Asymptotics of generalized P\'olya urns with non-linear feedback

Generalized P\'olya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement, a generic example being competition of agents in evolving markets. It is well known which conditions on the feedback mechanism lead to monopoly where a single agent achieves full market share, and various further results for particular feedback mechanisms have been derived from different perspectives. In this paper we provide a comprehensive account of the possible asymptotic behaviour for a large general class of feedback, and describe in detail how monopolies emerge in a transition from sub-linear to super-linear feedback via hierarchical states close to linearity. We further distinguish super- and sub-exponential feedback, which show conceptually interesting differences to understand the monopoly case, and study robustness of the asymptotics with respect to initial conditions, heterogeneities and small changes of the feedback mechanisms. Finally, we derive a scaling limit for the full time evolution of market shares in the limit of diverging initial market size, including the description of typical fluctuations and extending previous results in the context of stochastic approximation.


Introduction
In the near future, customers who intend to buy a new car will have the choice between several different technologies like modern cars powered by fossile or synthetic fuels, hydrogen or batteries. Although electric cars seem to be in the pole position in the race for the future car market, it is still open which technology will win or whether there will be a mixture of different technologies. The economist Brian R. Arthur suggests in [4] to model the competition between technologies as a generalized Pólya urn, which was basically introduced by Hill, Lane and Sudderth in [20]. In this model the decision which technology to choose depends on three factors. First, it supposes that each technology has an intrinsic deterministic attractiveness or fitness. Second, the decision depends on the choice of earlier customers. For example, if many bought an electric car before, there will be a dense charging infrastructure and thus electric cars get more attractive for future customers. A second argument for this reinforcement is that high revenues in the past provide financial means for a faster technological development as well as cheaper prices because of lower production costs per unit. The resulting overall attractiveness of technology i is now modeled as a hypothetical feedback-function F i (X i ) ≥ 0 depending on the number X i ∈ N = {1, 2, 3 . . .} of customers, who chose technology i before. High values of F i (X i ) indicate high attractiveness of technology i. A typical example is F i (k) = α i k β , where α i > 0 models the intrinsic attractiveness and β > 0 the reinforcement effects in the market. The third determinant of customers decision is their personal preference, which is difficult to include in a deterministic model and probabilistic approaches are more appropriate. We assume that customers enter the market sequentially and have full information. Given the current state (X 1 , . . . , X A ) of the market, a customer will opt for technology i with probability where A ≥ 2 is the number of different technologies. The market size X 1 +. . .+X A increases by one in each step. If F i (k) = k, then this corresponds to the original Pólya urn, which was introduced by Pólya and Eggenberger in [15]. Depending on the feedback function, monopoly may occur where one technology achieves full market share, as well as random or deterministic non-zero asymptotic market shares for several technologies. The monopolist is in general random and depends on the behaviour of the young market. Analyzing which feedback function leads to which regime provides an understanding of the determinants of the long-time behavior of markets. Mathematically, this setup corresponds to a discrete-time Markov process, which is called a (generalized) non-linear Pólya urn in the following and introduced in detail in the next Section. Apart from the competition of technologies, many other interpretations and applications of generalized Pólya urns are possible. An obvious one is the competition of companies in the same market for new customers or the competition between regions for new companies to settle. The dynamics of household wealth is another growth process with reinforcement (see e.g. [16] and references therein) that can be modelled with urns. [35] summarizes further applications in psychology or evolutionary biology, and more recently, [38,36] use Pólya urns in the context of cryptocurrencies. In the following we will adapt the more general terminology of agents {1, . . . , A} instead of technologies.
Mathematical properties of non-linear Pólya urns have been examined before, often focused on polynomial feedback functions [25,13,26,33,20,24,10,29] or homogeneous models with F i ≡ F [34,32,30]. In applications, the feedback functions are usually a hypothetical construction that can barely be measured in real systems similar to utility functions in economic situations, thus a general mathematical understanding without restrictive conditions on F i is important. This paper investigates the long-time behavior of non-linear Pólya urns for a very general class of feedback functions. F i could even be decreasing or exponentially increasing, which reveals some surprising differences to the usually studied polynomial case. An important restriction is, however, that F i depends only on X i , which excludes stationary limit cycles as studied e.g. in [11].
In the monopoly case, we present in Section 4 an asymptotic result for large initial market sizes on who will win the competition, extending previous results for particular feedback functions. In the non-monopoly case we present in Section 5 a novel approach to compute the deterministic long-time market shares, which do not depend on the initial condition or early dynamics. In Section 6, we also study in detail the transition between both cases for almost linear feedback functions, which are particularly relevant in certain applications [16]. Moreover, we derive in Section 7 a law of large numbers for the dynamics of the process for large initial market size, which is asymptotically described by an ordinary differential equation and has previously been studied for particular feedback functions in the context of stochastic approximation [9,35,37]. Extending these results, we also establish a functional central limit theorem to describe typical dynamic fluctuations by a system of random ODEs in the Sections 8 and 9. The question of a Gaussian approximation of the dynamics of a Pólya urns has also been addressed in recent research, see [8] and [12]. Predictable behaviour can only be expected for large initial market size, the behavior of very young markets is intrinsically random. While bounds on the probabilities of certain events can be obtained, we focus here mostly on asymptotic results and provide a rather complete account of the possible dynamic and long-time behaviour of generalized non-linear Pólya urns. In Section 10 we provide again a detailed summary of the main results and novelties of the paper.
(3) These are independent birth processes with Ξ i (0) = X i (0), where the time between the k-th and (k +1)-th event of Ξ i is given by τ i (X i (0)+k). If 0 = t 0 < t 1 < t 2 < ... is the sequence of jump-times of the process Ξ(t) = (Ξ 1 (t), . . . , Ξ A (t)), i.e. t n+1 = min {t > t n : Ξ(t) = Ξ(t n )} , then Rubin's theorem (proven in e.g. [34]) states, that the jump chain (Ξ(t n ) : n ∈ N 0 ) has the same distribution as the process (X(n) : n ∈ N 0 ). Thus we can define: In fact, the birth processes Ξ i (t) can explode as the sum ∞ k=X i (0) τ i (k) might be finite. We therefore define the random explosion times In the following we are especially interested in the occurrence of monopoly, which requires some definitions.
i.e. the market share of agent i converges to one; i.e. agent i wins in all but finitely many steps; 3. total monopoly i.e. agent i wins in all steps.
Obviously, a total monopoly is also a strong monopoly and a strong monopoly always implies a weak monopoly. Via exponential embedding one can express the event sM on i (χ(0), N ) by the explosion times through as equality of finite explosion times has probability zero (see below). With the observation one can easily derive the following generally known criterion for the occurrence of strong monopoly (see e.g. [34]).
otherwise the probability is zero.
If (M) holds, the density of the explosion time T i (X i (0)) (computed in [40]) as a sum of exponential variables has support on the whole positive real line for all choices of F i . So the probability of sM on i (χ(0), N ) is positive if and only if agent i fulfills (M) and the monopolist is random among all agents i ∈ [A] that satisfy (M). For the polynomial case On the other hand, when no agent fulfills (M) we have the following consistency property. Proposition 2.3. Assume that none of the F i satisfies (M). Define a 'partial' Pólya urn process X(n) for a subset B ⊂ [A] of agents with the same feedback functions F i and initial conditioñ X(0) = (X i (0) : i ∈ B). Then the process X (n) n∈N 0 can be identified as a (random) subsequence of X i (n) : i ∈ B n∈N 0 .
Proof. The independence property of the exponential embedding provides a canonical coupling of the processesX and X. For that, define recursively s 0 = 0 and s n+1 := inf{s > s n : ∃i ∈ B : Ξ i (s) = Ξ i (s n )} .
Note that s n < ∞ is well defined for all n ≥ 0, since none of the F i fulfill (M). Then setX i (n) := Ξ i (s n ), which directly implies the claim since (s n ) is a subsequence of (t n ).
In particular, if one of the limits exists, then so does the other and both have the same distribution. This implies further neutrality of χ(∞) in the sense of [22], so that it has a (possibly degenerate) Dirichlet distribution on ∆ A−1 , whenever it exists. According to [22], any degenerate Dirichlet distribution is either deterministic or concentrated on the vertices of ∆ A−1 , i.e. P A i=1 wM on i (χ(0), N ) = 1. This will be discussed in several examples in Sections 5 and 6.

