Pierre-Yves Louis
email: pierre-yves.louis@agrosupdijon.fr

Meghdad Mirebrahimi
email: me.mirebrahimi@gmail.com

Synchronization and Fluctuations for Interacting Stochastic Systems with Individual and Collective Reinforcement

Keywords: 2010 Mathematics Subject Classification. Primary 60K35, 60F15, 60F05, Secondary 62L20, 62P35 Reinforced stochastic processes, Interacting random systems, Almost sure convergence, Central limit theorems, stable convergence, synchronization, Fluctuations. 1

The Pólya urn is the most representative example of a reinforced stochastic process. It leads to a random (non degenerated) time-limit. The Friedman urn is a natural generalization whose almost sure (a.s.) time-limit is not random any more. In this work, in the stream of previous recent works, we introduce a new family of (finite size) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one component-wise, one collective) are tuned through (possibly) two different rates. In special cases, these reinforcements are of Pólya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. Different parameter regimes need to be considered.

We state two kind of results. First, we study the time-asymptotics and show that L 2 and a.s. convergence always holds. Moreover all the components share the same time-limit (so called synchronization phenomenon). We study the nature of the limit (random/deterministic) according to the parameters' regime considered. Second, we study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into many different rates of convergence. In particular, we identify the regimes where synchronization is faster than convergence towards the shared time-limit.

Introduction and model's definition

In urn models, it is well known that the bicolor Pólya reinforcement rule (reinforcement of the chosen color) leads to a random limiting a.s. proportion whereas the Friedman rule (reinforcement of the chosen color as well as the non chosen color) leads to a deterministic time-asymptotics proportion. This somewhat surprising fact is explained for instance through a fix point [START_REF] Hill | A Strong Law for Some Generalized Urn Processes[END_REF]. See [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF] too. Following many recent works (see Section 3 for details), this paper is motivated by the study of asymptotics time behavior of models of (discrete time) stochastic processes interacting through a reinforcement rule.

Let us define the following new model. Let (Z n (i), 1 ≤ i ≤ N ) n≥0 be an N -dimensional stochastic process with values in [0, 1] N where N ≥ 2. For each component indexed by i (1 ≤ i ≤ N ) we consider the stochastic dynamics defined through the recursive equation, for any n ∈ N,

Z n+1 (i) = (1 -r n -r g n )Z n (i) + r n ξ n+1 (i) + r g n ξ g n+1 , (1.1)
where Z 0 (i) = 1 2 and where ξ n+1 (i) and ξ g n+1 denote local (component-wise) and collective reinforcements. Let (F n ) n be the natural filtration associated with (Z n (i), 1 ≤ i ≤ N ) n≥0 . Given the σ-field F n , they have independent Bernoulli distributions with

P(ξ n+1 (i) = 1|F n ) = ψ 1 (Z n (i)) := (1 -2λ 1 )Z n (i) + λ 1 ,
(1.2)

P(ξ g n+1 = 1|F n ) = ψ 2 (Z n ) := (1 -2λ 2 )Z n + λ 2 , (1.3)
where ψ k : [0, 1] -→ [0, 1] (k ∈ {1, 2}) are linear maps with Z n := 1 N N i=1 Z n (i) (so called mean field ) and where λ 1 , λ 2 ∈ [0, 1] are parameters. The local (resp. the collective) reinforcement rate (r n ) n (resp. (r g n ) n ) are real sequences such that 0 ≤ r n < 1 and 0 ≤ r g n < 1, and lim

n n γ 1 r n = c 1 > 0 and lim n n γ 2 r g n = c 2 > 0 (1.4)
where γ j ∈ ( 1 2 , 1] for j ∈ {1, 2}. This assumption gives that (r n ) n (resp. (r g n ) n ) satisfy the following usual assumptions for processes defined through recursive equations like (1.1)

r n = +∞, (r n ) 2 < +∞.
(1.5) Note (r n ) 2 stand for the square of r n . As emphasized in [START_REF] Aletti | Networks of reinforced stochastic processes: Asymptotics for the empirical means[END_REF], in order to state the CLTs, we actually assume the slightly stricter following assumptions.

Assumption 1.1. There exist real constants γ 1 , γ 2 and c 1 > 0,

c 2 > 0 with (γ 1 , γ 2 ) ∈ ( 1 2 , 1] 2 , such that when n → ∞ r n = c 1 n γ 1 + O 1 n 2γ 1 and r g n = c 2 n γ 2 + O 1 n 2γ 2 .
(1.6)

When the two reinforcement rules compete through the different rates r n , r g n , one individual rule, one collective (in the sense all the components are involved), is there one leading? In particular, is there loss of synchronization which means, is there an almost sure limit depending on the component? Moreover, in the case were Pólya and Friedman type of reinforcement rule do compete through the system, and if there is synchronization, may the shared time-limit be random?

As emphasized in the previous works, there are many applications these stochastic models may be useful for. Urn models are well known [START_REF] Mahmoud | Polya Urn Models[END_REF] to have applications in economy, in contagion models [START_REF] Harrington | Infection-Curing Games over Polya Contagion Networks[END_REF], in clinical trials adaptive design [START_REF] Laruelle | Urn Model-Based Adaptive Multi-arm Clinical Trials: A Stochastic Approximation Approach[END_REF], random networks Hofstad (van der). In general, the reinforcement rate (r n ) n , may be such that lim n n γ r n = c > 0. The dynamics is nothing but a vector-valued stochastic algorithm [START_REF] Benaïm | Dynamics of Stochastic Approximation Algorithms[END_REF]; [START_REF] Duflo | Random Iterative Models[END_REF]. Such processes have many applications like in the framework of stochastic optimization (see for instance [START_REF] Delyon | Convergence of a Stochastic Approximation Version of the EM Algorithm[END_REF][START_REF] Gadat | A Stochastic Algorithm for Feature Selection in Pattern Recognition[END_REF]. In [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF] and in [START_REF] Aletti | Synchronization of reinforced stochastic processes with a network-based interaction[END_REF] an application of these processes as opinion dynamics was introduced. We will briefly explain it in our context in Section 3 : each random variable Z n (i) may be considered as an agent's inclination towards one of two possible choices.

In the new family of models we introduce and study in this paper, following previous recent works Dai [START_REF] Dai Pra | Synchronization via interacting reinforcement[END_REF]; Sahasrabudhe (2016); [START_REF] Crimaldi | Fluctuation theorems for synchronization of interacting Pólya's urns[END_REF]; [START_REF] Aletti | Synchronization of reinforced stochastic processes with a network-based interaction[END_REF][START_REF] Aletti | Networks of reinforced stochastic processes: Asymptotics for the empirical means[END_REF][START_REF] Aletti | Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means[END_REF]; [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF]; [START_REF] Mirebrahimi | Interacting stochastic systems with individual and collective reinforcement[END_REF], we consider a (finite) system of reinforced stochastic processes defined though recursive equations (1.1). Two kind of reinforcement are involved. One depending only on the component (ξ n+1 (i)) n (see next section for the notations), one creating the interaction (ξ g n+1 ) n and depending on the average over all components. The interaction holds through the reinforcement. This is modeling a collective reinforcement effect that may compete with the individual/local/component-wise reinforcement. For the sake of simplicity, we choose to consider a mean field interaction, in the sense that collective reinforcement depends on the arithmetic mean over the system at previous time step. Each reinforcement has its own rate r n (resp. r g n ). Each rate may have its own asymptotic behavior: r n ∼ c 1 n -γ 1 (resp. r g n ∼ c 2 n -γ 2 ). If 1/2 < γ 1 < γ 2 ≤ 1 (for instance), one may expect the collective reinforcement to become negligible in the long run. A naive guess could be, the system behaves for large time like a system with independent components, thus leading to a possible loss of synchronization. We prove later this does not happen: L 2 /a.s. synchronization holds (meaning, each component dynamics shares the same random time limit) like in Dai [START_REF] Dai Pra | Synchronization via interacting reinforcement[END_REF]. Additional issues we address, are: nature (deterministic/random, diffuse or atomic) of the almost sure (a.s.) time limit distribution according to the type of reinforcement, scales of fluctuations with respect to this limit, which are stated by proving central limit theorems (CLT) w.r.t stable convergence. We prove, according to the parameters' cases, that the rate of synchronization is quicker, or the same, as the speed of convergence to the limit. In the models considered in [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF], synchronization quicker than convergence towards the asymptotics value Z ∞ holds only in cases where Var(Z ∞ ) > 0. In the following models it may happen even when Z ∞ is deterministic.

The paper is organized as follows. In Section 2 we give some preliminary remarks about the model's definition. In Section 3 we compare the model with related families studied in previous works and give some interpretation as models for opinion dynamics. In Section 4 we prove that L 2 and a.s. convergence holds towards a limiting value Z ∞ in R shared by all the components (synchronization). Two main cases are to be distinguished: Theorems 4.1 and 4.2 deals with cases where Z ∞ = 1/2 (the word synchronization is abusive in this situation) whereas Theorem 4.4 deals with Z ∞ random. In Section 5, in the different cases, we state central limit theorems about the fluctuations of (Z n -Z ∞ ) and (Z n -Z n (i)). Scaling factors are worth of interest. Th. 5.1 consider the case where each individual and collective reinforcement leads to a deterministic limiting value. Theorem 5.2 consider the special case when γ 1 = γ 2 = 1 reminiscent of the Friedman urn context, in the regime where fluctuations are known not to be gaussian (c 1 λ 1 + c 2 λ 2 < 1 4 ). Theorem 5.3 deals with the mixed cases where individual and reinforcement type are of different nature. Th. 5.4 consider the case where both the individual and the collective reinforcement lead to a random limit. Section 6 is dedicated to comments on the model from a stochastic approximation perspective. Section 7 is dedicated to the proof of the synchronization. Section 8 deals with the proofs of the CLTs. An appendix A states and recalls for the sake of completeness some technical results.

Preliminary remarks

The model is define through the recursive equations (1.1), with (1.2), and general assumption (1.2) about the rates' asymptotic behavior. We refer to [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF] for a discussion on the case 0 < γ ≤ 1/2 in another model. We believe that the case 0 < γ ≤ 1/2 leads to a drastically different behavior, and therefore leave this case for further studies.

For the sake of simplicity, we choose to have some symmetry in the model with respect to 1/2. For the same reason, according to the previous works cited in the Section 3, we consider the starting conditions all equal to 1/2 without loss of generality.

Remark 2.1. In the particular case when γ 1 = γ 2 =: γ, we can rewrite the model as

Z n+1 (i) = (1 -2r n )Z n (i) + r n ξn+1 (i),
where ξn+1 (i) = ξ n+1 (i) + ξ g n+1 , therefore ξn+1 (i) ∈ {0, 1, 2}. The other probabilities in (1.2) and their subsequent relations may be computed in an analogous way. The reinforcement rate remains such that r n ∼ cn -γ when n → ∞.

Remark 2.2. In this paper the parameters λ 1 , λ 2 are kept fixed. Cases where λ 1 , λ 2 may converge to 0 depending on n, N will be considered in a forthcoming work. Note that when λ 1 = λ 2 = 1/2 (see (1.2)), there is no reinforcement. Moreover, when λ 1 = 1/2 and λ 2 = 1/2, although we still have reinforcement on each component individually, we loose the interaction between components.

Remark 2.3. If 0 ≤ λ 1 < 1/2 and 0 ≤ λ 2 < 1/2, then ψ 1 and ψ 2 are increasing maps and occurrence of events {ξ n (i) = 1} increases the probability of having {ξ n+1 (i) = 1} at next time step. This is the basic original meaning of reinforcement.

The following relationships hold.

