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www.imstat.org/aihp 2008, Vol. 44, No. 2, 293–323

DOI: 10.1214/07-AIHP130

© Association des Publications de l’Institut Henri Poincaré, 2008

An algebraic approach to Pólya processes

Nicolas Pouyanne

Département de mathématiques, LAMA UMR 8100 CNRS, Université de Versailles – Saint-Quentin, 45, avenue des Etats-Unis, 78035 Versailles cedex, France. E-mail: pouyanne@math.uvsq.fr

Received 18 May 2006; revised 7 October 2006; accepted 3 January 2007

Abstract. Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results forlargeprocesses (a Pólya process is calledsmallwhen 1 is a simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part≤1/2; otherwise, it is called large).

Résumé. Les processus de Pólya sont une généralisation naturelle des modèles d’urnes de Pólya–Eggenberger. Cet article présente une nouvelle approche de leur comportement asymptotique via les moments, basée sur la décomposition spectrale d’un opérateur aux différences finies sur des espaces de polynômes. En particulier, elle fournit de nouveaux résultats sur lesgrandsprocessus (un processus de Pólya est ditpetitlorsque 1 est valeur propre simple de sa matrice de remplacement et lorsque toutes les autres valeurs propres ont une partie réelle≤1/2 ; sinon, on dit qu’il est grand).

MSC:60F15; 60F17; 60F25; 60G05; 60G42; 60J05; 68W40

Keywords:Pólya processes; Pólya–Eggenberger urn processes; Strong asymptotics; Finite difference transition operator; Vector-valued martingale methods

1. Introduction

Take an urn (with infinite capacity) containing first finitely many balls ofs different colours named 1, . . . , s. This initial composition of the urn can be described by ans-dimensional vectorU1, thekth coordinate ofU1being the number of balls of colourkat time 1. Proceed then to successive draws of one ball at random in the urn, any ball being at any time equally likely drawn. After each draw, inspect the colour of the ball, put it back into the urn and add new balls following at any time the same rule. This rule, summed up by the so-calledreplacement matrix

R=(ri,j)1i,jsMs(Z)

consists in adding (algebraically), for anyj∈ {1, . . . , s},ri,j balls of colourj when a ball of colourihas been drawn.

In particular, a negative entry of R corresponds to subtraction of balls from the urn, when it is possible. The urn processis the sequence(Un)n1of random vectors with nonnegative integer coordinates, the kth coordinate ofUn being the number of balls of colourkat timen, i.e. after the(n−1)st draw.

Such urn models seem to appear for the first time in [7]. In 1930, in its original articleSur quelques points de la théorie des probabilités[17], G. Pólya makes a complete study of the two-colour urn process having a replacement matrix of the formS·Id2,S∈Z1.

We will only considerbalancedurns. This means that all rows ofR have a constant entries’ sum, sayS. Under this assumption, the number of added balls isSat any time, so that the total number of balls at timenis non-random.

Furthermore, we will only consider replacement matrices having nonnegative off-diagonal entries. Any diagonal entry

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may be negative but subtraction of balls of a given colour may become impossible. In order to avoid this extinction, one classically adds an arithmetical assumption to the column of any negative diagonal entry inR(see Definition 1.1 and related comments). An urn process submitted to all these hypotheses will be called Pólya–Eggenberger, in reference to the work of these authors.

A Pólya–Eggenberger urn process can be viewed as a Markovian random walk in the first quadrant ofRs with finitely many possible increments (the rows ofR), the conditional transition probabilities between timesnandn+1 being linear functions of the coordinates of the vector at timen. This point of view leads to the following natural generalization: we will namePólya processsuch a random walk in Rs with normalized balance (S=1), even if it does not come from an urn process, i.e. even ifU1andR have non-integer values. Note that a Pólya process as it is defined just below looks very much like a Pólya–Eggenberger urn process, with the only difference that instead of counting a number of balls, we deal with a positive real quantitylk(Xn)associated with each colourk(corresponding to the “number of balls” of this colour at timen), which gives the propensity to pick this colour at the next step. In this setting,wkis the vector inRs defined by the fact that, when colourkhas been drawn, then for allj∈ {1, . . . , s}, one addslj(wk)“balls” of colourj to the urn. Pólya processes generalize Pólya–Eggenberger urns only because this propensity may be real-valued (see comments after Definition 1.1).