Literature Review
As already described in the introduction, generalisations of Pólya urns have been studied in numerous papers. In this section, we shortly present a selection of results related to our work. To our knowledge, the most comprehensive result concerning the long time limit of the process (χ(n)) n of market shares is the following.  (2)) exists for all x ∈ ∆ A−1 and that even holds. Moreover, assume that there is a twice differentiable Lyapunov function for the vector field Then χ(n) converges almost surely for n → ∞ and the limit is either in {x ∈ ∆ A−1 : G(x) = 0} or the border of a connected component of this set.
Note that a Lyapunov function does always exist in the case A = 2 and when p is differentiable with equal feedback functions for all agents. Moreover, [9] shows under mild technical assumptions that each stable fixed point of the vector field G is attained in the limit n → ∞ with positive probability, whereas unstable fixed points are never attained.
Theorem 3.1 allows to compute the long time market shares in generic situations, like F i (k) = α i k β . Nevertheless, condition (6) is not fulfilled e.g. for F i (k) = log(k) or F i (k) = e k .
In the monopoly case described in Theorem 2.2, the monopolist is in general random. Consequently, one is interested in the probability that a specific agent is the monopolist, at least in the limit N → ∞. [30] derives such a result in a situation with only two symmetric agents.
Moreover, suppose that there is a constant C > 0 such that for all ∈ (0, 1 2 ) and all x > 0 large enough holds. Let X(0) = (N + λq(N ), N − λq(N )) for N, λ > 0 and q(a) := a 4a d da log F (a)−2 . Then the probability of agent 1 being the monopolist converges to Φ(λ) for N → ∞, where Φ denotes the cumulative distribution function of the normal distribution.
For F (x) = x β , β > 1 these assumptions are fulfilled and q(a) = √ a √ 4β−2 . Under similar assumptions as in Theorem 3.2, [34] shows that the number of steps, in which the looser wins, has a heavy tailed distribution. Moreover, if χ i (0) < 1 2 for an agent i, then P(sM on i (χ(0), N ) is exponentially decreasing in N , i.e. the first steps of the process decide who wins. [14] provides similar results for the asymmetric case [29], a result for polynomial feedback with different exponents was shown.
In addition, [29] provides a result for the critical case α = α cr , ν = ν cr . For F i (k) = k β , i ∈ [A] with β < 1, we know from Theorem 3.1 that lim n→∞ χ i (n) = 1 A almost surely for all i ∈ [A] irrespective of the initial configuration χ(0). The rate of convergence is specified in [26].
for a random, nonzero vector C.
where N denotes a Gaussian distribution.
The convergence in part 2 and 3 can be extended to the vector χ(n). Part 1 implies that the leading agent does only change finitely often. According to [32,Theorem 1], this happens in general if and only if

Asymptotics for the monopoly case
We assume that at least one agent i fulfills (M), so that a random strong monopoly occurs with probability one. To characterize the asymptotics, we have to distinguish two different types of feedback functions with slightly different behavior.
and of type E (for exponential) if lim sup For the rest of this section we assume that all agents with feedback functions that fulfill (M) are either of type P or type E. Of course it is possible to construct counter-examples (see Example 4.3), but these two types still cover a very large range, including most previous results.
then F is of type P, and if lim inf then F is of type E.
Proof. First we assume (9) and observe that Consequently, for any given > 0 there exists k 0 such that ∀k ≥ k 0 : F (k + 1)/F (k) ≤ 1 + . Then we get for k ≥ k 0 : The result for type E follows similarly.
This means that functions that grow exponentially or faster are of type E whereas functions that grow slower than exponential (like polynomials) are of type P. Note that Oliveira's "valid feedback functions" in [34] or [32] are of type P, which includes furthermore all regular varying functions. 1. The conditions from Proposition 4.2 are not necessary for being type P resp. E. For instance take any function F of type E and defineF (2k) =F (2k + 1) = F (k). Theñ F is also of type E, but does obviously not fulfill (10).