E(Z n+1 (i) -Z n (i)|F n ) = λ 1 r n 1 -2Z n (i) + λ 2 r g n 1 -2Z n + r g n Z n -Z n (i) , (2.1)
and by averaging over i in {1, . . . , N }, we have

E(Z n+1 -Z n |F n ) = (λ 1 r n + λ 2 r g n ) (1 -2Z n ).
(2.2)

Related models and application motivations

As emphasized in [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF], the evolution of proportions in urn models satisfies recursive equation like (1.1) with r g n = 0 (i.e. without interaction) and with γ 1 = 1. This family of models we introduce is related to some other models that were studied recently. We briefly present them in this section.

3.1. Interacting urns models. The models considered in this paper were introduced in Mirebrahimi (2019). When r g n ≡ 0 and γ 1 = 1, the model is a non interacting system where each component's value Z n (i) can be interpreted as the proportion of balls of a given color in a bicolor balanced (same deterministic number of balls are added whatever the chosen color is) urn classic model. The case λ 1 = 0 leads to ψ 1 (x) = x which is the basic Pólya reinforcement rule, where a fix number of balls is added, whose color is the same as the chosen one. It is well known lim n→∞ Z n (i) exists a.s. and defines a Beta distributed random variable Z ∞ (i), whose parameters depend on the initial number of balls of each colors. The case λ 1 = 0 leads to a Friedman urn model. The unique fix point of ψ 1 is 1/2. It is known, lim n→∞ Z n (i) exists a.s. and is equal to this fix point Z ∞ (i) = 1/2 := Z ∞ . See for instance [START_REF] Hill | A Strong Law for Some Generalized Urn Processes[END_REF] where very interesting cases of less regular ψ 1 maps are considered. This can be proven using stochastic approximation results [START_REF] Benaïm | Dynamics of Stochastic Approximation Algorithms[END_REF]; [START_REF] Duflo | Random Iterative Models[END_REF]. In our context we use the terminology Pólya type when λ 1 = 0 and Friedman type when λ 1 = 0. A similar remark holds for the collective reinforcement effect ruled by ψ 2 and tuned through γ 2 for the asymptotic behavior of the reinforcement rate r g n and λ 2 for the reinforcement's type. In Dai [START_REF] Dai Pra | Synchronization via interacting reinforcement[END_REF], the system introduced and studied is related to the case γ 1 = γ 2 = 1 and λ 1 = λ 2 = 0. It solves Equation (1.1) with r g n ≡ 0 and with

P(ξ n+1 (i) = 1|F n ) = (1 -α)Z n (i) + αZ n
where α ∈ [0, 1] is a parameter. A.s. synchronization towards a random shared limit Z ∞ (Var(Z ∞ ) > 0) was proved as soons as α > 0. Fluctuations were studied in [START_REF] Crimaldi | Fluctuation theorems for synchronization of interacting Pólya's urns[END_REF] by proving central limit theorems. In Sahasrabudhe (2016), a similar interacting model was studied, whose components dynamics can be interpreted as urn models (reinforcement's rate behaving like n -1 ) with a Friedman reinforcement rule. A.s. convergence holds towards a unique deterministic value. Moreover, it is known that Friedman urns can exhibit non gaussian fluctuations [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized Polya urns[END_REF]; [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF]. See [START_REF] Flajolet | Some exactly solvable models of urn process theory[END_REF]; [START_REF] Chauvin | Smoothing equations for large Pólya urns[END_REF]; [START_REF] Lasmar | Multiple drawing multi-colour urns by stochastic approximation[END_REF]; [START_REF] Mailler | Describing the asymptotic behaviour of multicolour Pólya urns via smoothing systems analysis[END_REF]; [START_REF] Chauvin | Limit distributions for large Pólya urns[END_REF] for more specific recent results about urn models and generalizations. In Sahasrabudhe (2016), this was proven to have consequences for the mean-field interacting system, where different speed of convergence may happen. In relationship with systems of interacting urns, some variations with different kind of urns/reinforcements bias towards one or the other color, were considered in [START_REF] Louis | Probabilistic Cellular Automata. Theory, Applications and Future Perspectives[END_REF].

For the model considered in this paper, similar interpretation as bicolor balanced urn model can be made, when γ 1 = γ 2 = 1. Each Z n (i) can be interpreted as the proportion of one chosen color in an urn i. Two reinforcement mechanism hold which can be related to the following one, applied to each urn i of the systems, independently, between two iterations. One (for instance) ball is chosen uniformly at random in the urn i and one (for instance) ball of the chosen color is added into urn i (Pólya reinforcement type, λ 1 = 0); resp. one ball of the non chosen color, Friedman type, λ 1 = 0). Additionally, one ball is chosen uniformly at random in the whole system (proportion Z n at previous time step) and one ball of the chosen color is added (collective reinforcement of Pólya type, λ 2 = 0) into urn i; resp. one ball of the non chosen color, collective reinforcement of Friedman type, λ 2 = 0). In fact adding a ball of the non chosen color is similar in general to add a ball of the chosen color as well as the non chosen color. The reinforcement matrices (for local reinforcement, resp. for collective reinforcement) defining these numbers are then giving the exact values of λ 1 (resp. λ 2 ).

3.2. General reinforcements' rates. Generalizing the reinforcement rate r n asymptotic behavior from r n ∼ cn -1 to r n ∼ n→∞ cn -γ leads to systems of stochastic processes with reinforcement which can be considered as interacting stochastic algorithms of Robbins-Monro type. In [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF] several cases of reinforcement (like Pólya/Friedman) were considered. A.s. synchronization was stated towards different kind of limit Z ∞ (deterministic or not) and speed of convergence studied through functional central limit theorems (FCLT) for Z n (i)-Z n and Z n -Z ∞ . It was proved that in parameters' regime where the time limit Z ∞ is random (in the sense Var(Z ∞ ) > 0), synchronization happens quicker than convergence to the time limit.

Building a reinforcement with the average proportion Z n (mean field) helps in these interacting systems since it is enough to deal with closed uni-dimensional recursive equations for (Z n ) n and (Z n -Z n (i)) n . The interaction was generalized from mean field to network based interaction in [START_REF] Aletti | Synchronization of reinforced stochastic processes with a network-based interaction[END_REF], with a reinforcement of Pólya type. The system dynamics is defined, for i ∈

V := {1, • • • , N }, through Z n+1 (i) = (1 -r n )Z n (i) + r n ξ n+1 (i) (3.1)
where for any n ≥ 0, the random variables {ξ n+1 (i) : i ∈ V } are conditionally independent given F n with

P(ξ n+1 (i) = 1 | F n ) = N j=1 w j,i Z n (j) (3.2)
with F n := σ(Z n (i)). The non negative matrix W = [w j,i ] j,i∈V ×V is considered as a weighted adjacency matrix of the graph G = (V, E) with V = {1, ..., N } as the set of vertices and E⊂V × V as the set of directed edges. Each edge (j, i) ∈ E represents the fact that the vertex j has a direct influence on the vertex i. The weight w i,j ≥ 0 quantifies how much j can influence i. The weights are assumed to be normalized W 1 = 1 where 1 denotes (1, . . . , 1) ∈ R N . The matrix W is assumed to be irreducible and diagonalizable. The reinforcement rate r n is assumed to satisfy (1.4) with γ ∈ (1/2, 1] or a more restrictive condition as (1.6). Synchronization is proven to hold and CLT's were stated. The empirical means N n (i

) := N -1 N i=1 ξ n (i)
i ∈ V . To each vertex i ∈ V is associated a value Z n (i) ∈ [0, 1]. The quantity Z n (i) (resp. 1 -Z n (i))
is interpreted as the inclination to adopt the choice 1 (resp. 0) at time n. The recursive equation (1.1) means the inclination of agent i at the next time step, is a convex combination of

• the inclination Z n (i) with self-reinforcement weight 1 -r n -r g n ,
• a choice ξ n+1 (i) made with a probability ψ 1 (Z n (i)) for opinion 1 related to the personal inclination, with a weight r n , • and a collective choice ξ g n+1 made with a probability ψ 2 (Z n ) for opinion 1 related to the collective inclination Z n , average of the personal inclinations, with a weight r g n . As time goes, the rates r n and r g n vanish. For larger n, the self-reinforcement leads. The different speed of convergence towards 0 for r n and r g n , tuned by γ 1 , γ 2 could mean the influence of collective actions may disappear quicker than the influence of individual choices. On the contrary, the a.s. synchronization phenomenon towards a shared inclination Z ∞ could be interpreted as the emergence of a consensus, in the sense every individual shares the same inclination. The special case Z ∞ = 1/2 may be interpreted as a complete undetermined "fifty-fifty" inclination towards the two actions.

Issues we addressed at the beginning of this paper may be reconsidered through this interpretation.

Main results: convergence and a.s. synchronization

In this section we state result on the convergence of (Z n ) n and the synchronization phenomenon. Indeed we obtain different kind of time-limit (deterministic or (truly-)random) for (Z n ) n according to the nullity of λ 1 , λ 2 . Moreover L 2 and a.s. synchronization are stated in all the cases. As previously mentioned, the choice of the mean-field instead of a network-based interaction allows us to address the proofs by studying Z n and Z n -Z n (i) instead of dealing with R N valued recursive equations.

4.1. Case of a deterministic time-asymptotics. We call the case deterministic, if the time limit Z ∞ (n → ∞) is not random (Var(Z ∞ ) = 0). This behavior corresponds to cases where at least one of the following assumptions is true: λ 1 > 0 or λ 2 > 0. The mean field process (Z n ) n is not a martingale. In order to investigate the behavior of the interacting system, we first consider the time limits of Var(Z n ). Second we show that L 2 -synchronization holds i.e. lim n→∞ Var(Z n (i)-Z n ) = 0. We get the rates of convergence too. Finally, we prove that the synchronization holds almost surely and the deterministic limit is Z ∞ := 1 2 . Theorem 4.1. For any λ 1 > 0 and λ 2 > 0 the following results hold:

i) Asymptotics of variance as n → ∞:

Var(Z n ) = O( 1 n γ * ) where γ * := min(γ 1 , γ 2 ); ii) behavior of the L 2 -distance between Z n and Z n (i) when n → ∞: a) if γ 1 ≤ γ 2 , then E([Z n -Z n (i)] 2 ) = O( 1 n γ 1 ), b) if γ 2 < γ 1 , then E([Z n -Z n (i)] 2 ) = O( 1 n 2γ 1 -γ 2 ); iii) almost sure convergence holds i.e. ∀i ∈ {1, ..., N }, lim n→+∞ Z n (i) = lim n→+∞ Z n = 1 2 =: Z ∞ a.s.
Two others choices of parameters λ 1 , λ 2 lead to the following results.

Theorem 4.2. In the following cases: either (λ 1 > 0 and λ 2 = 0) or (λ 1 = 0 and λ 2 > 0) it holds lim n→+∞ Z n (i) = lim n→+∞ Z n = 1 2 a.s. Moreover, the following table summarizes the L 2 speed of convergence.

λ 1 = 0, λ 2 = 0 λ 1 = 0, λ 2 = 0 γ 1 ≤ γ 2 Var(Z n ) = O( 1 n γ 1 ) Var(Z n ) = O( 1 n 2γ 1 -γ 2 ) E([Z n -Z n (i)] 2 ) = O( 1 n γ 1 ) E([Z n -Z n (i)] 2 ) = O( 1 n 2γ 1 -γ 2 ) γ 2 < γ 1 Var(Z n ) = O( 1 n 2γ 2 -γ 1 ) Var(Z n ) = O( 1 n γ 2 ) E([Z n -Z n (i)] 2 ) = O( 1 n 2γ 1 -γ 2 ) E([Z n -Z n (i)] 2 ) = O( 1 n 2γ 1 -γ 2 )
Remark 4.3. (Comparison of convergence and synchronization rates)

In the case λ 1 > 0, λ 2 > 0, when γ 1 < γ 2 , the L 2 convergence rate of (Z n ) n to 1 2 and the L 2 rate of convergence of (Z n (i) -Z n ) n to 0 are the same. However, when γ 2 < γ 1 , we obtain that synchronization happen faster than convergence.