Definition 1.1. LetV be a real vector space of finite dimension s≥1. Let X1, w1, . . . , ws be vectors of V and (lk)1ks be a basis of linear forms onV satisfying the following assumptions:

(i) (initialization hypothesis)

X1=0 andk∈ {1, . . . , s}, lk(X1)≥0; (1) (ii) (balance hypothesis)for allk∈ {1, . . . , s},

s j=1

lj(wk)=1; (2)

(iii) (sufficient conditions of tenability1)for allk, k∈ {1, . . . , s},

k=klk(wk)≥0, (3.a)

lk(wk)≥0orlk(X1)Z+s

j=1lk(wj)Z=lk(wk)Z. (3.b) (3)

The(discrete and finite dimensional) Pólya processassociated with these data is theV-valued random walk(Xn)n∈Z1 with increments in the finite set{w1, . . . , ws},defined byX1and the induction:for everyn≥1andk∈ {1, . . . , s},

Prob(Xn+1=Xn+wk|Xn)= lk(Xn)

n+τ1−1, (4)

whereτ1is the positive real number defined by

τ1= s k=1

lk(X1). (5)

The process is defined on the space of all trajectories ofX1+

1ksZ0wkendowed with the natural filtration (Fn)n0whereFnis theσ-field generated byX1, . . . , Xn. It is Markovian2and the transition conditional probabilities between timesn andn+1 depend linearly on the state at timen, as stated in Eq. (4). Conditions (1) and (2) are

1Some authors prefer the vocableviabilityinstead of tenability. This last word has been chosen in reference to recent literature on the subject.

2The time-homogeneity of the process is more explicit when one reads condition (4) with denominator

klk(Xn)instead ofn+τ11 (use relation (6)).

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necessary and sufficient for the random vectorX2 to be well defined by relation (4); a readily induction shows the deterministic relation

n≥1, s k=1

lk(Xn)=n+τ1−1. (6)

Condition (3) suffices to guarantee that the process is well defined, i.e. that the numberslk(Xn)do not become nega- tive so that the process does not extinguish as can be checked by an elementary induction. The arithmetical assump- tion (3.b), which has become classical (compare with [9,11,14] for urns) is equivalent to the following one:lk(wk)is nonnegative, or it divideslk(X1)and all thelk(wj)as real numbers. Actually, if conditioned on non-extinction, all the results about Pólya processes in this article remain valid when condition (3) is removed from the definition.

Pólya processes are natural generalizations of Pólya–Eggenberger urns in the following sense (see [2,9,11,19] for base references on Pólya–Eggenberger urns). Take a Pólya–Eggenbergers-colour urn process having replacement matrixR and vectorU1as initial composition; letS be the common sum ofR’s rows, assumed to be nonzero. The data consisting in taking the rows of S1Ras vectorswk’s, the coordinate forms as formslk’s andX1=1SU1as initial vector define a Pólya process(Xn)nonRs, the random vectorXnbeing 1/Stimes the 1×smatrixUnwhose entries are the numbers of balls of different colours aftern−1 draws. We will name this process thestandardized urn process.

Conversely, if one considers the forms lk of a Pólya process as being the coordinate forms of V (choice of a basis ofV), the matrix whose rows are the coordinates of the wk’s satisfies all hypotheses of a Pólya–Eggenberger urn’s replacement matrix with balance S=1, except that its entries are not integers but real numbers. This matrix will still be called thereplacement matrixof the process. Note that the balance property is expressed in relation (2). The definition of Pólya processes is readily stable after a linear change of coordinates, when urn processes do not have this property.

The present text deals with Pólya processes, so that all its results are valid for Pólya–Eggenberger urn processes.

Such a process being given, different natural questions arise: What is the distribution of the vector at any timen? Can the random vector be renormalized to get convergence? What kind (and speed) of convergence is obtained? What is the asymptotic distribution of the process?

Since the work of Pólya and Eggenberger, many authors have considered such models, sometimes with more general hypotheses, often with restrictive assumptions. Direct combinatoric attacks in some particular cases were first intended [7,10,17], for example. In the last years, they have been considerably refined by analytic considerations on generating functions in low dimensions by much more general methods [9,19]. A second approach was first introduced in [1] and developed in [14] and [16], viewing such urns as multitype branching processes. It consists in embedding the process in continuous time, using martingale arguments and coming back to discrete time. This method provides convergence results. One can find in [9,14] and [19] good surveys and references on the subject.

A Pólya process will be calledsmallwhen 1 is a simple eigenvalue of the replacement matrixR and when every other eigenvalue ofRhas a real part≤1/2. Otherwise, it will be saidlarge.

Under some assumptions of irreducibility onR, it is well known that if(Xn)nis a small Pólya process, a normal- ization(Xnnv1)/

nlogνnconverges in law to a centered Gaussian vector,v1being a deterministic vector andν a nonnegative integer that depends only on the conjugacy class ofR– see [14] for a complete statement of that fact.