2.
A possible construction of a feedback function that is neither of type P nor type E, but satisfies (M), is the following. Take a function F such that holds, e.g. F (k) = e k . Then define a new feedback functionF by replacing each F (k) by k elements that all equal kF (k), i.e. One can easily check thatF i has the desired properties.

Asymptotic attraction domains
If at least one agents fulfills the monopoly condition (M), we know by Theorem 2.2 that there is a strong monopoly, where all agents satisfying (M) have a positive probability of being the monopolist. Thus, the monopolist is in general random. Nevertheless, in most situations it is possible to predict the winner with high probability for large initial market size.
Obviously, the asymptotic attraction domains are disjoint, since P(sM on j (χ(0), N )) ≤ 1 − P(sM on i (χ(0), N )) for j = i. The main result of this section states that the asymptotic attraction domains cover the whole simplex up to boundaries under mild regularity conditions. Theorem 4.5. Let at least one agent satisfy (M) and all agents satisfying (M) are either of type P or type E. Moreover, assume that one of the following conditions holds: 1. At least one agent is of type E and for all 2. No agent is of type E and all agents of type P (there is at least one) fulfill lim sup In addition, suppose that lim N →∞ Then the asymptotic attraction domains are polytopes that dissect the simplex up to boundaries, i.e.
where (·) is the topological closure. If agent i does not satisfy (M) then D i = ∅.
As a direct consequence of the exponential embedding, P(sM on i (χ(0), N ) = 0 for all agents that do not fulfill (M). Hence, their attraction domains are empty. The rest of Theorem 4.5 basically follows from the results presented in the following subsections, where e.g. explicit conditions for only for agents of type P, but not for type E, so we need to study these two types of feedback functions separately. The technical conditions in each case are mild and will be discussed in the following subsections. Another characteristic of type E is, that a strong monopoly is typically even a total monopoly, at least when N is large.
Theorem 4.6 is a direct consequence of Theorem 4.7 given below. As explained in Corollary 4.11, total monopoly does in general not occur, if χ(0) is on the boundary of the attraction domain. In adddition, it turns out that in generic situations the probability of total monopoly is bounded away from one, if all agents are of type P.

Agents of type E and total monopoly
This subsection examines the process, when at least one agent is of type E. The following results basically imply the first part of Theorem 4.5 as well as Theorem 4.6 as described in Section 4.4. The main result of this subsection provides a useful lower and upper bound for the probability of total monopoly.
Proof. Direct calculation yields using log(x + y) − log(x) ≥ y x+y in . An immediate consequence of Theorem 4.7 is, that for any agent fulfilling (M) the probability of a total monopoly is positive but less than one. In addition, the theorem reveals a significant behavioural difference between agents of type E and type P (see figure 3): whereas total monopoly is very likely for type E agents when the initial market size N is large, it is rather untypical for type P, which is explained in the following corollary and example.  The type E case (left) reveals a total, the type P case (right) a strong monopoly. 1. If agent i is of type E , then for all χ(0) ∈ ∆ o A−1 the following are equivalent: 2. If agent i fulfills (M), then for all is sufficient for (15). If in addition F i (k) is monotone for large k, (16) is equivalent to (15).
Proof. 1. If i is of type E, then (14) implies (8), and (15) follows from the lower bound of Theorem 4.7. The necessity of (14) follows from 2. (16) implies that the lower bound of Theorem 4.7 converges to one so that (15) holds. Now we assume that (16) does not hold. If F i (χ i (0)N ) F j (χ j (0)N ) does not converge to infinity for some j = i, then with 1., (15) cannot hold. Thus we can assume (14) for all j = i, which implies c N N →∞ − −−− → 1 for the upper bound in Theorem 4.7 due to asymptotic monotonicity of F i (χ i (0)N + k) as N → ∞. The upper bound then implies that P(tM on i (χ(0), N )) does not converge to one.

When
Remarkably for type E agents, if F i (k) = α i F (k) for all i and a function F fulfilling (8), then for large N the almost surely deterministic monopolist does not depend on the attractivenessparameters α i , but is only determined by the initial condition due to the strong feedback effect of type E functions.
Moreover, Theorem 4.7 provides information about the rate of convergence in (15) and (22). If agent i is of type E, then Theorem 4.7 states together with 1 + x ≤ e x and k because of (8). Thus the convergence can be considered as quite fast. Example given, in the situation of Figure 3(a), the bounds in Theorem 4.7 are: 0.652 ≈ e −2/(e(e−1)) ≤ P(tM on 1 (χ(0), N )) ≤ e −2e/((e−1)(2−e 2 )) ≈ 0.714 Indeed, condition (14) is fulfilled for an i in most generic cases, when at least one agent is of type E. To be more precise: If the expression in (14) neither converges to infinity nor to zero, then an arbitrarily small change in the initial market shares provides (14). 1. If j is of type P for all j = i, then (14) holds.
then for any > 0: for some c ∈ (0, 1) and k large enough, thus the sequence converges to zero faster than (1 − c) N/χ i (0) . For an agent j = i of type P we have by (7) for any d > 0 ∞ l=k+1 exponentially fast as d is arbitrarily small. Finally (14) follows from F j (l) → ∞ slower than exponentially. 2. Now let agent j = i be of type E and assume (17). Then with (8): and as a consequence Once again, the estimate in (19) proves the claim together with (8).

Corollary 4.8 implies that for any agent
Due to Proposition 4.10, these sets are even equal up to boundaries under Assumption 1 of Theorem 4.5. Moreover, the first part of Proposition 4.10 states that the attraction domains of all agents of type P are empty, if there is at least one agent of type E. Recall that for finite N the probability of monopoly is positive for all agents satisfying (M). Finally, one can ask what happens for large N and critical market shares, i.e. for χ(0) lying exactly on the edge between the asymptotic attraction domains. It stands to reason that in this situation the monopolist remains random even for large N . Nevertheless, the exact limiting behaviour depends on whether the feedback functions grow exponentially or even super-exponentially.
Corollary 4.11. Let all agents be of type E and consider . Then the following holds:

If for all agents i ∈ [A]
we have super-exponentially growing feedback, i.e.

