,1

j=1,...,N

F (l

1

, l

2

, . . . , l

N

)

N

Y

r=1

(a

i−r

− ¯a)

l

r

.

(48)

It is a general form for the arbitrary binary fun tion, be ause

P (·|·)

an be thought of a linear fun tion of ea h argument

a

j

.Werefertofun tion

F (l

1

, l

2

, . . . , l

N

)

asthegeneralized memoryfun tion.

Equation(48) anbeapplied totheadditiveMarkov haindes ribedbyEq.(30),namely,

F (0, 0, . . . , 0) = ¯

a,

F (0, . . . , 0

| {z }

r−1

, 1, 0, . . . , 0

| {z }

N −r

) = F (r),

r = 1, 2, . . . , N,

(49)

F (l

1

, l

2

, . . . , l

N

) = 0,

l

1

+ l

2

+ . . . + l

N

> 1.

Nowweobtainthere urren erelationforthe orrelationfun tionwiththe oeffi ientsexpressedviathegeneralized memoryfun tion.a ordingtothepro edureperformedinSubse tionIIIA,wesubstitutethelastsymbol

a

r

1

+...+r

s−1

in Eq.(9) (for

r

j

> 0

,

j = 1, . . . , s − 1

)for its onditionalprobability

P (·|·)

in theform ofEq. (48).Asaresultone gets,

K

s

(r

1

, . . . , r

s−1

) = (F (0, 0, . . . , 0) − ¯a)K

s−1

(r

1

, . . . , r

s−2

)+

(50)

+

X

l

j

=0,1

j=1,...,N

F (l

1

, l

2

, . . . , l

N

)K

s+k−1

(r

1

, . . . , r

s−2

, r

s−1

− ρ

k

, ρ

k

− ρ

k−1

, . . . , ρ

2

− ρ

1

),

where

k = l

1

+ . . . + l

N

6= 0

.Thein reasingnumbers

ρ

j

,

j = 1, . . . , k

aretheindexes,forwhi h

l

ρ

j

6= 0

.

Note that somesummands in Eq. (50) an have non-positivearguments if

r

s−1

6

N

. For this ase, one should applythe ollating pro eduretothisequation.

Thus,thealgorithmforfindingofthe orrelationfun tions anbeformulatedasfollows:

1.To obtain the orrelation fun tion

K

s

(r

1

, r

2

, . . . , r

s−1

)

at

r

i

> 0

,

i = 1, 2, . . . , s − 1

, we should use Eq. (50) and exe ute the ollating pro edure.We find that fun tion

K

s

(r

1

, . . . , r

s−1

)

is expressed viathe orrelation fun tions withthesumoftheirargumentslessthaninitial sum

r

1

+ . . . + r

s−1

.

2.Using Eq. (50) and performing the  ollating pro edure

m

times with respe t to the obtained orrelation fun tions,wederivearelationbetweenfun tion

K

s

(r

1

, . . . , r

s−1

)

andsomeother orrelationfun tions.

(a) If all of them have the order less than

s

, we obtainthe re urren e relation for the orrelation fun tion ofthe

s

th order.This relation anbesolved bythemethodof hara teristi equationswithoutexe uting item3.(Asimilar asetakespla efortheadditiveMarkov hain.)

(b) In the opposite ase, for

m > r

1

+ r

2

+ . . . + r

s−1

− N

, fun tion

K

s

(r

1

, . . . , r

s−1

)

is expressed via the orrelation fun tions with the sums of their arguments less than

N

. Some of these fun tions are of the ordergreaterthan

s

.Thenwe shouldgotothenextitemofthealgorithm.

3.All orrelationfun tionswiththesumofargumentslessthan

N

shouldbefoundfromasetoflinearequations. To thisendwe writeEq.(50)foreverysu h orrelationfun tion.Afterexe utingthe ollating pro edurewe obtainthesetof

2

N −1

_{− 1}

linearequationsfor

2

N −1

_{− 1}

sought valuesofthe orrelationfun tions. Insomeparti ular ases,itis onvenientto define

K

s

(·)

in Eq.(9) and

P (·|·)

inEq. (48)usinganarbitraryfixed value

˜

a

(forexample,

a = 1/2

˜

)insteadoftheaverage

¯

a

.Inthisin ident everyreasoningremainvalidprovidedthat thevalueof

¯

a

is hanged to

˜

a

in Eqs.(50)and(35),and

K

1

= 0

is hanged to

K

1

= ˜

a − ¯a

. Toobtainaverage

¯

a

we should addEq.(50),written for

K

1

, totheset ofequationsin item3ofthealgorithm.Then wearriveat theset of

2

N −1

linearequationsand

2

N −1

− 1

sought values ofthe orrelation fun tionsand one ofthe average

a

¯

. Besides, one anuse

˜

a

insteadof

¯

a

in thedifferentphysi alproblems,whi hare relevanttotheMarkov hain. Forinstan e, theenergyin theIsingmodelismore onvenientto expressintermsorfun tion

K

2

(r) = (a

i

− 1/2)(a

i+r

− 1/2)

.

IV. CONCLUSION

Thus,wehavedemonstratedtwoapproa hestodeterminingthestatisti alpropertiesofthebinaryMarkov hains. Thefirstofthemshouldbeusediftheprobabilitiesof

N

-wordso urring anbeeasilyfound.The orrelationfun tions ofdifferentorders an beexpressedviatheseprobabilities.Theexamplesofthese hainsis thepermutative Markov sequen es. The se ond approa h allows one to find the orrelation fun tions dire tly from the re urren erelations onne tingthem with thememoryfun tion.Inthe general ase,these relations ontain the orrelationfun tions of differentordersand,hen e,theyarediffi ulttosolve.Inthe aseofadditive hains,thisrelationsaresimplifiedand theirusehelpstofindthesolutions.

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