In the case of reducible small processes, convergence in law after normalization has been shown for several fami- lies of processes in low dimensions; this concerns for instance urns with a triangular replacement matrix ([9,16,19], example (2) in Section 7.2). Found limit laws in these studies are most often non-normal.

In the case of large Pólya processes, a suitable normalization of the random vectorXn leads to an almost sure asymptotics, as shown in Theorems 3.5 and 3.6, the main results of the paper. These results do not require any irreducibility assumption. This asymptotics is described by finitely-many random variablesWk that appear as limits of martingales. Joint moments of theWk are computed in terms of so-called reduced polynomials(Qα)α(Z0)s that will be defined later and initial conditions of the process. We give hereunder a simplified version of the result: suppose that the replacement matrixRhas 1 andλ2assimpleeigenvalues and that any other eigenvalue is the conjugateλ2or has a real part< 2). Such a process will be calledgeneric.3

3Note that such a processisgeneric in the sense that almost all (in the strong sense of algebraic geometry) replacement matrices of Pólya processes satisfy this assumption.

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Asymptotics of generic large Pólya processes. If(Xn)nis a generic large Pólya process,there exist some complex- valued random variableW and non-random complex vectorsv1andv2such that

Xn=nv1+ nλ2W v2

+o n 2)

,

the smallobeing almost sure and in anyLp,p≥1.Furthermore,any joint moment of the variableWand its complex conjugateW is given by the formula

E

WpWq

= 1)

1+2+2)Q(0,p,q,0,...)(X1), whereis Euler’s function.

The positive numberτ1, defined by (5), depends on initial conditionX1. Vectorsv1andv2are here eigenvectors of the replacement matrix respectively associated with the eigenvalues 1 andλ2. In particular, the second-order term is oscillating whenλ2is non-real, giving a complete answer to the already observed non-convergence of any non-trivial normalization(XnEXn)/nz,z∈C(see [5] and related papers for example).

The method used here to establish the general asymptotics of large Pólya processes also leads to results on dis- tributions at a finite time (exact expressions for moments for example) but we do not focus on this point of view. It relies on asymptotic estimates of suitable moments ofXn. Hence, the first step is to express, for general functionsf, the expectationEf (Xn)in terms of initial conditionX1and of iterations of a finite difference operatorΦ, namely, by Proposition 4.1,

Ef (Xn)=γτ1,n(Φ)(f )(X1),

whereγτ1,nis the polynomial defined byγτ1,1=1 and, for anyn≥2, γτ1,n(t )=

n1 k=1

1+ t

k+τ1−1

; (7)

Φ is the transition operator associated with the process, defined on the space of all functionsf:V →R(or more generally on the space of all functionsf:VW whereW is any real vector space) by:∀vV,

Φ(f )(v)=

1≤k≤s

lk(v)

f (v+wk)f (v)

. (8)

The second step is to study this linear operatorΦon its restriction to the space of linear forms onV, which leads to set a corresponding Jordan basis(uk)1ksof this space, with corresponding eigenvaluesk)1ks(Definition 2.3). The third step consists in observing, as done in Proposition 3.1, thatΦstabilizes, for anyα(Z0)s, the finite dimensional polynomial subspaceSα=Span{uβ, βα}where, for allβ=1, . . . , βs)(Z0)s,uβ =

1ksuβkk and≤is the degree-antialphabetical order ons-uples of integers, defined below by (18). Therefore, it is subsequently possible to decompose anyu-monomialuα,α(Z0)s as a sum of functions in the characteristic subspaces4 ker(Φ−z)=

n0ker(Φ−z)n,z∈C.

If one denotesλ=1, . . . , λs)andα, λ =

1ksαkλk for anyα(Z0)s, it turns out that the eigenvalues of the restriction ofΦ to stable finite dimensional polynomial spaces are precisely theα, λ, as justified in Section 3.

The projection of anyuα on ker(Φ− α, λ)parallel to

z=α,λker(Φ−z)will be denoted byQα and named thereduced polynomialofΦof rankα. The reduced polynomials of rankαconstitute a basis ofSαand anyuαcan be written

uα=Qα+

β<α,β,λ=α,λ

qα,βQβ (9)

4When the context is unambiguous, ifzis a complex number,zwill also denotezIwhereIis the identity endomorphism.

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as proved in Proposition 4.8.

This leads to an asymptotic estimate of the momentsEuα(Xn)(Theorem 3.4) since, for anyz∈Cand anyf ∈ ker(Φ−z), there exists an integerν≥0 such that

Ef (Xn)

n→+∞

nzlogνn ν!