If for all agents i ∈ [A]
we have at most exponentially growing feedback, i.e.
lim sup Proof. 1. This follows directly from Theorem 4.7 and (8): and then apply Theorem 4.7 and (8): Similarly to the second part, this follows from . According to Corollary 4.11, we have in this case We summarize the main conclusions for total mononpoly in the limit of large initial market size N → ∞: If for all agents the feedback functions grow super-exponentially, the winner of the first step will win all steps. This does not hold for any χ(0) ∈ ∆ o A−1 if all feedback functions grow at most exponentially. In general, total monopoly of an agent i can occur with probability one according to Corollary 4.8: if i is of type E and (14) holds, or if (16) holds.

Agents of type P
Let us now turn to the more widely studied case when all agents are of type P. We already saw in Example 4.9 that in this case a total monopoly is rather untypical. Since the definition of type P includes the monopoly condition (M), strong monopoly still occurs with probability one. Again, it is possible to predict the monopolist in the limit N → ∞.
Note that condition (21) can be replaced by the easier, but stricter condition due to de l'Hospital's Theorem. This implies that for regular varying F i (k) = α i k β L(k), where β > 1 and L is a slowly varying function, the attraction domains are equal to the polynomial case, where F i (k) = α i k β . Moreover, the attraction domains do not change if F i is replaced by another Proof. This is an immediate consequence of the following Lemma 4.14 and the exponential embedding representation (5) of the strong monopoly via . Lemma 4.14. If agent i is of type P, then: Proof. We can find an appropriate regular extension of F i , such that for all n ≥ 1 By the theorem of de L'Hospital and (7) this implies Thus, in contrast to the type E case (Example 4.9), the attractivenessparameters α i affect the monopolist. Lemma 4.14 uncovers another behavioral difference between type P and type E agents: For type P agents the explosion time concentrates on its expectation, whereas the variance of T i (χ i (0)N )/ET i (χ i (0)N ) remains bounded from below for type E agents by an analogous argument, using (8). For many type P agents, including It is now natural to look for an analogy to Proposition 4.10 for type P agents in order to make sure that (21) is fulfilled for almost all initial market shares χ(0). Unfortunately, this attempt is meant to fail as the example F i (k) = F j (k) = k(log k) α for α > 1 shows. In this case (21) is not fulfilled for all choices of Nevertheless, with a further condition we can find a similar result as Proposition 4.10.

If
and (23) holds for one choice of χ i (0), χ j (0), then (23) holds for all choices Proof. 1. We have by (12) and iterated application of this yields Finally, this implies 2. The second part follows by similar arguments, using Condition (24) for an "≥"-estimate in (25), where C = C(N ) is arbitrarily large.
If all agents are of type P, Theorem 4.13 implies that for any agent i ∈ [A] holds. Assuming 2. in Theorem 4.5, we get from Proposition 4.16 that the sets are equal up to boundaries.
In the situation of the second part of Proposition 4.16, the explosion times concentrate asymptotically on the same value, i.e.
Thus, it is not possible to predict the monopolist for large N by the means of this section. If α 1 = . . . = α A and χ i (0) = 1 A for all i, then P(sM on i (χ(0), N )) = 1 A holds for all N for symmetry reasons, i.e. χ(0) does not belong to any attraction domain as the monopolist remains random even in the limit N → ∞. The following example underlines that this property does not hold in general, because in some cases the boundary between the attraction domains belongs to one of them.
for N → ∞. Moreover, set χ 1 (0) = χ 2 (0) = 1 2 , such that (23) holds, and define N := N 3 4 −β . Chebyshev's inequality yields In addition, we have for large enough N that Thus for large N We finish this subsection with a result on the rate of convergence in (22). [14] presents a bound for P(sM on i (χ(0), N )) in the case F i (k) = F j (k) = k α , but a straight-forward generalization of this procedure is possible. 19. Let all agents be of type P with monotone feedback functions, such that (9) holds in addition. If (21) holds for agent i, i.e. χ(0) ∈ D i , we have This means that the rate of convergence in (7) gives a lower bound for the rate of convergence of P(sM on i (χ i (0)).
Proof. Once again, the proof uses the exponential embedding from Section 2. Let t > 0 and Then the Markov-inequality and monotone convergence yield for all j ∈ [A] and t > 0: where c j (k) := exp The second estimate uses F j (l) ≥ F j (k) by monotonicity. Analogously, one can show which will be used only for j = i. Both estimates then imply for large enough N together with (5): for j = i because of (7) and c j (k) k→∞ −−−→ 1 due to (9). Finally, this leads to ) can be considered as fast. Hence, P(sM on i (χ(0), N )) is close to one even for moderate N , when χ(0) ∈ D i is in the asymptotic attraction domain.
In the type E case we saw that a total monopoly is very likely whereas in the type P case the losers might also win in some steps. It is now a question of interest how many steps the losers win, i.e. the value of X j (∞) = lim n→∞ X j (n) if agent j is not the monopolist. Results on this question can be found in [34] and [40]. It is remarkable that for polynomially growing feedback functions the distribution of X j (∞) has heavy tails. [40] also presents results on the time when the monopoly occurs. Further asymptotic results on strong monopoly, mainly in the type P case, can be found e.g. in [26,30,33,14,13,29,24].

Proof of Theorem 4.5 and Theorem 4.6
Finally, we shortly explain how Theorem 4.5 and Theorem 4.6 follow from the results of the previous sections.
First, assume that Assumption 1 of Theorem 4.5 is satisfied, i.e. at least one agent is of type E. Then Corollary 4.8 implies that for any agent i ∈ [A] of type Ẽ is an intersection of half-spaces and the simplex, i.e. a polytope. Moreover,D 1 , . . . ,D A cover the whole simplex up to boundaries, since the "winning"-relation lim N →∞ Thus, D 1 , . . . , D A cover the simplex up to boundaries as well andD i equals D i up to boundaries. According to Corollary 4.8, we even have P(tM on i (χ(0), N )) − → 1 for N → ∞, if χ(0) ∈D i . Hence, Theorem 4.6 is proven, too.
If Assumption 2 of Theorem 4.5 is satisfied, the proof is analogous using Theorem 4.13 and Proposition 4.16. Note that due to the Theorem of de l'Hospital. In summary, for finite N the monopolist is random and even disadvantageous agents can win. If the initial market size N is large, it is possible to predict the winner with high probability depending on the initial market shares.