Moreover in the case λ 1 > 0, λ 2 = 0 and when γ 1 < γ 2 , the speed of convergence and synchronization are the same (n -γ 1 ). While when γ 2 < γ 1 , the synchronization is faster than convergence.

Similarly, in the case λ 1 = 0, λ 2 > 0 and when γ 1 ≤ γ 2 , the speed of convergence and synchronization are the same (n -(2γ 1 -γ 2 )), while when γ 2 < γ 1 , the speed of synchronization is faster than convergence (n -(2γ 1 -γ 2 ) and n -γ 2 respectively). 4.2. Case of a shared random time-asymptotics. Unlike to the previous cases, the case λ 1 = λ 2 = 0 yields (Z n ) n is a martingale. We will prove it leads to a random time-asymptotics Z ∞ (Var(Z ∞ ) > 0). We will study the system's time-asymptotics behavior in a similar way as in the previous cases. First we show that lim n→∞ Var(Z n ) = 0. Second we prove that L 2 -synchronization holds. Third we state the almost sure synchronization holds.

Theorem 4.4. When λ 1 = λ 2 = 0, i) it holds (n → ∞) Var(Z n ) > 0.
In particular (Z n ) n converges a.s. to a non-degenerated random limit denoted by Z ∞ (Var(Z ∞ ) > 0). ii) The L 2 -distance between the mean field Z n and each component Z n (i) behaves as follows,

E[(Z n (i) -Z n ) 2 ] = O 1 n 2γ 1 -γ 2 and synchronization holds almost surely. It means, for all i ∈ {1, . . . , N }, lim n→∞ Z n (i) = Z ∞ a.s.

Main results: fluctuations through CLT

In this section we study the fluctuations of (Z n (i) -Z n ) n (synchronization) w.r.t 0 and also fluctuations of (Z n ) n w.r.t its limit Z ∞ . These are studied by stating Central Limit Theorems. Pay attention to the fact that different scalings hold according to (γ 1 , γ 2 ) relationship. We follow the proof's techniques initiated for these models in [START_REF] Crimaldi | Fluctuation theorems for synchronization of interacting Pólya's urns[END_REF] based on Theorem A.5 in Appendix, which leads to stable convergence results.

We first study cases where Z ∞ = 1 2 . Theorems 5.1, 5.2 deal with the case λ 1 > 0 and λ 2 > 0. Moreover, we show that there is some special regime when 0 < (c 1 λ 1 + c 2 λ 2 ) < 1 4 . The Theorem 5.3 describes the results of the cases where exactly one of the λ j is 0.

Finally we state the behavior when Var(Z ∞ ) > 0 in Theorem 5.4.

The following statements hold, where the generic symbol σ 2 denotes the variances (depending on N and λ 1 , λ 2 ) make more precise in proofs. In the proofs of Sections 7 and 8 we used c 1 = c 2 = 1 to simplify. Following statements are nevertheless formulated in full generality.

Theorem 5.1. Let λ 1 > 0, λ 2 > 0; let γ * := min(γ 1 , γ 2 ). i) It holds a) when γ 1 ≤ γ 2 , n γ 1 2 (Z n -Z n (i)) stably ---→ n→∞ N 0, σ2 1 , b) when γ 2 < γ 1 , n 2γ 1 -γ 2 2 (Z n -Z n (i)) stably ---→ n→∞ N 0, σ2 2 . ii) When γ * < 1, it holds n γ * 2 (Z n - 1 2 ) stably ---→ n→∞ N 0, σ2 . iii) When γ * = 1, a) if (c 1 λ 1 + c 2 λ 2 ) > 1 4 , √ n(Z n - 1 2 ) stably ---→ n→∞ N 0, σ * 2 1 . b) if (c 1 λ 1 + c 2 λ 2 ) = 1 4 , √ n √ ln n (Z n - 1 2 ) stably ---→ n→∞ N 0, σ * 2 2 .
Theorem 5.2. Let λ 1 > 0, λ 2 > 0. When γ * = 1 and when (c

1 λ 1 + c 2 λ 2 ) < 1 4 , the following statement holds n 4(c 1 λ 1 +c 2 λ 2 ) (Z n - 1 2 ) a.s./L 1 ----→ n→∞ X,
for some real random variable X such that P( X = 0) > 0.

This regime is related to the known non gaussian fluctuation regime of the Friedman urn (see for instance Th. 2.9 (ii) in [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF] or Th. 4 and 5 in Sahasrabudhe (2016) were additive assumptions need to be used).

Two other main cases leads to following results. For the sake of readability, the asymptotic variances are detailed in the proofs.

Theorem 5.3. In the following cases: either (λ 1 > 0, λ 2 = 0) or (λ 1 = 0, λ 2 > 0), the stable convergence towards some Gaussian distribution holds for the quantities (Z n -Z n (i)) n and (Z n -1 2 ) n . The following tables summarizes the different scales according to the relationship between γ 1 , γ 2 . The first table deals with γ * := min(γ 1 , γ 2 ) < 1.

λ 1 = 0, λ 2 = 0 λ 1 = 0, λ 2 = 0 γ 1 ≤ γ 2 n γ 1 2 (Z n -Z n (i)) stably ---→ N 0, σ2 3 n 2γ 1 -γ 2 2 (Z n -Z n (i)) stably ---→ N 0, σ2 4 n γ 1 2 (Z n -1 2 ) stably ---→ N 0, σ2 1 n 2γ 1 -γ 2 2 (Z n -1 2 ) stably ---→ N 0, σ2 2 γ 2 < γ 1 n 2γ 1 -γ 2 2 (Z n -Z n (i)) stably ---→ N 0, σ2 5 n 2γ 1 -γ 2 2 (Z n -Z n (i)) stably ---→ N 0, σ2 6 n 2γ 2 -γ 1 2 (Z n -1 2 ) stably ---→ N 0, σ2 3 n γ 2 2 (Z n -1 2 ) stably ---→ N 0, σ2 4
The following second table holds when γ * = 1. The indices i and j are different and belongs to {1, 2}.

λ i = 0, λ j > 1 4 λ i = 0, λ j = 1 4 λ i = 0, λ j < 1 4 √ n(Z n -1 2 ) stably ---→ N 0, σ * 2 3 √ n √ ln n (Z n -1 2 ) stably ---→ N 0, σ * 2 4 n 4(λ 1 +λ 2 ) (Z n -1 2 ) a.s./L 1 ----→ χ Theorem 5.4. Assume λ 1 = λ 2 = 0.
The stable convergence towards some Gaussian kernel holds for the quantities

(Z n -Z n (i)) n and (Z n -1 2 ) n with the following scalings. (i) It holds n 2γ 1 -γ 2 2 (Z n -Z n (i)) stably ---→ n→∞ N 0, ϑ Z ∞ (1 -Z ∞ ) . (ii) With γ * := min(γ 1 , γ 2 ), it holds n 2γ * -1 2 (Z n -Z ∞ ) stably ---→ n→∞ N 0, ϑ Z ∞ (1 -Z ∞ ) ,
where ϑ denotes a constant, whose dependency according to N , γ 1 , γ 2 is given in the proofs.

Remark 5.5. (analogous to Theorem 3.2 in [START_REF] Crimaldi | Fluctuation theorems for synchronization of interacting Pólya's urns[END_REF]).

We have P(Z ∞ = 0) + P(Z ∞ = 1) < 1 and P(Z ∞ = z) = 0 for each z ∈ (0, 1). Indeed, it guarantees that these asymptotic Gaussian kernels are not degenerate.

Proof : The first part immediately follows from the relation E

[Z 2 ∞ ] < E[Z ∞ ] by Lemma 7.2.
The second part is a consequence of the almost sure conditional convergence stated in Th. 5.4 (ii) (for details see proof of Theorem 2.5 in [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF].

Stochastic approximation point of view

The recursive equations (1.1) may be written in the following stochastic approximation forms:

Z n+1 (i) = Z n (i) + r n λ 1 (1 -2Z n (i)) + r n ∆ M n+1 (i) + r g n λ 2 (1 -2Z n ) + r g n (Z n -Z n (i)) + r g n ∆M g n+1 (6.1)
and

Z n+1 = Z n + (r n λ 1 + r g n λ 2 )(1 -2Z n ) + r n ∆M n+1 + r g n ∆M g n+1 (6.2)
where

∆ M n+1 (i) := ξ n+1 (i) -E(ξ n+1 (i)|F n ), (6.3
)

∆M n+1 := 1 N N i=1 ∆ M n+1 (i), (6.4) ∆M g n+1 := ξ g n+1 -E(ξ g n+1 |F n ) (6.5) are martingale differences. Similarly, it holds for X n (i) := Z n -Z n (i), X n+1 (i) = X n (i) -(2λ 1 r n + r g n )X n (i) + r n (∆M n+1 -∆ M n+1 (i)). (6.6)
We refer to the general theorems about asymptotic behavior as stated in Laruelle and Pagès (2013); [START_REF] Lasmar | Multiple drawing multi-colour urns by stochastic approximation[END_REF]; [START_REF] Laruelle | Nonlinear randomized urn models: a stochastic approximation viewpoint[END_REF] and classical references therein like [START_REF] Duflo | Random Iterative Models[END_REF]; [START_REF] Benaïm | Dynamics of Stochastic Approximation Algorithms[END_REF]. According to the cases either γ 1 ≤ γ 2 or γ 2 < γ 1 and λ i = 0 or not (i ∈ {1, 2}), then system Z n = (Z n (1), . . . , Z n (N )) satisfies the following framework.

Let Z = (Z n ) n≥0 be an N -dimensional stochastic process with values in [0, 1] N , adapted to a filtration F = (F n ) n≥0 . Suppose that Z satisfies

Z n+1 = Z n + r n F(Z n ) + r n ∆M n+1 + r n ζ n+1 , (6.7)
where (r n ) n is such that (1.5) hold; F is a bounded 2016) covers all parameters' cases, including the one when λ 1 = λ 2 = 0; and they give L 2 rates. These are useful to prove the scales of fluctuations stated in Section 5 thanks to CLT's w.r.t stable convergence [START_REF] Crimaldi | Introduzione alla nozione di convergenza stabile e sue varianti (Introduction to the notion of stable convergence and its variants)[END_REF]; [START_REF] Häusler | Stable convergence and stable limit theorems[END_REF].

C 1 vector-valued function on an open subset O of R N , with [0, 1] N ⊂ O; (∆M n ) n is a bounded martingale difference with respect to F; and (ζ n ) n is a [0, 1] N -valued F n+1 -

Proof of a.s. synchronization and rates of convergences

This section is devoted to the proofs of Th. 4.1, Th. 4.2, Th. 4.4. As discussed in Section 6 cases need indeed to be distinguished according to the nullity of λ 1 , λ 2 (λ 1 + λ 2 > 0 or λ 1 = λ 2 = 0).

(i) = 1 2 leads to ∀n ∈ N, E(Z n ) = E(Z n (i)) = 1
2 thanks to (2.1) and (2.2). We then state the following relationships.