1)

1+z)(Φz)ν(f )(X1) (10)

as it is proven in Corollary 4.2. The asymptotic estimate in Theorem 3.4 is based on the determination of the indicesβ in expansion (9) that contribute to the leading term ofEuα(Xn); this is the object of the whole of Sections 4.4 and 4.5.

To this end, Theorem 4.20 enables us to refine relation (9): it implies that a coefficientqα,β does not vanish only if β belongs to a convex polyhedron(AαΣ )(R0)s ofRs, whereAα is a the set of nonnegative integer points of a certain rational cone with vertexα that depends on the Pólya process andΣ a universal rational cone (universal means here thatΣ is the same one for any Pólya process). Definitions ofΣ andAα are respectively given by (35) and (39). Formula (9) can thus be refined into

uα=Qα+

βAαΣ,β,λ=α,λ

qα,βQβ (11)

which is the same as relation (44).

We will say thatα=1, . . . , αs)(Z0)s is apower of large projectionswheneverαk=0 for all indicesksuch that k)≤1/2; similarly,αwill be called apower of small projectionswheneverαk=0 for all indicesksuch that k) >1/2. Now, ifαis a power of large projections, Propositions 4.15(1) and 4.19 imply that β, λ< α, λ wheneverβAαΣ,β, λ = α, λ. Therefore, thanks to relation (10), the leading term ofEuα(Xn)in formula (11) will come fromEQα(Xn)only, with an order of magnitude of the formnα,λlogνn, the number α, λbeing greater than|α|/2. Similarly, Proposition 4.15(2) implies that, ifαis a power of small projections, this order of magnitude never exceeds n|α|/2logνn for some nonnegative integerν. A precise statement of these moments’ asymptotics is given in Theorem 3.4. Note that the intervention ofΣ can be bypassed by a self-sufficient argument that has been suggested by the anonymous referee (see Remark 5.5).

Section 2 is devoted to Jordan decomposition ofΦ’s restriction to linear forms and related definitions and notations.

The main results of the paper are introduced and completely stated in Section 3 while the action of transition operator Φ on polynomials is studied in Section 4. This is done in three steps: first, the stability of the filtration (Sα)α of subspaces is established as well as its consequences on reduced polynomials; cone Σ and polyhedraAα are then introduced in the space(R0)s of exponents; afterwards, consequences of these geometrical considerations are drawn to refineΦ’s action. Main Theorems 3.4–3.6 are proved in Sections 5 and 6. At last, Section 7 contains diverse remarks and examples.

2. Preliminaries, notations and definitions

The definition of Pólya processes in a real vector spaceV of finite dimensions≥1 was given in Definition 1.1. We associate with any process itsreplacement endomorphismthat will be denoted byAin reference to literature on the subject (see [1,14] for example). LetVC=VRCbe the complexified space ofV.

Definition 2.1. If(Xn)n is a Pólya process,its replacement endomorphism is,with notations of Definition1.1,the endomorphismA=

1kslkwkVVEnd(V ),defined as A(v)=

1ks

lk(v)wk

for everyvinV.

Note that the transpose ofAis the restriction of the transition operatorΦ to linear forms onV. When the process is a Pólya–Eggenberger urn process, the matrix ofAin the dual basis of(lk)kis the transpose of the normalized urn’s replacement matrix 1SR(notations of Section 1).

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With this definition, the expectation ofXn+1conditionally toXnis readily expressed as(I+A/(n+τ1−1))Xn, so that the expectation ofXnequals

EXn=γτ1,n(A)(X1) (straightforward induction).

One of the first tools used to describe the asymptotics of a Pólya process is the reduction of its replacement endomorphismA(or of its transpose on the dual vector space ofV). Because of condition (2), the linear formu1= s

k=1lksatisfiesu1A=u1, which shows that 1 is always eigenvalue ofA. The whole of assumptions (1)–(3), allows us to say more onA’s spectral decomposition. Even if these properties can be proved using the Perron–Frobenius theory, we give a proof’s hint of Proposition 2.2.

Proposition 2.2. Any complex eigenvalueλofAequals1or satisfies λ <1.Moreover, dim ker(A−1)equals the multiplicity of1as an eigenvalue ofA.

Proof. Replace Aby its matrix in the dual basis of(lk)k. Suppose first that all entries ofAare nonnegative. The space of alls×s matrices having nonnegative entries and columns with entries’ sum 1 is bounded (for the norms’

topology) and stable for multiplication. This forces the sequence(An)n0to be bounded, which implies both results (for the second one, consider Jordan’s decomposition ofAand note that the positive powers ofI+N constitute an unbounded sequence ifNis a nilpotent nonzero matrix). IfAhas at least one negative diagonal entry, apply the results to(A+a)/(1+a)for any positiveasuch thatA+ahas nonnegative entries.