The non-monopoly case
Now we consider the case when no agent fulfills (M), such that no strong monopoly occurs. It is known that in the case of a standard Pólya urn, i.e. F i (k) = k for all agents, the limit exists almost surely and χ(∞) has a Dirichlet-distribution with parameter X(0) (see e.g. [17]). Thus, in the long run all agents have a stable, non-zero, random market share.
It is basically known (e.g. from [9]) that if the feedback functions grow significantly slower than linear, then χ(∞) is deterministic. We present an alternative approach to the sub-linear case, which allows some additional insights. For example, the case F i (k) = log(k) is not included in the results of [9]. In addition, our approach allows to construct feedback functions such that χ(n) does not even converge for n → ∞. In order to get deterministic limits in our approach, we will need a condition, which ensures that the feedback functions grow slow enough. We will mainly use: Note that this already implies that i does not fulfill (M). We add some examples to gain an understanding of this restriction.
In fact, condition (26) contains a monotonicity in the following sense. Proof.
The assumption implies via (11) for all k, l ∈ N and hence: In general, our approach even allows feedback functions that converge to zero as long as this convergence is not to fast, which is ensured by the condition lim inf Note that (27) is fulfilled for any feedback function with lim inf k→∞ F i (k) > 0 as well as for F i (k) = k −α , α > 0, but not for F i (k) = e −k . In analogy to Proposition 5.2 we get a monotonicity here in the sense that if F i fulfills (27) and then F j fulfills (27), too. We are now prepared for the main result of this section regarding the counting processes (3) of the exponential embedding from Section 2. (26) and (27). Then Note that a −1 i exists as a i is strictly monotone. The asymptotics of birth processes have been studied in the literature before, e.g. in [6]. One main result of [6] will be used for a special case in Section 6 to abandon condition (26). The following lemma provides the first step of the proof of Theorem 5.3, using standard ideas from the renewal theory.
According to the Kolmogorov criterion (see e.g. [19], Section 6.2) this is sufficient for where S i (k) := k l=1 τ i (l). We use this and Ξ i (t) → ∞ a.s. for the final estimate: Now Theorem 5.3 is easy to prove.
Proof. Lemma 5.4 states almost surely (using the Landau o-notation). It thus remains to show that The condition (26) implies using a i (k) ∼ k and hence, replacing t by a −1 i (t) (note: a −1 i (t) → ∞), we get: which includes (28). Theorem 5.3 implies that the market shares in the exponential embedding are asymptotically given by for t → ∞.
Via (4) we can now conclude for the discrete-time urn model.
Corollary 5.5. Let all agents fulfill (26) and (27). If the limit exists for an i ∈ [A], then If the limit in (29) does not exist, then χ i (n) does not converge for n → ∞.
If the limit in (29) exists for all i ∈ [A], then χ(n) Note that the a −1 i do not depend on N and χ(0), thus the long time behavior of market shares (χ(n)) n∈N in the generalized Pólya urn does not depend on initial conditions if (26) and (27) are satisfied. If the limit in (29) exists, a market modeled by a Pólya urn under the assumptions of the corollary reveals stable and deterministic market shares in the long run and these market shares do not depend on the current market situation and can also take values in (0, 1). If the limit χ(∞) exists it is in ∆ A−1 , since ∆ A−1 is compact and therefore the laws of χ(n) form a tight sequence. The corollary provides a way to explicitly calculate these long-time market shares. Consequently, the impact of the fitness parameters α i in the long-time limit increases with β, where The limiting case β → 1 will be discussed later in Proposition 5.9.