Proposition 7.1. The following recursive equation hold:

Var(Z n+1 ) = [1 -4 λ 1 r n + λ 2 r g n -2λ 1 λ 2 r n r g n -λ 2 1 (r n ) 2 -λ 2 2 (r g n ) 2 + (r g n ) 2 4 (1 -2λ 2 ) 2 ]Var(Z n ) + (r n ) 2 N [(1 -2λ 1 ) 2 1 2 - 1 N N i=1 E(Z 2 n (i)) + λ 1 -λ 2 1 ] + (r g n ) 2 4 . (7.1)
Proof : From (1.1), we get

Z n+1 = (1 -r n -r g n )Z n + r n N i ξ n+1 (i) + r g n ξ g n+1 .
Using (1.1) and (1.2), we compute:

Var(Z n+1 |F n ) = Var (1 -r n -r g n )Z n + r n N i ξ n+1 (i) + r g n ξ g n+1 F n = (r n ) 2 N 2 N i=1 Var(ξ n+1 (i)|F n ) + (r g n ) 2 Var(ξ g n+1 |F n ),
then using (2.1),(2.2) and the law of total variance ( ), we have

Var(Z n+1 ) = E[Var(Z n+1 |F n )] + Var[E(Z n+1 |F n )] = (r n ) 2 N (1 -2λ 1 ) 2 E(Z n ) - 1 N N i=1 E(Z 2 n (i)) + λ 1 -λ 2 1 + (r g n ) 2 (1 -2λ 2 ) 2 (E(Z n ) -E(Z 2 n )) + λ 2 -λ 2 2 + (1 -2λ 1 r n -2λ 2 r g n ) 2 Var(Z n )
and using the fact Z 0 (i) = 1/2 leading to the result.

Lemma 7.2. When λ 1 = λ 2 = 0, it holds lim n→∞ Var(Z n ) < 1 4 . Moreover, sup n E(Z 2 n ) < 1 2 .
Remark this implies

lim n→∞ 1 2 - 1 N N i=1 E(Z 2 n (i)) > 0. (7.2) Proof : Since for all i, E(Z 2 n (i)) ≤ E(Z n (i)) = 1 2 , it holds obviously Var(Z n ) ≤ 1 4 . Using (7.1) with λ 1 = λ 2 = 0 gives: Var(Z n+1 ) = 1 -(r g n ) 2 Var(Z n ) + (r g n ) 2 4 + (r n ) 2 N 1 2 - 1 N N i=1 E(Z 2 n (i)) .
Then using the fact that

1 2 - 1 N N i=1 E(Z 2 n (i)) ≤ 1 2 (7.3)
we have

Var(Z n+1 ) ≤ 1 -(r g n ) 2 Var(Z n ) + (r g n ) 2 4 + (r n ) 2 2N . Let x n := Var(Z n ) -1 4 ≤ 0, one gets x n+1 ≤ 1 -(r g n ) 2 x n + (r n ) 2 2N . With K n = 1 2N , take x n+1 = 1 -(r g n ) 2 x n + K n (r n ) 2 . Since n (r g n ) 2 < +∞, we obtain lim n→∞ x n = 0 by lemma A.2. As claimed, lim n→∞ Var(Z n ) < 1 4 , that equivalents to lim n E(Z 2 n ) < 1 2 . Moreover, since E(Z 2 n+1 |F n ) = Z 2 n + Var(Z n+1 |F n ), it holds E(Z 2 n+1 |F n ) ≥ Z 2 n so, (Z 2 n ) n is a sub-martingale. Consequently, sup n E(Z 2 n ) = lim n E(Z 2 n ) < 1 2 .
To treat the cases when (λ 1 , λ 2 ) = (0, 0), we need the following lemma:

Lemma 7.3. If λ 1 > 0 or λ 2 > 0, then it holds lim n→∞ Var(Z n ) = 0.
In particular, lim n→∞ Var(Z n ) < 1/4 and

lim n→∞ ( 1 2 - 1 N N i=1 E(Z 2 n (i))) > 0.
Proof : We re-write (7.1) as:

Var(Z n+1 ) = (1 -4ε n )Var(Z n ) + K 1 n (r n ) 2 + (r g n ) 2 4 ,
where

ε n := λ 1 r n + λ 2 r g n -2λ 1 λ 2 r n r g n -λ 2 1 (r n ) 2 -λ 2 2 (r g n ) 2 + (r g n ) 2 4 (1 -2λ 2 ) 2 ,
and

K 1 n := 1 N (1 -2λ 1 ) 2 1 2 - 1 N N i=1 E(Z 2 n (i)) + λ 1 -λ 2 1 .
In all the considered cases, n ε n = +∞. Using (7.3), we get

0 ≤ K 1 n ≤ 1 2 (1 -2λ 1 ) 2 + λ 1 -λ 2 1 ≤ 1.
It follows Var(Z n ) ≤ y n where (y n ) n is the sequence defined in the proof of lemma A.1 (see (A.3)) with the same ε n and Kδ n is 1

N (r n ) 2 + (r g n ) 2
4 . Thus, using Lemma A.1, we get lim n→∞ Var(Z n ) = 0. Remark that using the same argument as previously, (Z 2 n ) n≥0 is a sub-martingale. Thus we get

sup n E(Z 2 n ) < 1 2 as claimed.
7.2. Proofs of L 2 and a.s. convergence. We now prove the theorems of Section 4 about convergence and synchronization.

Proof : Theorem 4.1 (i) ♣ First consider the equation (7.1) summarized as

Var(Z n+1 ) = (1 -4λr * n + o(r * n ))Var(Z n ) + K n (r * n ) 2 + o((r * n ) 2 ), where λ = λ 1 if γ 1 < γ 2 λ 2 if γ 1 > γ 2 . and r * n = 1 n γ * .
We now give the values of A and K n , depending on γ 1 , γ 2 , λ 1 and λ 2 for which this equation holds, and show how one can apply Lemma A.1 in all different cases.

• When γ 1 < γ 2 , then A = 4λ 1 and

K n = 1 N ((1 -2λ 1 ) 2 ( 1 2 - 1 N N i=1 E(Z 2 n (i))) + λ 1 -λ 2 1 )
is bounded and

lim n→∞ K n > 0. Indeed, since E(Z 2 n ) < 1 2 , we get N i=1 E(Z 2 n (i)) = E(Z 2 n (j)) + N -1 i=1, i =j E(Z 2 n (i)) < 1 2 + N -1 2 = N 2 .
By Lemma A.1 we get lim n→∞ Var(Z n ) = 0. Moreover, by Lemma A.2, it holds Var(Z n ) = O( 1 n γ 1 ).

• When γ 1 > γ 2 , it holds A = 4λ 2 and K n = 1 4 thus, by Lemma A.1 it holds lim n→∞ Var(Z n ) = 0 and Var(Z n ) = O( 1 n γ 2 ) by Lemma A.2.

• When γ 1 = γ 2 (=: γ), we have

Var(Z n+1 ) = 1 -4(λ 1 + λ 2 )r n -((2λ 1 + 2λ 2 ) 2 -(1 -2λ 2 ) 2 )r 2 n Var(Z n ) + r 2 n K n where A = 4(λ 1 + λ 2 ) and K n = 1 N ((1 -2λ 1 ) 2 ( 1 2 - 1 N N i=1 E(Z 2 n (i))) + λ 1 -λ 2 1 + N 4 ),
which is bounded and lim n→∞ K n > 0, which implies by Lemma A.1 lim n→∞ Var(Z n ) = 0 where by Lemma A.2, it holds Var(Z n ) = O( 1 n γ ). In the case γ = 1 and

λ 1 +λ 2 = 1 4 , Var(Z n ) = O( log n n ). Proof : Theorem 4.1 (ii)
Consider the following recursive equation satisfied, for any i ∈ {1, . . . , N }, by the L 2 -distance between one component and the mean field. For symmetry reasons, the following quantity x n does not depend on the specific choice of the component i. With

x n := E[(Z n (i) -Z n ) 2 ] = Var(Z n (i) -Z n ),
recall means law of total variation and it holds

x n+1 = E(Var[(1 -r n -r g n )(Z n (i) -Z n ) + r n (ξ n (i) - 1 N j ξ n (j)) F n ]) + Var Z n (i) -2λ 1 r n Z n (i) + r g n (Z n (i) -(1 -2λ 2 )Z n ) -Z n (1 -2λ 1 r n -r g n ) = (r n ) 2 E Var(ξ n (i) - 1 N i ξ n (i)|F n ) + Var (1 -2λ 1 r n -r g n )(Z n (i) -Z n ) = (1 -2λ 1 r n -r g n ) 2 Var(Z n (i) -Z n ) + (r n ) 2 (1 - 1 N ) 2 + ( N -1 N 2 ) E Var(ξ n (i)|F n ) = (1 -2λ 1 r n -r g n ) 2 x n + N -1 N (r n ) 2 (1 -2λ) 2 [E(Z n (i)) -E(Z 2 n (i))] + λ 1 -λ 2 1
Therefore we obtain (7.4) where When γ 1 = γ 2 , we have from (7.4)

x n+1 = 1 -4λ 1 r n -2r g n + 4λ 2 1 (r n ) 2 + (r g n ) 2 + 2λ 1 r n r g n x n + (r n ) 2 J n ,
J n = N -1 N (1 -2λ 1 ) 2 [ 1 2 -E(Z 2 n (i))] + λ 1 -λ 2 1 is bounded and not equal zero for N > 1. (a) When γ 1 < γ 2 the relation (7.4) gives x n+1 = [1 -4λ 1 r n -o(r n )]x n + (r n ) 2 J n . It
x n+1 = [(1 -r n -2λ 1 r n ) 2 ]x n + r 2 n J n = [1 -(2 + 4λ 1 )r n + o(r n )]x n + r 2 n J n
which implies by Lemma A.1 lim n→∞ x n = 0 and it holds, by Lemma A.2,

x n = O( 1 n γ ) where A = 2 + 4λ 1 . (b) When γ 2 < γ 1 , x n+1 = (1 -2r g n + o(r g n ))
x n + J n (r n ) 2 where A = 2 implies by Lemma A.1 lim n→∞ x n = 0 and it holds, by Lemma A.2,

x n = O( 1 n 2γ 1 -γ 2 ).
Proof : (iii) Theorem 4.1

• To prove that, in this case, a.s. convergence holds towards 1/2, we use (6.2) and consider

E[(Z n+1 - 1 2 ) 2 |F n ] = (Z n - 1 2 ) 2 [1 + 4(r n ) 2 λ 2 1 + 4(r g n ) 2 λ 2 2 -4r n λ 1 -4r g n λ 2 + 4r n r g n λ 1 λ 2 ] + (r n ) 2 E[(∆M n+1 ) 2 |F n ] + (r g n ) 2 E[(∆M g n+1 ) 2 |F n ] = (Z n - 1 2 ) 2 1 -4r n λ 1 -4r g n λ 2 + o(r n ) + o(r g n ) + (r n ) 2 W n + (r g n ) 2 W g n .
with

W n := 4λ 2 1 (Z n -1 2 ) 2 + E[(∆M n+1 ) 2 |F n ] and W g n := 4λ 2 2 (Z n -1 2 ) 2 + E[(∆M g n+1 ) 2 |F n ]. Thus, E[(Z n+1 -1 2 ) 2 |F n ] ≤ (Z n -1 2 ) 2 + (r n ) 2 W n + (r g n ) 2 W g n . Since (γ 1 , γ 2 ) ∈ (1/2, 1] 2 , we get that ((Z n+1 -1 2 ) 2
) n is a positive almost super-martingale and a.s. convergence holds. It is enough to consider L 2 convergence in order to identify the (deterministic) limit.