In the whole paper, a Pólya process with replacement endomorphismAbeing given, we will denote byσ2the real number≤1 defined by

σ2=

1 if 1 is multiple eigenvalue ofA,

max

λ, λ∈Sp(A), λ=1

otherwise, (12)

where Sp(A)is the set of eigenvalues ofA.

2.1. Jordan basis of linear forms of the process

The present subsection is devoted to notations and vocabulary related to spectral properties of the replacement endo- morphismA.

Definition 2.3. If(Xn)nis a Pólya process of dimensions,a basis(uk)1ks of linear forms onVCis called a Jordan basis of linear forms of the process or abbreviated a Jordan basis when

(1) u1=

1≤k≤slk;

(2) ukA=λkuk +εkuk1 for allk≥2,where the λk are complex numbers(necessarily eigenvalues of A)and where theεk are numbers in{0,1}that satisfyλk=λk1εk=0.

In other words, the matrix of the transposed endomorphismtAin a Jordan basis of linear forms has a block-diagonal form Diag(1, Jp1k1), . . . , Jptkt)), whereJp(z)denotes thep-dimensional square matrix

Jp(z)=

⎜⎜

⎜⎝ z 1

z . .. . .. 1

z

⎟⎟

⎟⎠.

A (real or complex) linear formukwill be called aneigenformof the process whenukA=λkuk, i.e. whenεk=0.

An eigenform of the process is an eigenvector of tA; some authors call these linear forms left eigenvectors ofA, referring to matrix operations.

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Definition 2.4. A Jordan basis of linear forms being chosen with notations as above,a subsetJ⊆ {1, . . . , s}is called a monogenic block of indices whenJhas the formJ= {m, m+1, . . . , m+r}(r≥0,m≥1,m+rs)withεm=0, εk=1for every k∈ {m+1, . . . , m+r}andJ is maximal for this property.Any monogenic block of indices J is associated with a unique eigenvalue ofAthat will be denoted byλ(J ).

In other words,J is monogenic when the subspace Span{uj, jJ}isA-stable and when the matrix of the en- domorphism of Span{uj, jJ}induced by tA in the Jordan basis is one of the Jordan blocks mentioned above with numberλ(J )on its diagonal. The adjectivemonogenichas been chosen because this means that the subspace Span{uj, jJ} =C[tA] ·um+r is a monogenic sub-C[t]-module of the dual spaceVC for the usualC[t]-module structure induced bytA.

Definition 2.5. A monogenic block of indicesJ is called a principal block when λ(J )=σ2andJ has maximal size among the monogenic blocksJsuch that λ(J)=σ2(see(12)forσ2’s definition).

A Jordan basis(uk)1ks of linear forms of the process being chosen,

(vk)1ks (13)

will denote its dual basis, made of the vectors ofVCthat satisfyuk(vl)=δk,l(Kronecker notation) for anykandl, and

λ=1, . . . , λs) (14)

thes-uple of eigenvalues (distinct or not) respectively associated withu1, . . . , us (orv1, . . . , vs). In particular,λ1=1 for any Jordan basis of linear forms. The eigenvaluesλ1, . . . , λs ofAare calledrootsof the process. For anyk, we also denote byπk the projection on the lineCvk relative to the decompositionVC=

1lsCvl; these projections satisfy

Id=

1ks

πk and πk=uk·vk. (15)

Note that theπk commute with each other (πkπl=δk,lπk) but do not commute withA. Nevertheless,Acommutes with

jJπj, the sum being extended to any monogenic block of indicesJ (these sums are polynomials inA). This fact will be used in the proofs of Theorems 3.5 and 3.6. The lines spanned by the vectorsvk can be seen as principal directions of the process, the word principal being here used in physicists’ sense.

2.2. Semisimplicity, large and small projections

For every Jordan basis(uk)1ksof linear forms, and for everyα=k)1ks∈Zs, we adopt the notations

|α| =

1≤k≤s

αk (total degree), α, λ = (16)

1ks

αkλk

and, when all theαkare nonnegative integers uα=

1ks

uαkk,

uαbeing a homogeneous polynomial function of degree|α|.

Given a Jordan basis(uk)1ks of linear forms of the process, we adopt the following definitions.

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Definition 2.6. A Pólya process is called semisimple when its replacement endomorphismAis semisimple,i.e.when Aadmits a basis of eigenvectors inVC(this means that all theuk are real or complex eigenforms ofA).The process is called principally semisimple when all principal blocks have size one(for any choice of a Jordan basis).