If
and thus: χ i (∞) = α i α 1 + ... + α A Note that this is the same asymptotic market share as if the customers' decisions were independent (with constant feedback functions as for β = 0 above), so that the strong law of large numbers applies.
It is also possible to find examples where the limit (29) does not exist. In the following situation the market share of the agents oscillates with constant amplitude but increasing period.
We now add a criterion that ensures the existence of the limit in (29).
Corollary 5.8. Suppose that for an agent i the following tightening of (26) holds, and that the limits Then the limit in (29) exists and In particular, P(wM on i (χ(0), N )) = 1 if and only if all c j are infinity, otherwise P(wM on i (χ(0), N )) = 0. If all c j are one, then the condition (30) can be replaced by (26) and χ j (∞) = 1/A for all j = 1, ..., A.
Proof. Recall that a i (t) = t 1 dx F i (x) is strictly increasing. For a fixed j = i we show that Ξ j (t)/Ξ i (t) converges to c −1 j . First, we assume 0 < c j < ∞, such that agnets i and j fulfill (26) and (27). (31) implies via the theorem of de l'Hospital a j (t)/a i (t) → c j for t → ∞ and consequently a −1 Thus: For agents j with c j = 0 the asymptotic market share is for sure bigger than in a situation where F j is replaced by CF i , C > 0, i.e. Ξ j (t)/Ξ i (t) is for t → ∞ larger than any C. Hence, it converges to infinity. Similarly for agents with c j = ∞.
Note that in the case c = 1 (including e.g. feedback functions such as log k, 1/ log k or functions converging in (0, ∞)) the limit χ i (∞) is equal to the case F i (k) = const., i.e. draws from the urn are independent and the usual strong law of large numbers applies. So this weak reinforcement does not play any role on the long run.
So far, we did not consider cases near the classical Pólya urn with F i (k) = k, where random limits χ i (∞) are possible. Nevertheless, as Lemma 5.4 does not require (26), our approach provides some insight into such asymmetric cases as well. The symmetric case with feedback functions close to the classical Pólya urn is treated in Section 6.
Proposition 5.9. Let an agent i fulfill but not (M), i.e.
Proof. First note that via exponential embedding, the event wM on i (χ(0), N ) is equivalent to Obviously, agent i fulfills (27). First, we assume that agent j does, too. Define and thus a −1 i (t) = e ψ −1 (t) . Assumption (33) implies that for any j = i there is a constant c < 1 with a i (t) ≤ ca j (t) for large enough t and consequently a −1 If agent j does not fulfill (27), then F j is bounded from above and hence Ξ j is stochastically dominated by a homogeneous Poisson process (with constant rate). Consequently, Ξ j (t) grows asymptotically not faster than linear and hence Ξ i (t)/Ξ j (t) → ∞ almost surely.
Condition (32) includes feedback functions of the form F i (k) = α i k(log k) β for all β ≤ 1, including the linear case F i (k) = α i k for β = 0. If in addition (33) holds, i.e. α i > α j for an agent i and all j = i, then we have an almost sure weak monopoly for agent i. This is consistent with the strong monopoly for β > 1 as described in Example 4.17. Note that the weak monopoly in Proposition 5.9 is almost sure even for finite N , in contrast to the results on strong monopoly derived in Section 4, where the strong monopolist is random and can only be predicted in the limit N → ∞.
On the other hand, condition (26) includes sublinear feedback functions of the form F i (k) = α i k β with β < 1, which have positive long-time market shares for all agents as discussed in Example 5.6.
Exponentially decreasing feedback functions were not taken into account so far as they do not fulfill (27). Since such cases do not seem to be of great importance for the mentioned interpretations of the model, we are content with an example. Other cases where (27) is not fulfilled can be treated similarly.
Example 5.10. Let A = 2 and F i (k) = α i e −β i k , α i , β i > 0, i = 1, 2. As explained in detail in Section 7, we can write where (M 1 (n)) n∈N 0 is an almost sure convergent martingale and H 1 (n) := n−1 k=0 G 1 (N + k, χ 1 (k)) N + k + 1 is predictable with G 1 (k, x) := p 1 (k, (x, 1 − x)) − x, x ∈ (0, 1) given by centered transition probabilities (2). In the case of exponentially decreasing feedback, we have the following convergence: The convergence is locally uniform in (0, 1) apart from the point x = x 0 := β 2 β 1 +β 2 . Take > 0. For large enough k, G 1 (k, (·)) is sufficiently close to G 1 outside an -neighborhood of x 0 . If for a large n, |χ 1 (n) − x 0 | > , then the process (χ 1 (n)) n enters the -neighborhood of x 0 in finite time because of the convergence of the martingale. As the same holds for /2 instead of , we get that the process leaves this -neighborhood only finitely often. This yields Thus, the limit is not only independent of the initial market shares, but also of the fitness-parameters α i (in contrast to polynomially decreasing feedback). Note that these findings are consistent with Corollary 5.5, i.e. (29) still holds. Because of the independence property in the exponential embedding in Section 2, this can easily be extended to general A. For different (at least) exponentially decreasing feedback, we basically only need a convergence as in (34) for an analogous result.
Remarkably, Example 5.10 reveals the following behavioural difference between exponentially decreasing and polynomial feedback. Suppose that there are agents i, j such that Then agent i is marginalized, i.e. lim n→∞ χ i (n) = 0, if F i satisfies (30), in particular if F i (k) = α i k β i for β i < 1. On the other hand, for exponentially decreasing feedback like in Example 5.10, we might still have lim n→∞ χ i (n) > 0. We conclude the presentation with a short overview of further related results. [31,33,25] discuss another change of behaviour that is not apparent from our approach. Consider the case for F i (k) = k, then the leading agent changes only finitely often with probability one, whereas in the case [26] that χ i (n) converges to 1/A at rate n β−1 for 1/2 < β < 1 (almost surely), at rate n −1/2 for 0 < β < 1/2 and at rate log(n)/n for β = 1/2 (in a weak sense). For 0 < β ≤ 1/2, a central limit theorem holds.
In the case A = 2 and F i (k) = α i k β [24] derives the tail distributions of the number and last times of ties X 1 (n) = X 2 (n).

Feedback functions close to the classical Pólya urn
We know from Theorem 2.2 that a generalized Pólya urn reveals strong monopoly if and only if at least one feedback function grows significantly faster than linear, i.e. fulfills (M). As described in Section 5, linear feedback functions imply random long-time market shares, whereas a deterministic limit occcurs for feedback functions growing significantly slower than linear, i.e. those fulfilling (26). Nevertheless, some feedback functions that are close to linear (like F i (k) = k(log k) β , β = 0) are not covered by our results so far. To our knowledge, the literature does not provide results on the long time behaviour of a generalized Pólya urn with almost linear feedback. For instance, if F i (k) = kL(k) for a slowly varying function L, then Theorem 3.1 does not determine the longtime limit, since lim N →∞ p(N, x) = x for all x ∈ ∆ A−1 . We approach this question exploiting general results on birth processes, which require that F i does not fulfill (M) but inverted squares are summable, i.e.
Recall the exponential embedding from Section 2 and notations introduced therein. For this section, it is convenient to adapt previous definitions using and to extend F i on (0, ∞) by a right-continuous step function. The key to the desired results is provided by the following result in [6].  (35). Then t − a i (Ξ i (t)) and S i (k) − a i (k) converge almost surely for t → ∞ resp. k → ∞ to the same random variable U i ∈ R. Moreover, σ 2 i is the variance of U i . We can now apply this general result in our situation. Corollary 6.2. Assume that F i fulfills (35). Then: Proof. Theorem 6.1 implies t − a i (Ξ i (t)) = U i + o(1) and hence and consequently ds .
Now, be aware that lim t→∞ a −1 i (t) = ∞ as F i does not fulfill (M) and that the limit of F (k + const.)/k for k → ∞ is equal to the limit of F (k)/k. Then all parts of the corollary follow directly from their assumptions.
Like in Section 5, we can now conclude from the exponential embedding to the evolution of market shares in the Pólya urn via lim n→∞ χ i (n) provided that the limit exists.
We are now particularly interested in cases with equal feedback functions for all agents, since agents with different attractiveness are already covered by Proposition 5.9. Corollary 6.3. Assume that all agents have the same feedback function F i ≡ F and that F fulfills (35). Then for all χ(0) ∈ ∆ o A−1 : 2. If lim k→∞ F (k) k = c ∈ (0, ∞), then the limit χ(∞) = lim n→∞ χ(n) exists almost surely and has a non-degenerate Dirichlet distribution on ∆ A−1 .
3. If lim k→∞ F (k) k = ∞, then χ(∞) = lim n→∞ χ(n) exists almost surely and the process exhibits a weak monopoly, i.e In other words: If the feedback function grows any slower than the identity, then the market shares converge to a deterministic limit as time tends to infinity, and the limit does not depend on the initial condition. If the feedback functions grow any faster than the identity, the process exhibits weak monopoly, which is not strong as (M) is necessary in Theorem 2.2. In contrast to the non-symmetric situation of Proposition 5.9, the monopolist is random with probability depending on the initial condition χ(0).
Proof. Note that the U i from Theorem 6.1 are independent with distribution depending on χ(0) and N . In addition, their distribution is continuous as U i emerges from a sum of independent, centered exponentially distributed random variables. By definition (36) we get a i (t) = a j (t + const.) + const. with constants depending on the initial conditions and F , and after inversion we have a −1 and note that a −1 j (t) → ∞ in all cases. 1. In this case (38) implies that a −1 i (t) ∼ a −1 j (t). Then the claim follows directly from Corollary 6.2 via lim n→∞ χ i (n) 2. Here, again with (38), a −1 i (t)/a −1 j (t) converges to a finite, non-zero constant for all i, j ∈ [A], such that Corollary 6.2 yields Hence, lim n→∞ χ(n) exists almost surely and has a continuous distribution, which is of Dirichlet type according to Proposition 2.3 and [22]. 3. Due to Lemma 6.4, we can assume X(0) = (1, . . . , 1), so that a i = a j and h i = h j . Then: Recall that lim s→∞ h i (s) = ∞. Again by Lemma 6.7, the unboundedness of the U i implies that Lemma 6.4. For all choices of F 1 , . . . , F A , we have Proof. The implication ⇐ is trivial. Thus, assume that the process X(n) starts in X(0) = (1, . . . , 1) and that P The following example presents a class of feedback functions, for which four different regimes are possible. Example 6.5. Let F i (k) = k(log k) β for all i ∈ [A] and β ∈ R. Depending on β, four different regimes occur for n → ∞: 1. For β < 0, χ i (n) for each agent i converges almost surely to 1 A independently of χ(0).