E(E[(Z n+1 - 1 2 ) 2 |F n ]) = E(Z n - 1 2 ) 2 1 -4r n λ 1 -4r g n λ 2 + 4r n r g n λ 1 λ 2 + (r n ) 2 K n + (r g n ) 2 K g n . With y n := E(Z n -1
2 ) 2 , one gets

y n+1 = 1 -4r n λ 1 -4r g n λ 2 + λ 2 1 (r n ) 2 + λ 2 2 (r g n ) 2 + 4r n r g n λ 1 λ 2 y n + (r n ) 2 K n+1 + (r g n ) 2 K g n+1 (7.5)
where 0 < K n+1 := E[(∆M n+1 ) 2 ] ≤ 1, and 0 < K g n+1 := E[(∆M g n+1 ) 2 ] ≤ 1. By lemma A.1 we get lim n→∞ y n = 0.

• When γ 1 = γ 2 , the proof holds similarly. Indeed,

E[(Z n+1 - 1 2 ) 2 |F n ] = (Z n - 1 2 ) 2 [1 -2r n (λ 1 + λ 2 )] 2 + r 2 n E[∆ Mn+1 (i) 2 |F n ]. Thus, E[(Z n+1 -1 2 ) 2 |F n ] ≤ (Z n - 1 2 ) 2 +r 2 n Wn , where Wn = 4(λ 1 +λ 2 ) 2 (Z n -1 2 ) 2 +E[(∆ Mn+1 (i)) 2 |F n ].
To prove the a.s. synchronization, use (6.6) with X n (i

) := Z n (i) -Z n which means in the considered case E(X n+1 (i)|F n ) = (1 -2λ 1 r n -r g n )X n (i). Thus, we obtain E(X n+1 (i)|F n ) ≤ X n (i) and therefore (Z n (i) -Z n ) n
is a bounded super-martingale and its a.s. limit exists.

Proof : Theorem 4.2

As expected, we shall consider two different situations of nullity or not for λ 1 , λ 2 and different relationships between γ 1 and γ 2 .

The proof of following cases follows along the same arguments as above. We only sketch the proof and list the essential elements below. Then it is easy to get the results.

♣ Case λ 1 = 0, λ 2 = 0. Consider the recursive equations (7.1) and (7.4) and letting λ 2 = 0 satisfied by Var(Z n ) and L 2distance respectively.

Proofs of the CLTs

We now prove the central limit theorems in order to study the scales of the fluctuations. Recall we are using the notation a n b n when lim n→∞ a n b n exists and is a constant. We will use Th. A.5 in order to prove the CLT's w.r.t. stable convergence.

8.1. Proofs of the CLTs (Theorem 5.1). Consider the following definitions.

Define X k (i) := Z k -Z k (i). Set L 0 (i) = X 0 (i) and define L n (i) := X n (i) - n-1 k=0 (E[X k+1 (i)|F k ] -X k (i)). (8.1)
As (6.6), we get

X n+1 (i) = [1 -2λ 1 r n -r g n ]X n (i) + ∆L n+1 (i) (8.2) where ∆L n+1 (i) := L n+1 (i) -L n (i).
Note that (L n ) n is an F-martingale by construction. Iterating the above relation, we can write

X n (i) = c 1,n X 1 (i) + n-1 k=1 c k+1,n ∆L k+1 (i) (8.3) where c n,n = 1 and c k,n = n-1 h=k (1 -2λ 1 r h -r g h ) for k < n. Proof : Theorem 5.1 (i-a) Case γ 1 < γ 2 . Note that in this case γ 1 = 1 because (γ 1 , γ 2 ) ∈ (1/2, 1]. We get lim n→∞ n γ 1 2 c 1,n = 0 since, c 1,n = n-1 h=1 [1 -2λ 1 r h -r g h ] = n-1 h=1 [1 - 2λ 1 c 1 h γ 1 - c 2 h γ 2 -O( 1 h 2γ 1 )] = exp[- n-1 h=1 2λ 1 c 1 h γ 1 - c 2 h γ 2 + O(1) ] = O exp[- 2λ 1 c 1 1 -γ 1 n 1-γ 1 (1 - c 2 1 -γ 2 1 -γ 1 2λ 1 c 1 1 n γ 2 -γ 1 )] = O exp( -2λ 1 1 -γ 1 n 1-γ 1 ) .
Therefore, using the same argument as previously for c k,n we obtain

c k,n = n-1 h=k [1 -2λ 1 r h -r g h ] = O exp[- 2λ 1 c 1 1 -γ 1 (n 1-γ 1 -k 1-γ 1 )(1 - c 2 1 -γ 2 1 -γ 1 2λ 1 c 1 1 n γ 2 -γ 1 )] = O exp -2λ 1 1 -γ 1 (n 1-γ 1 -k 1-γ 1 )
(agree with Lemma A.3).

It is then enough to prove the convergence n

γ 1 2 k c k+1,n ∆L n+1 (i) → N (0, (1 -1/N )/16λ 1 ). First, let us define U n,k = n γ 1 2 c k+1,n ∆L k+1 (i) and G n,k = F k+1 . Thus {U n,k , G n,k : 1 ≤ k ≤ n} is a square-integrable martingale difference array. Indeed we have E(U 2 n,k ) < +∞ and E(U n,k+1 |G n,k ) = n γ 1 2 c k+1,n E(∆L k+1 (i)|F k+1 ) = 0.
In order to conclude, we use the Theorem recalled as Th. A.5. We will prove the following three statements for

U n,k := n γ 1 2 c k+1,n ∆L k+1 (i). a) max 1≤k≤n |U n,k | → 0. b) E[max 1≤k≤n U 2 n,k ] is bounded in n. c) n k=1 U 2 n,k → (1 -1/N )/16λ 1 a.s. • It holds a) since ∆L n+1 (i) -(X n+1 (i) -X n (i)) = 2λ 1 X n (i) n -γ 1 , |∆L n+1 (i)| = O(n -γ 1 ).
• To state b), we use a) and

E[ max 1≤k≤n U 2 n,k ] ≤ E[ n k=1 U 2 n,k ] = n γ 1 n k=1 c 2 k+1,n E[(∆L k+1 (i)) 2 ] n γ 1 n k=1 e -4λ 1 1-γ 1 (n 1-γ 1 -k 1-γ 1 ) O(k -2γ 1 ) = n γ 1 e -4λ 1 1-γ 1 n 1-γ 1 n-1 k=1 e 4λ 1 1-γ 1 k 1-γ 1 O(k -2γ 1 ) + n 2 O(n -2γ 1 ) n .
Thus, E[max 1≤k≤n U 2 n,k ] is bounded in n.

• Finally, in order to prove c), we have

n k=1 U 2 n,k = n γ 1 k c 2 k+1,n (∆L n+1 (i)) 2 n γ 1 n k=1 k -2γ 1 e 4λ 1 1-γ 1 k 1-γ 1 e 4λ 1 1-γ 1 n 1-γ 1 (∆L k+1 (i)) 2 k 2γ 1 .
From a) we obtain

∆L k+1 (i) 2 = (X k+1 -X k + 2λ 1 r k X k ) 2 = [(Z k+1 -Z k ) -(Z k+1 (i) -Z k (i))] 2 + 4λ 2 1 (r k ) 2 (Z k -Z k (i)) 2 + (r k )(Z k -Z k (i))[(Z k+1 -Z k ) -(Z k+1 (i) -Z k (i))]. Since Z n -Z n (i) → 0 a.s. and (r k ) 2 X 2 k = O(k -2γ 1 ) thus, n k=1 U 2 n,k = n γ 1 n k=1 c 2 k+1,n [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i) -Z k (i))]. Let V k = k 2γ 1 [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i) -Z k (i))
] and setting the

sequences b n := 1 n γ 1 e +4λ 1 n 1-γ 1 1-γ 1 and a k := k 2γ 1 c 2 1,n e -4λ 1 k 1-γ 1 1-γ 1 . Hence, it holds 1 b n n k=1 1 a k ---→ n→∞ 1 4λ 1 . Indeed, lim sup n 1 b n n k=1 1 a k = lim sup n n γ 1 e -4λ 1 1-γ 1 n 1-γ 1 n 1 u -2γ 1 e 4λ 1 u 1-γ 1 1-γ 1 du = lim sup n ( 1 4λ 1 + γ 1 n γ 1 4λ 1 e -4λ 1 n 1-γ 1 1 -γ 1 n 1 1 u 1+γ 1 e 4λ 1 u 1-γ 1 1-γ 1 du) = 1 4λ 1 .
The same holds for the limit inferior. Then

lim n 1 b n n k=1 1 a k = 1 4λ 1 . It implies by Lemma A.4, that n k=1 U 2 n,k converges to V 4λ 1 a.s., where V is (deterministic random variable) defined as lim k E(V k+1 |F k ) =
V. Indeed, we compute

E(k 2γ 1 (Z k+1 (i) -Z k (i)) 2 |F k ) = k 2γ 1 (r k ) 2 E((ξ k+1 (i) -Z k (i)) 2 |F k ) + (r g k ) 2 E((ξ g k+1 -Z k (i)) 2 |F k ) + 2r k r g k E (ξ k+1 (i) -Z k (i))(ξ g k+1 -Z k (i))|F k = k 2γ 1 (r k ) 2 (Var(ξ k+1 (i)|F k ) + E((ξ k+1 (i) -Z k (i)) 2 |F k )) + (r g k ) 2 Var(ξ g k+1 |F k ) + E((ξ g k+1 -Z k (i)) 2 |F k ) + 2r k r g k E (ξ k+1 (i) -Z k (i))(ξ g k+1 -Z k (i))|F k which behaves like k 2γ 1 ( (r k ) 2 4 + (r g k ) 2 4 ) when k → ∞. Similarly, E[k 2γ 1 (Z k+1 -Z k ) 2 |F k ] = k 2γ 1 (r k ) 2 E ( 1 N i ξ k+1 (i) -Z k ) 2 |F k + (r g k ) 2 E (ξ g k+1 -Z k ) 2 |F k + 2r k r g k E ( 1 N i ξ k+1 (i) -Z k )(ξ g k+1 -Z k )|F k = k 2γ 1 (r k ) 2 Var[ 1 N i ξ k+1 (i)|F k ] + E 2 ( 1 N i ξ k+1 (i) -Z k |F k ) + (r g k ) 2 Var[ξ g k+1 |F k ] + E 2 (ξ g k+1 -Z k |F k ) + 2r k r g k E ( 1 N i ξ k+1 (i) -Z k )(ξ g k+1 -Z k )|F k behaves like k 2γ 1 ( (r k ) 2 4N + (r g k ) 2
4 ) when k → ∞. And it holds

E[k 2γ 1 (Z k+1 -Z k )(Z k+1 (i) -Z k (i))|F k ] = k 2γ 1 ((r k ) 2 E (ξ k+1 (i) -Z k (i))( 1 N i ξ k+1 (i) -Z k )|F k + (r g k ) 2 E (ξ g k+1 -Z k (i))(ξ g k+1 -Z k )|F k ) which behaves like k 2γ 1 ( (r k ) 2 4N + (r g k ) 2 4 ). It follows E(V k+1 |F k ) = k 2γ 1 [(r k ) 2 Var[ξ k+1 (i)|F k ] + Var[ 1 N i ξ k+1 (i)|F k ] -2E (ξ k+1 (i) -Z k (i)) 1 N i ξ k+1 (i) -Z k F k ] a.s --→ 1 4 (1 - 1 N ). Thus, V k a.s --→ 1 4 (1 -1 N ) and therefore, σ2 1 = (1-1 N ) 16λ 1 .
The proof of next parts follows similarly as previously. We sketch the proofs and mention the essential steps below.