The four following assertions are readily equivalent:

(i) the process is principally semisimple;

(ii) for anyk∈ {1, . . . , s},( λk=σ2uk is eigenform);

(iii) the induced endomorphism(

{k, λk=σ2}πk)Ais diagonalizable overC;

(iv) ifr≥1 and if{λk, kr+1}are the roots of the process having a real part less thanσ2, the matrix oftAin the Jordan basis has a block-diagonal form Diag(1, λ2, . . . , λr, Jp1k1), . . . , Jptkt)).

Note that Proposition 2.2 asserts that anyukassociated with root 1 is an eigenform ofA.

Definition 2.7. A root of the process is called small when its real part is less than or equal to1/2;otherwise,its is said large.The process is called small when σ2≤1/2,which means that1 is a simple root and all other roots are small;when the process is not small,it is said large.

Definition 2.8. Letα=1, . . . , αs)(Z0)s.

(1) αis called a power of large projections whenuαis a product of linear forms associated with large roots,i.e.when for allk∈ {1, . . . , s},k=0⇒ λk>1/2).

(2) αis called a power of small projections whenuα is a product of linear forms associated with small roots,i.e.

when for allk∈ {1, . . . , s},k=0⇒ λk≤1/2).

(3) αis called a semisimple power whenuαis a product of eigenforms,i.e.when for allk∈ {1, . . . , s},k=0⇒uk

is an eigenform of the process).

(4) αis called a monogenic power when its support is contained in a monogenic block of indices.

In the whole text, the canonical basis ofZs(or ofRs) will be denoted by

k)1ks (17)

and the symbol

αβ (18)

ons-uples of nonnegative integers will denote thedegree-antialphabetical(total)order, defined byα=1, . . . , αs) <

β=1, . . . , βs)when(|α|<|β|)or(|α| = |β|and∃r∈ {1, . . . , s}such thatαr< βr andαt=βtfor anyt > r). For this order,δ1< δ2<· · ·< δs<1< δ1+δ2· · ·.

Whenα=1, . . . , αs)is as-uple of reals, the inequality α≥0

will mean that all the numbersαkare≥0.

3. Main results

As it was briefly explained in Section 1, the method used to study the asymptotics of a Pólya process(Xn)nrelies on estimates of its moments in a Jordan basis, namelyEuα(Xn),α(Z0)s. To this end, as it is developed in Section 4.1, it is natural to consider the transition operatorΦas it was defined by Eq. (8). Proposition 3.1 is the first result on the action ofΦ on polynomials. One can find a proof of it in Section 4.2.

Proposition 3.1. For any choice of a Jordan basis(uk)1ks of linear forms of a Pólya process and for everyα(Z0)s,

Φ uα

α, λuα∈Span

uβ, β < α .

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The complex numbersα, λwere defined in (16). An immediate consequence of this proposition is theΦ-stability of the finite-dimensional polynomial subspace

Sα=Span

uβ, βα

(19) for anyα(Z0)s. These subspaces form an increasing sequence whose union is the spaceS(V )of all polynomial functions on V, so that Proposition 3.1 asserts that the eigenvalues ofΦ on S(V )are exactly all numbersα, λ, α(Z0)s (in the (ordered) basis(uβ)β≤α of anySα, the matrix ofΦis triangular).

Notation: ifΨ is an endomorphism of any vector space, we will denote by kerΨthe characteristic space ofΨ associated with zero, that is

kerΨ=

p0

kerΨp. (20)

We will use the notationΦ to refer toΦ itself as well as to the endomorphism induced byΦ onS(V )or on some stable subspace. Decomposition of allSα as direct sums of characteristic subspaces ofΦleads to the splitting

S(V )=

z∈C

ker(Φ−z).

As it was announced in Section 1, we can now properly define the reduced polynomials.

Definition 3.2. For any choice of a Jordan basis(uk)1ks of linear forms of a Pólya process and for anyα(Z0)s, the reduced polynomial of rankαis the projection ofuα onker(Φ− α, λ)parallel to

z=α,λker(Φ−z).It will be denoted byQα.

Properties of the reduced polynomials will be further developed in Section 4. In particular, it will be explained how one can compute them inductively (see (32)). They admit sometimes closed formulae (see (33), [18] and (58)).

It follows from its definition thatQα belongs to ker(Φ− α, λ); the numberνα defined just below is its index of nilpotence in this characteristic space. In particular,να=0 if and only ifQα is an eigenvector ofΦ. Proposition 5.6 in Section 5.2 shows how one can easily compute this number for any power of large projections.