2.
For β = 0, the market shares χ(n) converge almost surely to a random limit χ(∞) ∈ ∆ A−1 , which is not a corner point and its distribution depends on the initial condition χ(0).
3. For β ∈ (0, 1], the process exhibits a weak monopoly which is not strong, i.e. all agents win in infinitely many steps, but the market share of one agent converges to one. The monopolist is random, and the distribution of χ(∞) on the corner points of ∆ A−1 depends on χ(0).

4.
For β > 1, there is a strong monopoly. The monopolist is random and the distribution of χ(∞) on the corner points of ∆ A−1 depends on the initial condition χ(0) as well.
According to Theorem 6.1, we have t n − a i (X i (n)) n→∞ −−−→ U i by definition of the exponential embedding with jump times t n . If lim k→∞ F (k)/k = ∞, this convergence can be specified by replacing t n by a deterministic function and by computing the distribution of U i . Theorem 6.6. Assume that F i ≡ F does not fulfill (M) and that F (k)/k k→∞ −−−→ ∞ holds. Then there exist independent random variables U 1 , . . . , U A such that almost surely. Moreover, the cumulant generating function (CGF) of each U i is given by (39) and the radius of convergence is min k≥X i (0) F (k).
In particular, there is exactly one agent, namely the weak monopolist, such that the limit of a i (n) − a i (X i (n)) is zero. For the proof, we characterize the distribution of U i by computing its CGF. For that, we exploit that U i is also the limit of S i (k) − a i (k) for k → ∞ according to Theorem 6.1. Lemma 6.7. Assume that F i ≡ F fulfills (35). Then the CGF of U i is given by (39) and the radius of convergence is min k≥X i (0) F (k).
Proof. The CGF of the limit is the pointwise limit of the CGFs: We now use the series representation of x → log(1 + x) and change the order of summation due to absolut convergence: The radius of convergence of the power series representation of the CGF is given by In particular, EU i = 0 since the first term in the series is λ 2 , and the l- For the proof of Theorem 6.6, it remains to show that t n − a i (n) converges as desired.
Lemma 6.8. In the situation of Theorem 6.6 we have Proof. By definition of t n and a i (n) and by Theorem 6.1, we have
According to Theorem 6.6, X i (n) is asymptotically well described by . Now, consider two distinct agents i, j and assume for simplicity of notation that X i (0) = X j (0), such that a i (k − X i (0)) = a j (k − X j (0)) =: a(k). Then Theorem 6.6 states that Moreover, the CGF of U j − U i is the sum of the CGFs of U j and U i due to independence. Hence, Ee λ(U j −U i ) is finite if and only if |λ| < min k≥X i (0) F (k). Thus, the distribution of U j − U i has exponential tails, and these findings can be used as follows.
Example 6.9. Let F i (k) ≡ F (k) = k log(k) and X i (0) = X j (0) = 1 for two agents i, j ∈ [A], so that a i (t) = a j (t) = log log t. Then the continuous mapping theorem yields where e U j −U i has a power-law distribution due to the explanations above. Remarkably, the logratios log X i (n) log X j (n) and log X i (n) log X j (n) are asymptotically also independent for distinct pairs of agents (i, j), (i , j ).
An important application of Theorem 6.6 is its implication for the rate of convergence. In fact, the convergence of the process of market shares χ(n) to an edge of the simplex can be considered as logarithmically slow. Proof. Since the limit in Theorem 6.6 is finite, there is a constant c > 0 such that Since lim n→∞ for an updated constant c, which proves the claim.
In particular, χ i (n) converges to zero slower than any polynomial when lim n→∞ L(n)/ log(n) = 0. The following example discusses that bound in a generic situation. Example 6.11. Let F i (k) ≡ F (k) = k(log k) β for β ≥ 0. For β = 0, the lower bound e −cL(n) is constant since χ i (n) does converge to a non-zero limit. For β ∈ (0, 1), the bound converges to zero slower than any polynomial, whereas it is of order n −c for β = 1. Note that c is random and unbounded. Finally for β > 1, the process reveals strong monopoly such that χ(n) converges to an edge of the simplex at rate 1/n. In that specific case for β ≤ 1, we can also derive an upper bound for χ i (0), provided that agent i is not the monopolist. Since the limit in Theorem 6.6 is non-zero and a i (t) ∼ (log t) 1−β , there is a positive constant such that 0 < const. ≤ (log n) 1−β − (log(X i (n))) 1−β and consequently log(X i (n)) ≤ (log n) 1−β − const.
Note that 1− (n) > 0 converges to zero at rate 1/n 1−β , so that we finally get the following estimate: Thus, the bound in Corollary 6.10 can be considered as sharp.
If the second part of (35) is not fulfilled, i.e. σ 2 fulfills the Lindeberg condition. Hence, Theorem 6.1 and its implications are wrong if we drop the condition σ 2 i < ∞. As already described at the end of Section 5, [26] derives a central limit theorem for polynomial feedback functions with σ 2 i = ∞. Moreover, [31,33,25] present another transition between functions satisfying this condition and those who do not.
Another remarkable property is the following: The proof of part 3 of Corollary 6.3 reveals that X j (n) → 0 or ∞ for n → ∞ for all i = j. This corresponds to a hierarchical structure of asymptotic market shares consistent with weak monopoly and the consistency property in Proposition 2.3, such that within each subset of agents a weak monopolist has full relative market share. Such hierarchical structures are often observed at phase transition points, in our case the transition between strong monopoly and deterministic limit shares.