• Case γ 1 = γ 2 (=: γ). We obtain c 1,n = O exp[ -(1+2λ 1 ) 1-γ n 1-γ ] . Therefore lim n→∞ n γ 2 c 1,n = 0. Let b n = 1 n γ e 2(1+2λ 1 ) 1-γ n 1-γ , a k = k 2γ c 2 1,n e - 2(1+2λ 1 ) 1-γ k 1-γ and V k = k 2γ [(Z k+1 -Z k ) 2 +(Z k+1 (i)-Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i)-Z k (i))]. Therefore, σ2 1 = (1-1 N ) 4(1+2λ 1 ) . • Case γ 1 = γ 2 = 1. We obtain c 1,n := n h [1 -(1 + 2λ 1 )r h ] = O(n -(1+2λ 1 ) ). Then √ nc 1,n → 0. Let b n := n 1+4λ 1 , a k := k -4λ 1 and V k = k 2 [(Z k+1 -Z k ) 2 +(Z k+1 (i)-Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i)-Z k (i))]. Therefore, σ2 1 = (1-1 N ) 2(1+4λ 1 ) . Proof of Theorem 5.1 (i-b) Since c 1,n = n-1 h=1 [1 -2λ 1 r h -r g h ] = O(exp[ -1 1-γ 2 n 1-γ 2 ]) therefore, n γ 1 -γ 2 2 c 1,n → 0. Let b n := n -(2γ 1 -γ 2 ) exp( 2 1-γ 2 n 1-γ 2 ) and a k := k 2γ 1 c -2 1,n exp(-2 1-γ 2 k 1-γ 2 ) and V k = k 2γ 1 [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i) -Z k (i))]. Therefore, σ2 2 = 1 8 (1 -1 N ). Proof of Theorem 5.1 (ii) • When γ 1 < γ 2 , let X k := Z k - 1 2 so, L n = X n - n-1 k=0 E(Z k+1 - 1 2 |F k ) -(Z k - 1 2 ) = X n + 2(λ 1 r n + λ 2 r g n ) n-1 k=0 X k and X n+1 = [1-2λ 1 r n -2λ 2 r g n ]X n +∆L n+1 . So c 1,n = O(exp[ -2λ 1 1-γ 1 n 1-γ 1 ]) and therefore n γ 1 2 c 1,n -→ 0. Let b n := 1 n γ 1 e 4λ 1 1-γ 1 n 1-γ 1 and a k := k 2γ 1 c 2 1,n e -4λ 1 1-γ 1 k 1-γ 1 and V k = k 2γ 1 (Z k+1 -Z k ) 2 therefore, σ2 = 1 16λ 1 . • When γ 2 < γ 1 , it holds c 1,n = O(exp[ -2λ 2 1-γ 2 n 1-γ 2 ]) and therefore n γ 2 2 c 1,n -→ 0. Let b n := 1 n γ 2 e 4λ 2 1-γ 2 n 1-γ 2 , a k := k 2γ 2 c 2 1,n e -4λ 2 1-γ 2 k 1-γ 2 and V k = k 2γ 2 (Z k+1 -Z k ) 2 therefore, σ2 = 1 16λ 2 . • When γ 1 = γ 2 (=: γ), it holds c 1,n = O(exp[ -2(λ 1 +λ 2 ) 1-γ n 1-γ ]) and therefore n γ 2 c 1,n → 0. Let b n = 1 n γ e 4(λ 1 +λ 2 ) 1-γ n 1-γ 1 , a k = k 2γ c 2 1,n e -4(λ 1 +λ 2 ) 1-γ k 1-γ and V k = k 2γ (Z k+1 -Z k ) 2 . Therefore, σ2 = 1 16(λ 1 +λ 2 ) .
Proof of Theorem 5.1 (iii)

• When γ 1 = γ 2 = 1, it holds c 1,n = n h [1 -2(λ 1 + λ 2 )r h ] = O(n -2(λ 1 +λ 2 )
). We then consider the following sub-cases.

-

When (λ 1 + λ 2 ) > 1 4 , √ n c 1,n = n -2(λ 1 +λ 2 )+ 1 2 -→ 0 then we get c k,n = O k n 2(λ 1 +λ 2 ) . Moreover, k U 2 k,n = n k ( k n ) 4(λ 1 +λ 2 ) (∆L k+1 ) 2 k 2 k -2 and therefore using A.4 taking suitable (a n ) n and (b n ) n , 1 bn n k=1 1 a k → 1 1-4(λ 1 +λ 2 ) and thus, (∆L n+1 ) 2 = (Z k+1 -Z k ) 2 then lim k→∞ k 2 E((Z k+1 -Z k ) 2 |F k ) = 1 4 a.s. and therefore, σ * 2 1 = 1 4(1-4(λ 1 +λ 2 )) . -When λ 1 +λ 2 = 1 4 , it holds √ n(log n) -1 2 c 1,n -→ 0. So c k,n = ( k n ) 1 2 and U k,n = √ n √ log n c k+1,n ∆L k+1 and n k=1 U 2 k,n = n log n n k=1 ( k n )(∆L k+1 ) 2 = 1 log n k 1 k k 2 (∆L k+1 ) 2 . Using Lemma A.4 leads to con- clusion with σ * 2 2 = 1 4 .
8.2. Proofs of the CLTs (Theorem 5.2). We now prove Theorem 5.2.

Proof : Let us define Xn := n 4(λ 1 +λ 2 ) (Z n -1 2

). Recall we state c 1 = c 2 = 1 for simplicity. Since

E[ X2 n ] < ∞, it is therefore enough to show that ( Xn ) n is a quasi-martingale. Indeed, we have k E(|E[ Xk+1 |F k ] -Xk ]|) = k E [(1 + 1 k ) 4(λ 1 +λ 2 ) (1 -2(λ 1 + λ 2 )r k ) -1] Xk = k O( 1 k 2 )8(λ 1 + λ 2 ) 2 E(| Xk |) < +∞.
Moreover, from the computations carried out in the proof of Theorem 5.1, E( X2 n ) < +∞ and so it converges a.s and in L 1 to some real random variable X. In order to prove that P( X = 0) > 0, we will prove that ( X2 n ) n is bounded in L p for a suitable p > 1. Indeed this fact implies that X2 n converges to X2 and so we obtain

E( X2 ) = lim n E( X2 n ) = lim n n 4(λ 1 +λ 2 ) E(X 2 n ) > 0.
To this purpose, we set p = 1 + /2, with > 0 and z n := E(|X n | 2+ ). Using the following recursive equation:

X n+1 = (1 -2r n )Z n + r n N N i=1 ξk+1 - 1 2
one gets

z n+1 = E(|X n | 2+ ) -(2 + )r n 2Z n E(|X n | 1+ ) + (2 + )r n E |X n | 1+ sign(X n ) (X n )( 1 N N i ξk+1 (i)) + R n where R n = O(n -2 ). Now since E[ 1 N i ξn+1 (i)|F n ] = 2Z n -2(λ 1 + λ 2 )(Z n -1 2 ), we have z n+1 = E(|X n | 2+ ) -2(2 + )r n Z n E(|X n | 1+ ) + (2 + )r n E[|X n | 1+ sign(X n ) (2Z n -2(λ 1 + λ 2 ))X n ] + R n = E(|X n | 2+ ) -(2 + )r n 2(λ 1 + λ 2 )E[|X n | 1+ sign(X n ) (X n )X n ] + R n = E(|X n | 2+ ) -(2 + )r n 2(λ 1 + λ 2 )E |X n | 2+ + R n = 1 -2(λ 1 + λ 2 )(2 + )r n z n + g(n)
with g(n) = O(n -2 ). Therefore, we have

z n+1 = 1 -2(λ 1 + λ 2 )(2 + )r n z n + g(n).
Since, for > 0 sufficiently small, we have α(2 + ) < 1 and for n large,

n-1 k=0 1 -2(λ 1 + λ 2 )(2 + )r k = exp[ n-1 k=0 (ln 1 -2(λ 1 + λ 2 )(2 + ) c k γ + O( 1 k 2γ )) ] = O(exp[-2(λ 1 + λ 2 )(2 + ) ln n]) = O(n -2(λ 1 +λ 2 )(2+ ) ).
Thus,

E[|X n | (2+ ) ] = O( 1 n 2(λ 1 +λ 2 )(2+ ) )
which it implies that X2 is bounded in L 1+ 2 . 8.3. Proofs of the CLTs from Theorem 5.3.

Proof : We organize the proof in two main cases according to nullity of λ 1 and λ 2 .

♣ Case λ 1 = 0, λ 2 = 0. In order to study the evolution of X n (i) := (Z n -Z n (i)), we consider two sub-cases.

• When γ 1 ≤ γ 2 , X n+1 (i) = (1 -2λ 1 r n )X n (i) + ∆L n+1 (i) and the proof follows like the part (i)(a) of Theorem 5.1 with σ2

3 = (1-1 N ) 16λ 1 when γ 1 < γ 2 , σ2 3 = (1-1 N )
4(1+2λ 1 ) when γ 1 = γ 2 (denoted by γ) and σ2

3 = (1-1 N ) 2(1+4λ 1 ) when γ 1 = γ 2 = 1. • When γ 2 < γ 1 , X n+1 (i) = (1 -r g n )X n (i) + ∆L n+1 (i), then the proof follows like part (i-b) of Theorem 5.1 with σ2 5 = 1 8 (1 -1 N ).
We further consider X n := (Z n -1 2 ) n .

• When γ 1 ≤ γ 2 , X n+1 = (1 -r g n )X n + ∆L n+1 , then the proof follows in a similar way as the part (ii) of Theorem 5.1 with σ2 1 = 1 16λ 1 when γ 1 < γ 2 and σ2 1 = 1 16(λ 1 +λ 2 ) when γ 1 = γ 2 = γ. • When γ 2 < γ 1 , the proof follows along the same lines as previously. We sketch the essential arguments in the following. We have

X n+1 = (1 -2λ 1 r n )X n + ∆L n+1 . therefore, c 1,n = O(e -2λ 1 1-γ 1 n 1-γ 1 ) and thus, n γ 2 -γ 1 2 c 1,n → 0.
Following the same steps as in the previous proof. Only showing that

n k=1 U 2 n,k = n 2γ 2 -γ 1 e -4λ 1 1-γ 1 n 1-γ 1 n k=1 k -2γ 2 e 4λ 1 k 1-γ 1 1-γ 1 k 2γ 2 (∆L k+1 ) 2
goes to a constant with lim n→∞ 1 b n n k=1

1 a k = 1 4λ 1 and k 2γ 2 (∆L k+1 ) 2 k 2γ 2 (Z k+1 -Z k ) 2 1 4 . Therefore, σ2 3 = 1 16λ 1 . ♣ Case λ 1 = 0, λ 2 = 0.
Concerning the evolution of (Z n -Z n (i)), for both cases γ 1 ≤ γ 2 and γ 2 < γ 1 , it is proved analogously as part (i)(b) of Theorem 5.1 with σ2

4 = σ2 6 = 1 8 (1 -1 N ).
We now consider X n := Z n -1 2 n . • When γ 1 ≤ γ 2 , the proof follows in a similar way. We sketch essential arguments below. We have

X n+1 = (1 -2λ 2 r g n )X n + ∆L n+1 , therefore it holds c 1,n = O(e -2λ 2 1-γ 2 n 1-γ 2 ) and thus, n γ 1 -γ 2 2 c 1,n → 0. It is enough to show that n k=1 U 2 n,k = n 2γ 1 -γ 2 e -4λ 2 1-γ 2 n 1-γ 1 n k=1 k -2γ 1 e 4λ 2 k 1-γ 2 1-γ 2 k 2γ 1 (∆L k+1 ) 2 .
tends to a constant with lim n 1 b n n k=1

1 a k = 1 4λ 2 and k 2γ 1 (∆L k+1 ) 2 k 2γ 1 (Z k+1 -Z k ) 2 1 4 . Therefore, σ2 2 = 1 16λ 2 .
The proof when γ 2 < γ 1 follows as in part (ii) of Theorem 5.1 with σ2 4 = 1 16λ 2 .