Definition 3.3. For everyα(Z0)s,the nonnegative integerναis defined by να=max

p≥0,

Φα, λp

(Qα)=0

. (21)

These facts, definitions and notations being given, we claim the following three main results of the article.

Theorem 3.4 (Joint moments of small or large projections). Let(uk)1ks be a Jordan basis of linear forms of a Pólya process(Xn)n.Letα(Z0)s.

(1) Ifαis a power of small projections,then there exists some nonnegative integerνsuch that Euα(Xn)∈O

n|α|/2logνn asntends to infinity.

(2) Ifαis a power of large projections,then there exists a complex numbercsuch that Euα(Xn)=cnα,λlogναn+o

n α,λlogναn asntends to infinity.

(3) Ifαis a semisimple power of large projections,then Euα(Xn)=nα,λ 1)

1+ α, λ)Qα(X1)+o n α,λ

asntends to infinity,whereQαis the reduced polynomial of rankαrelative to the Jordan basis(uk)1ks.

(10)

Constantcin assertion (2) has an explicit form given in Remark 5.3. The proof of Theorem 3.4 can be found in Section 5. It is based on a careful study of coordinates of theu-monomials in the basis of reduced polynomials, which is developed in Sections 4.4 and 4.5.

Although it is not formally necessary, we give two different statements on the asymptotics of large Pólya processes, respectively, when the process is principally semisimple or not. Their proofs can be found in Section 6. They are based on Theorem 3.4 and use martingale techniques (quadratic variation, Burkholder Inequality).

Theorem 3.5 (Asymptotics of large and principally semisimple Pólya processes). Suppose that a Pólya process (Xn)nis large and principally semisimple.Fix a Jordan basis(uk)1ks of linear forms such thatu1, . . . , ur (2≤rs)are all the eigenforms of the basis that are associated with roots5λ1=1, λ2, . . . , λr having a real partσ2.

Then,with notations(13)and(14)of Section2,there exist unique(complex-valued)random variablesW2, . . . , Wr such that

Xn=nv1+

2kr

nλkWkvk+o nσ2

, (22)

the smallobeing almost sure and inLp for everyp≥1.Furthermore,if one denotes by(Qα)α(Z0)s the reduced polynomials relative to the Jordan basis(uk)k,all joint moments of the random variablesW2, . . . , Wr exist and are given by:for allα2, . . . , αr∈Z0,

E

2kr

Wkαk

= 1)

1+ α, λ)Qα(X1), whereα=

2krαkδk=(0, α2, . . . , αr,0, . . .).

Theorem 3.6 (Asymptotics of large and principally nonsemisimple Pólya processes). Suppose that the Pólya process(Xn)nis large and principally nonsemisimple.Fix a Jordan basis(uk)1ks of linear forms;letJ2, . . . , Jr be the principal blocks of indices6andν+1the common size of theJk’s(ν≥1).

Then,with notations(13)and(14)of Section2,there exist unique(complex-valued)random variablesW2, . . . , Wr such that

Xn=nv1+ 1

ν!logνn

2≤k≤r

nλ(Jk)WkvmaxJk+o

nσ2logνn

, (23)

the smallobeing almost sure and inLp for everyp≥1.Furthermore,if one denotes by(Qα)α(Z0)s the reduced polynomials relative to the Jordan basis(uk)k,all joint moments of the random variablesW2, . . . , Wr exist and are given by:for allα2, . . . , αr∈Z0,

E

2kr

Wkαk

= 1)

1+ α, λ)Qα(X1), whereα=

2krαkδminJk.

5In short, if 1 is a multiple root,λ1= · · · =λr=1; otherwise,12< λ2= · · · = λr=σ2<1. See (12), definition ofσ2.

6In other words, ifJis any Jordan block ofAin theuk’s basis,Jis 1 or one of theJk’s, or the size ofJisν, or the root ofJhas a real part less thanσ2. See Definition 2.5 (principal blocks).

(11)

4. Transition operator

Let(Xn)nbe a Pólya process given by its increment vectors(wk)1ksand its basis of linear forms(lk)1ks submit- ted to hypotheses of Definition 1.1. We recall here the definition of its associated transition operatorΦas it was given in Section 1: iff:VW is anyW-valued function whereWis any real vector space,∀vV,

Φ(f )(v)=

1ks

lk(v)

f (v+wk)f (v) .

4.1. Transition operatorΦand computation of moments

Proposition 4.1 expresses the expectation of anyf (Xn)in terms off, of iterations of the transition operatorΦ and of X1, initial value of the process. Polynomials γτ1,n with rational coefficients and one variable were defined by Eq. (7).

Proposition 4.1. Iff:VW is any measurable function taking values in some real(or complex)vector spaceW, then for alln≥1,

Ef (Xn)=γτ1,n(Φ)(f )(X1). (24)

Proof. It follows immediately from (4) that the expectation off (Xn+1)conditionally to the state at timenis EFnf (Xn+1)=

1ks

1

n+τ1−1lk(Xn)f (Xn+wk)

=f (Xn)+ 1 n+τ1−1

1ks

lk(Xn)

f (Xn+wk)f (Xn) .

By definition of the transition operatorΦ, this formula can be written as EFnf (Xn+1)= Id+ 1

n+τ1−1Φ

(f )(Xn); (25)

taking the expectation leads to the result after a straightforward induction.

It follows from Proposition 4.1 that the asymptotic weak behaviour of the process, or at least the asymptotic behaviour of its moments is reachable by decompositions of the operatorΦon suitable function spaces. Corollary 4.2 is the first step in this direction, stating the result for functions that belong to finite dimensional stable subspaces.

Corollary 4.2. Letf :VW be a measurable function taking values in some real(or complex)vector spaceW.

(1) Iff is an eigenfunction ofΦ associated with the(real or complex)eigenvaluez,that is ifΦ(f )=zf,then Ef (Xn)=nz 1)

1+z)f (X1)+O nz1 asntends to infinity(is Euler’s function).

(2) Assume thatf is nonzero and belongs to someΦ-stable subspaceSof measurable functionsVW and that the operator induced byΦonSis a sumzIdS+ΦN,whereΦNis a nonzero nilpotent operator onSandza complex number.Letνbe the positive integer such thatΦNν(f )=0andΦNν+1(f )=0.Then

Ef (Xn)=nzlogνn ν!

1)

1+z)ΦNν(f )(X1)+O

nzlogν1n asntends to infinity.

(12)

Proof. (1) It follows from Proposition 4.1 that Ef (Xn)=γτ1,n(z)×f (X1). Note that, as soon as the terms are defined,

γτ1,n(t )= 1) 1+t )

(n+τ1−1+t )

(n+τ1−1) , (26)

so that the result is a consequence of the Stirling formula.

(2) The Taylor expansion ofγτ1,n(zId+ΦN)leads to Ef (Xn)=

p0

1

p!γτ(p)1,n(z)ΦNp(f )(X1)

(finite sum), whereγτ(p)1,ndenotes thepth derivative ofγτ1,n. Besides, ifpis any positive integer, γτ(p)1,n(z)=nzlogpn 1)

1+z)+O

nzlogp1n

(27) whenntends to infinity, as can be shown by the Stirling formula (see (26)) and an elementary induction starting from the computation ofγτ1,n’s logarithmic derivative. These two facts imply the result.

Remark 4.3. As it is written,Corollary4.2is valid only if the complex numberτ1+zis not a nonpositive integer.We adopt the convention1/ (w)=0whenw∈Z0,so that this corollary is valid in all cases.

Remark 4.4. Iff:VW is linear,formula(8)implies thatΦ(f )=fA.In that particular case,formula(24) givesEf (Xn)=fγτ1,n(A)(X1).This fact will be used in the proofs of Theorems3.5an3.6whenf is a linear combination of projectionsπk(see Section6).

4.2. Action ofΦon polynomials

Because of condition (2) in the definition of a Pólya process, none of the vectorswk is zero. For anyk, iff is a function defined onV, we denote by∂f/∂wk, when it exists, the derivative off along the direction carried by the vectorwk. With this notation, we associate with the finite difference operatorΦ the differential operatorΦ defined by

Φ(f )(v)=

1ks

lk(v) ∂f

∂wk(v) (28)

for every functionf defined onV and derivable at each point along the directions carried by the vectorswk’s. When f is differentiable,Φ(f )can be viewed as a “first approximation” ofΦ(f ). As derivation behaves well with respect to the product of functions when finite differentiation does not,Φ(f )is helpful for the understanding ofΦ’s action on polynomials.

Remark 4.5. The differential operator can be written asΦ(f )(v)=Dfv·Av for any differentiable function f, whereDfvdenotes the differential off at pointv.This can be readily seen from the formulaDfv·wk=∂w∂fk(v).

Proposition 4.6 (Action ofΦ on the u-monomials). For any choice of a Jordan basis(uk)1ks of linear forms of a Pólya process,

(1) for everyα(Z0)s, Φ

uα

α, λuα∈Span

uβ, β < α

;

(2) ifα(Z0)sis a semisimple power,thenΦ(uα)= α, λuα.

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