• The case γ 1 = γ 2 = 1 is proven similarly as in part (iii) Theorem 5.1 with σ * 2 3 = 1 4(1-4(λ 1 +λ 2 ))

when λ 1 + λ 2 > 1 4 , σ * 2 4 = 1 4 when λ 1 + λ 2 = 1 4 and Theorem 5.2 when λ 1 + λ 2 < 1 4 .

8.4. Proofs of the CLTs from Theorem 5.4.

Proof : Proof of Theorem 5.4 (i)

• Case γ 1 = γ 2 . Define X k (i) := Z k -Z k (i). Set L 0 (i) = X 0 (i) and let us rewrite L n (i) = X n (i) - n-1 k=0 ([1 -r g k ](Z k -Z k (i)) -(Z k -Z k (i))) = X n (i) + n-1 k=0 r g k X k (i). Then X n+1 (i) = [1-r g n ]X n (i)+∆L n+1 (i). Note that (L n (i))
n is an F-martingale by construction. Iterating the above relation, we can write

X n (i) = c 1,n X 1 (i) + n k=1 c k+1,n ∆L n+1 (i) where c n+1,n = 1 and c k,n = n h=k [1 -r g h ] for k ≤ n. It holds c 1,n = n h=1 [1 -r g h ] = O(exp[ -1 1-γ 2 n 1-γ 2 ]). Then n γ 1 -γ 2 2 c 1,n → 0 and c k,n = O(exp[ -1 1-γ 2 (n 1-γ 2 -k 1-γ 2 )]). So it is enough to prove that n γ 1 -γ 2 2 k c k+1,n ∆L n+1 (i) → N 0, (1 -1/N )(Z ∞ -Z 2 ∞ )
. Again, this can be proved using Theorem A.5 for U n,k+1 = n γ 1 -γ 2 2 k c k+1,n ∆L n+1 (i) and proving a), b) and c). It is easy to check that conditions a) and b) hold. Let us now consider 3). We have

n k=1 U 2 n,k = n 2γ 1 -γ 2 k c 2 k+1,n (∆L n+1 (i)) 2 n 2γ 1 -γ 2 n k=1 k -2γ 1 e 1 1-γ 2 k 1-γ 2 e 1 1-γ 2 n 1-γ 2 (∆L n+1 (i)) 2 k 2γ 1 .
From 1) we obtain

(∆L n+1 (i)) 2 [(Z k+1 -Z k ) -(Z k+1 (i) -Z k (i))] 2 + (r g k ) 2 (Z k -Z k (i)) 2 + (r g k ) 2 (Z k -Z k (i))[(Z k+1 -Z k ) -(Z k+1 (i) -Z k (i))]. Since Z n -Z n (i) a.s --→ 0 and (r g k ) 2 X k (i) 2 = O(k -2γ 2 ) so, n k=1 U 2 n,k = n 2γ 1 -γ 2 n k=1 c 2 k+1,n [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i) -Z k (i))]
where we use Lemma A.4 with b n :=

1 n 2γ 1 -γ 2 e 2 1-γ 2 n 1-γ 2 and a k := k 2γ 1 c 2 1,n e -2 1-γ 2 k 1-γ 2 . Let V k = k 2γ 1 [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i) -Z k (i))]. Thus lim n→∞ 1 bn n k=1 1 a k = 1 2 . This implies that n k=1 U 2 n,k converges to V 2 a.s., where V is such that lim k→∞ E(V k+1 |F k ) = V. Indeed, E(k 2γ 1 (Z k+1 (i) -Z k (i)) 2 |F k ) = k 2γ 1 (r k ) 2 E[(ξ k+1 (i) -Z k (i)) 2 |F k ] = k 2γ 1 (r k ) 2 Var[ξ k+1 (i)|F k ] = k 2γ 1 (r k ) 2 (Z k -Z 2 k ) a.s --→ Z ∞ -Z 2 ∞ .
Similarly

, E(k 2γ 1 (Z k+1 -Z k ) 2 |F k ) a.s --→ Z ∞ -Z 2 ∞ , and
E(k 2γ 1 (Z k+1 (i) -Z k (i))(Z k+1 -Z k )|F k ) a.s --→ Z ∞ -Z 2 ∞ N . Thus, lim k→∞ U 2 k = ϑ2(1 -1 N )(Z ∞ -Z 2 ∞ ) a.s. where ϑ = 1 2 . • Case γ 1 = γ 2 (=: γ). We get c 1,n = n h=1 [1 -r h ] = O(exp[ -1 1-γ n 1-γ ]). Then n γ 2 c 1,n → 0. Moreover lim n→∞ 1 bn n k=1 1 a k = 1 2 and V k = k 2γ [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 - Z k )(Z k+1 (i) -Z k (i))]. Thus, lim k→∞ U 2 k = ϑ4(1 -1 N )(Z ∞ -Z 2 ∞ ) a.s. where ϑ = 1 2 . • Case γ 1 = γ 2 = 1. c 1,n = n h=1 [1 -r h ] = O(n -1 ). Then √ n c 1,n → 0. Choosing b n := n and a k := 1 and let V k = k 2 [(Z k+1 -Z k ) 2 + (Z k+1 (i) -Z k (i)) 2 -2(Z k+1 -Z k )(Z k+1 (i) -Z k (i))]. Thus, lim k→∞ U 2 k = ϑ4(1 - 1 N )(Z ∞ -Z 2 ∞ ) a.s. where ϑ = 1.
Proof of Theorem 5.4 (ii)

• Case γ 1 < γ 2 . The process (Z n ) n is a (bounded) martingale. Therefore (Z n ) n converges a.s. We want to prove the following two statements

(1) E sup k k γ 1 -1 2 |Z k+1 -Z k | < +∞; (2) n 2γ 1 -1 k≥n (Z k+1 -Z k ) 2 a.s --→ 1 N (2γ 1 -1) (Z ∞ -Z 2 ∞ ). Indeed, the first condition immediately follows from |Z k+1 -Z k | = |r n ( 1 N i ξ k+1 (i) -Z k ) + r g n (ξ g k+1 -Z k )| = O(k -γ 1 ).
Concerning the condition 2), we observe that

n 2γ 1 -1 k≥n (Z k+1 -Z k ) 2 = n 2γ 1 -1 k≥n k -2γ 1 (r k ) 2 ( i ξ k+1 (i) N -Z k ) 2 k 2γ 1
and so the desired convergence follows by lemma A.4 with a

k := k -2γ 1 +2 , b n := n 2γ 1 -1 and U k = k 2γ 1 (r k ) 2 ( k ξ k+1 (i) N -Z k ) 2 , lim n→∞ b n k≥n 1 a k b 2 k = -1 1-2γ 1 so, E k ξ k+1 (i) N -Z k 2 F = Var( k ξ k+1 (i) N |F n ) = 1 N (Z ∞ -Z 2 ∞ ).
Finally, we take ϑ = 1 (2γ 1 -1) . • Case γ 2 < γ 1 . We want to prove the following two statements

1) E sup k k γ 2 -1 2 |Z k+1 -Z k | < +∞; 2) n 2γ 2 -1 k≥n (Z k+1 -Z k ) 2 a.s --→ 1 (2γ 2 -1) (Z ∞ -Z 2 ∞ ). The first result immediately follows from |Z k+1 -Z k | = |r n ( 1 N i ξ k+1 (i) -Z k ) + r g n (ξ g k+1 -Z k )| = O(k -γ 2 ).
To prove the second point, we observe that

n 2γ 2 -1 k≥n (Z k+1 -Z k ) 2 = n 2γ 2 -1 k≥n (r k ) 2 k -2γ 2 N i=1 ξ k+1 (i) N -Z k 2 k 2γ 2
and the desired convergence follows by lemma A.4 with a

k := k 2γ 2 +2 , b n := n 2γ 2 -1 and U k = k 2γ 2 (r g k ) 2 (ξ g k+1 -Z k ) 2 , lim n→∞ b n k≥n 1 a k b 2 k = -1 1-2γ 2 and E(ξ g k+1 (i) -Z k ) 2 |F) = Var(ξ g k+1 |F) = (Z ∞ -Z 2 ∞ ).
Finally, we take ϑ = 1 (2γ 2 -1) . • Case γ 1 = γ 2 (=: γ). The process (Z n ) n is a martingale and converges a.s. Indeed,

E(Z n+1 |F n ) = (1 -2r n )Z n + r n E N i=1 ξn+1 (i) N |F n = Z n .
We want to check the following two conditions:

1) E sup k k γ-1 2 |Z k+1 -Z k | < +∞; 2) n 2γ-1 k≥n (Z k+1 -Z k ) 2 a.s --→ 2 N (2γ-1) (Z ∞ -Z 2 ∞ ). The first result follows from |Z k+1 -Z k | = r n 1 N N i=1 ξk+1 (i) -2Z k ) = O(k -γ ).
And for the second point, we observe that

n 2γ-1 k≥n (Z k+1 -Z k ) 2 = n 2γ-1 k≥n r 2 k k -2γ i ξk+1 (i) N -Z k 2 k 2γ
and so the desired convergence follows by lemma A.5 with a k := k -2γ+2 , b n := n 2γ-1 and

U k = k 2γ r 2 k i ξk+1 (i) N -2Z k 2 , lim n→∞ b n k≥n 1 a k b 2 k = -1 1-2γ , and E N i=1 ξk+1 (i) N -2Z k 2 |F k = 2 N (Z ∞ -Z 2 ∞ )
. Finally, we have ϑ = 1 (2γ-1) .

• Case γ 1 = γ 2 = 1. As usual, we prove

1) E sup k k 1 2 |Z k+1 -Z k | < +∞; 2) n k≥n (Z k+1 -Z k ) 2 a.s --→ 2 N (Z ∞ -Z 2 ∞ ). First result follows from |Z k+1 -Z k | = |r n ( 1 N N i=1 ξk+1 (i) -2Z k ))| = O(k -1 ).
Second result comes from

n k≥n (Z k+1 -Z k ) 2 = n k≥n r 2 k k -2 k 2 ( i ξk+1 (i) N -Z k ) 2
and the desired convergence follows then by lemma A.4 with a k := 1, b n := n and

U k = k 2 r 2 k ( i ξk+1 (i) N -2Z k ) 2 , lim n→∞ b n k≥n 1 a k b 2 k = 1. Moreover, E( i ξk+1 (i) N -2Z k ) 2 |F) = 2 N Z ∞ (1 -Z ∞ )
. Finally, we have ϑ = 1.

Lemma A.2. Let (z n ) n be a sequence of positive reals satisfying the following equation: (A.6) where A > 0 and ∀n ∈ N, 0 < K n ≤ K. Assume that (ε n ) n and (δ n ) n are positive sequences of reals In particular, assuming lim inf n K n > 0 and

z n+1 = (1 -Aε n )z n + K n δ n ,
ε n = c 1 n κ 1 + O( 1 n 2κ 1 ), lim n n κ 2 δ n = c 2 > 0
where 1 2 < κ 1 ≤ 1 < κ 2 then, we get that

z n =          O( 1 n κ 2 -κ 1 ) if κ 1 < 1, O( log n n A ) if κ 1 = 1 and κ 2 -A = 1, O( 1 n κ 2 -1 ) if κ 1 = 1 and κ 2 -A < 1 O( 1 n A ) if κ 1 = 1 and κ 2 -A > 1.
. Proof : The case K = 0 is well-known and we will prove the statement K > 0. Let be such that Aε n < 1 for all n ≥ . Then for n ≥ we have z n ≤ y n , where (1 -ε k ).

Since n ε n = +∞, then lim n→∞ n-1 h= (1 -ε h ) = 0. Moreover, for every m ≥ ,

n-1 h= δ h n-1 k=h+1 (1 -ε k ) = m-1 h= δ h n-1 k=h+1 (1 -ε k ) + n-1 h=m δ h n-1 k=h+1 (1 -ε k ) ≤ n-1 k=m (1 -ε k ) m-1 h= δ h + +∞ h=m δ h .
Using the fact that n-1 k=m (1 -ε k ) -→ 0 and that n δ n < +∞, letting first n -→ +∞ and then m -→ +∞, the conclusion follows. We are left to prove if n ε n < +∞ then lim n z n = 0. From (A.6) we have

z n+1 ≥ 1 -ε n z n from which it follows z n ≥ z 0 n-1 k=0 1 -ε n .
Since by assumption, n ε n < +∞, we obtain lim n→∞ z n > 0.

Thus, lim n→+∞ z n = 0 ⇔ n ε n = +∞ (κ 1 ≤ 1). Otherwise, if n ε n < +∞ (κ 1 > 1), then lim n→+∞ z n = 0.

More specifically, one gets.

• When κ 1 < 1. Let z ,n := n-1 h= δ h n-1 k=h+1 (1 -ε k ), thus, assuming is large enough to replace ε n and δ n with their asymptotics, and using the monotony of their asymptotics,

z ,n = O n 1 s κ 2 exp - n s 1 u κ 1 du ds = O n 1 s κ 2 exp - 1 (1 -κ 1 )u κ 1 -1 n s ds = O n 1 s κ 2 exp[ 1 1 -κ 1 ( 1 s κ 1 -1 - 1 n κ 1 -1 )]ds = O e - 1 (1-κ 1 )n κ 1 -1 n 1 s κ 2 e 1 (1-κ 1 )s κ 1 -1 ds = O 1 n κ 2
n s -κ 2 e 1 (1-κ 1 )s κ 1 -1 ds n -κ 2 e 1 (1-κ 1 )n κ 1 -1 .

Letting n → ∞, using L'Hôpital rule, we obtain

z ,n = O 1 n κ 2 n -κ 2 e 1 (1-κ 1 )n κ 1 -1 [(-κ 2 )n -κ 2 -1 + n -κ 2 n -κ 1 ]e 1 (1-κ 1 )s κ 1 -1 = O 1 n κ 2 1 -κ 2 n + 1 n κ 1 = O 1 n κ 2 -κ 1 1 1 -κ 2 n κ 1 n . • When κ 1 = 1, set f n := z n n-1 k=0 (1 -ε k )
.

By (A.6) we obtain,

f n+1 = f n + F (n)
where

F (n) = δ n n k=0 (1-ε k )
. So, observing that f 0 = z 0 = 0, we obtain

f n = n-1 h=0 F (h),
or equivalently,

z n = n-1 k=0 (1 -ε k ) n-1 h=0 F (h).
Since n-1 k=0 (1 -ε k ) = O( 1 n A ) and therefore F (n) = O(n A-κ 2 ), then

z n = n-1 k=0 (1 -ε k ) n-1 h=0 F (h) = O n-1 h=0 1 h κ 2 -A n A = O n 1 h A-κ 2 n A n→∞ =      O( log n n A ) if κ 2 -A = 1, O( 1 n b-1 ) if κ 2 -A < 1, O( 1 n A ) if κ 2 -A > 1.
.

The conclusion follows.

As mentioned, previous result agrees with the next lemma which is proved as Lemma A.4 in [START_REF] Aletti | Synchronization of reinforced stochastic processes with a network-based interaction[END_REF].

Lemma A.3. Let γ be a real in ( 1 2 , 1], and c > 0. Let (r n ) n be a sequence of real numbers such that 0 < r n < 1. Assume that (1 -ar m ) and m 0 ,n = p -1 m 0 ,n .

It holds

p m 0 ,n = O(exp[-ca 1-γ n 1-γ ]) if 1 2 < γ < 1 O(n -ca ) if γ = 1 . and m 0 ,n = O(exp[ ca 1-γ n 1-γ ]) if 1 2 < γ < 1 O(n ca ) if γ = 1 .
Thus, setting F k+1,n := p m 0 ,n p m 0 ,k for m 0 ≤ k ≤ n , one gets

F k+1,n = O(exp( a 1-κ 1 (k 1-κ 1 -n 1-κ 1 ))) for 1/2 < κ 1 < 1 O(( k n ) a )
for κ 1 = 1.

Lemma A.4. Let G be an (increasing) filtration and (V k ) be an G-adapted sequence of real random variables such that E[V k |G k-1 ] → V a.s. for some real random variable V . Moreover, let (a k ) and (b k ) be two sequences of strictly positive real numbers such that

b k ↑ +∞, ∞ k=1 E[V 2 k ] a 2 k b 2 k < +∞.
Then we have: and the σ-fields are nested, i.e. F n,k ⊆ F n+1,k for 1 ≤ k ≤ k n , n ≥ 1. Then S n,kn = kn k=1 U n,k converges stably to a random variable with characteristic function φ(u) = E[exp(-σ 2 u 2 /2)], i.e. to the Gaussian kernel N (0, σ 2 ). Sahasrabudhe, N. Synchronization and fluctuation theorems for interacting Friedman urns. J. Appl. Probab., 53 (4), 1221-1239[START_REF] Crimaldi | Introduzione alla nozione di convergenza stabile e sue varianti (Introduction to the notion of stable convergence and its variants)[END_REF]. URL https://projecteuclid.org/euclid.jap/1481132848.

  are studied in Aletti et al. (2019): a.s. synchronization toward Z ∞ and CLT are proven. Weighted empirical means are studied analogously in Aletti et al. (2020). As considered in Crimaldi et al. (2019); Aletti et al. (2019), we may think about following context for the random evolutions we consider. Let us state in the case where S = {0, 1} represents two possible choices or actions made by "individuals" or agents

  adapted term such that lim n→∞ ζ n = 0 a.s. Thus a.s. convergence towards zeros of F gives the a.s. convergence towards 1/2 when λ 1 + λ 2 > 0 or towards a value belonging to the diagonal {z = (z 1 , . . . , z N ) ∈ [0, 1] N : ∀i ∈ {1, . . . , N }, z i = z 1 } when λ 1 = λ 2 = 0. The case λ 1 = λ 2 = 0 leads to non isolated zeros of F which is not a case covered by the general stochastic approximation theorems. The methods developped here, following Dai Pra et al. (2014);Crimaldi et al. (

7. 1 .

 1 First results about the variances. First remark the assumption ∀i ∈ {1, . . . , N }, Z 0

  implies by Lemma A.1 lim n→∞ x n = 0 and it holds, by Lemma A.2, x n = O( 1 n γ 1 ) where A = 4λ 1 c 1 .

y

  n+1 = (1 -Aε n )y n + Kδ n , y = z . . Set ε n = Aε n and δ n = Kδ n . It holds y n = y

  as n → ∞. Let a > 0. Denote with m 0 ≥ 2 an integer such that ∀m ≥ m 0 , a < 1 rm . Let p m 0 ,n := n m=m 0

a

  k → ϑ for some constant ϑ, then1 bn n k=1 V k a k → ϑV . b) If b n k≥n 1 a k b 2 k → ϑ for some constant ϑ, then b n k≥n V k a k b 2 k → ϑV . Theorem A.5. (Theorem 3.2 in[START_REF] Hall | Martingale limit theory and its application[END_REF] Let {S n,k , F n,k : 1 ≤ k ≤ k n , n ≥ 1} be a zero-mean, square-integrable martingale array with differences U n,k , and let σ 2 be an a.s. finite random variable. Suppose that1) max 1≤k≤kn |U n,k | P -→ 0; 2) E[max 1≤k≤kn U 2 n,k ] is bounded in n;

Acknowledgments

These results belongs to the PhD report Mirebrahimi (2019) of M. Mirebrahimi under supervision of P.-Y. Louis. They gratefully acknowledge G. Aletti, P. Cénac-Guesdon, and C. Mailler for many comments while refereeing the PhD thesis. Their comments, as well as anonymous referee's comments, helped to improve this paper.

• Case γ 1 < γ 2 . To study the variance,

1 and with A = 4λ 1 . To study the synchronization,

) and with A = 4λ 1 . • Case γ 2 < γ 1 . To study the variance

where K n = 1 4 with A = 4λ 1 .

To study the L 2 -distance's behavior,

To prove that a.s. convergence holds towards 1/2 consider to equation (6.2) and let λ 2 = 0,

To study the variance,

where

and with A = 4λ 1 .

To study the L 2 -distance's,

Finally, to prove the a.s. synchronization, using (6.6), E(X n+1 (i

• Case γ 1 < γ 2 . To study the variance,

where

where

where

and with A = 4λ 2 .

To study the L 2 -distance,

Proof : Theorem 4.4 (i)

When λ 1 = λ 2 = 0, (Z n ) n is a bounded martingale which therefore converges a.s. to a random variable Z ∞ . On the other hand, by Lemma 7.2, Var(Z ∞ ) n < 1 4 . Remark, it means we do not have Z ∞ ∈ {0, 1} a.s. which is a behavior that may happen with some reinforcements like in reinforced random walks.

Let us use γ * = min(γ 1 , γ 2 ) and r * n := max(r n , r g n ),

with the following developments.

• When γ 1 < γ 2 , then A = 1 and

] is bounded and not equal zero. Indeed, since E(Z 2 n ) < 1 2 by Lemma 7.2, we get

Using the first part of Lemma A.2, since n (r g n ) 2 < +∞ we get Var(Z n ) > 0. • When γ 2 < γ 1 then, A = 1 and K n = 1 4 , thus, by the first part of Lemma A.2, we get Var(Z n ) > 0.

• When γ 1 = γ 2 (=: γ),

where

To study the synchronization phenomenon, we consider the L 2 -distance x n between Z n (i) and Z n .

) is bounded and not equal zero for N > 1 and then A = 2. Thus, by Lemma A.1 it holds lim n→∞ x n = 0 and Lemma A.2 yields x n = O( 1 n 2γ 1 -γ 2 ), meaning in particular that the L 2 -synchronization holds as n → ∞.

Moreover when γ

In this section, we prove and recall some technical results. The following Lemma is adapted from [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF] to the more general cases considered in this work. It is used with

Lemma A.1. Let (x n ) be a sequence of positive such that following equation holds:

where a > 0, r n ≥ 0 and 0 ≤ K n ≤ K. Assume that (ε n ) n and (δ n ) n are positive sequences of reals

Then lim n→+∞ x n = 0.

Proof. The case K = 0 is well-known. We will prove the statement when K > 0. Let m 0 be such that ε n < 1 for all n ≥ m 0 . Then for n ≥ m 0 we have x n ≤ y n , where

Using that n-1 j=m (1 -ε j ) -→ 0 and that n δ n < +∞, letting first n → +∞ and then m → +∞ in (A.5), the conclusion follows.

We now present an extended version of the previous result, stating the rate of convergence. Following Lemma is adapted from [START_REF] Crimaldi | Synchronization and functional central limit theorems for interacting reinforced random walks[END_REF]. This is in agreement with Aletti et al. (2020, Lemma A.1), Aletti et al. (2019, Lemma A.1) given here as Lemma A.3 for completeness.