A law of large numbers for the dynamics
So far our investigations focused on the analysis of the long-time behavior of a generalized Pólya urn. This section examines the dynamics of the process in the limit for large initial market size N , based on the concept of stochastic approximation (see e.g. [9,35,37]). Note that X(n) and χ(n) depend on N , thus we establish the notation X (N ) (n) = X A (n) = χ(n) for this section and assume that χ (N ) (0) is equal for all N (up to roundings).
where we assume that G(k, (·)) converges for k → ∞ uniformly to a Lipschitz-continuous function G on an open neighborhood D ⊂ ∆ A−1 of the image of the solution Z : (0, ∞) → ∆ A−1 of the differential equation d dt Moreover, we define the following sequence of stochastic processes in ∆ A−1 : for all t, s ≥ 0, where · = · ∞ denotes the supremum norm. This implies by [21,Proposition VI.3.26] that the sequence (Z (N ) ) N is tight in D([0, ∞), ∆ A−1 ), with the additional property that all weak limits of converging subsequences are concentrated on the subspace of continuous functions. We now take any converging subsequence and show that the limit solves (41). As the solution of (41) is unique due to the assumed Lipschitz-continuity of G, this implies the claim. For simplicity of notation assume that the subsequence is (Z (N ) ) N itself. Then we can write the increments as n ) n≥0 is the filtration generated by the process (χ (N ) (n)) n≥0 . Furthermore, With uncorrelated and centered increments (M (N ) (n)) n≥0 is a centered martingale with respect to the filtration (F (N ) n ) n≥0 , thus Doob's inequality yields for any , t > 0: since ξ (N ) (k) ≤ 1 almost surely by definition. Hence, the sequence M (N ) (N t) : t ≥ 0 N of stochastic processes converges to zero weakly on D([0, ∞), R A ). Now we turn to the predictable part H (N ) . By the Skorochod representation theorem we can find a probability space such that the convergence of (Z (N ) ) N is almost sure. Then for fixed ω ∈ Ω (Z (N ) ) N converges with respect to the Skorochod norm to a processẐ on ∆ A−1 . AsẐ is continuous, the convergence is uniform on bounded time intervals. Denote t 0 ∈ (0, ∞] the stopping time, when Z first leaves D. Then for any t < t 0 and large enough N = N (t) we have Z (N ) (t) ∈ D and consequently This means, that (χ (N ) (n)) n≥0 is asymptotically deterministic and driven by the vector-field (G(x)) x∈∆ A−1 modulo a time change. Let Y : [0, ∞) → ∆ A−1 be the solution of the time-homogeneous differential equation so that Z(t) = Y (log(1 + t)). Then for large N the process (χ (N ) (n)) n≥0 is approximately given by Y log 1 + n N n≥0 . We can use this result e.g. to estimate the number of steps until the process reaches a given neighborhood of its long-time limit for large N .     This follows directly from the Theorem 7.1 via the continuous mapping theorem. Another interesting consequence of Theorem 7.1 is the following. In the monopoly case described in Section 4, we may start our process in an unstable fixed point χ(0) of the vector field G. Although we know that the process exhibits strong monopoly, we have Z(t) ≡ χ(0) for all times t ≥ 0 in Theorem 7.1. This implies that a linear scaling of time is not sufficient to capture the escape from an unstable equilibrium. with the convention inf ∅ = ∞. Then Proof. This follows from Theorem 7.1 via for all t > 0 since Z(s) ≡ χ(0).
Simulations for F i (k) = k β , β > 1 indicate that the escape from an unstable equilibrium is faster the larger β is. Recall that for superexponential feedback functions (see Corollary 4.11) the winner of the first step wins in all further steps with high probability if N is large. Hence, it only takes O(N ) time to escape from an unstable equilibrium in this case. Nevertheless, this does not pose a contradiction to Corollary 7.3 since the convergence of G(k, (·)) to G is not uniform in an unstable equilibrium. Thus, Theorem 7.1 is not applicable and the assumption of uniform convergence can not be removed. Figure 4 shows the dynamics of the process (χ(n)) n in various generic situations. The fixed points of the dynamics, i.e. the zeros of the vector-field G, are the long-time market-shares of our generalized Pólya-urn, but only the stable fixed points are attained with positive probability. Figure (a), (b) and (c) comply with the properties found in the sections before, i.e. monopoly in the superlinear case and stable, non-zero market-shares in the sublinear case. Figure (d) underlines that the set of stable fixed points is not necessarily discrete. Note that when F i (k) = kL(k) for all agents i ∈ [A] and a slowly varying function L, then the field G is constantly zero, such that all points are fixed points. In particular, this holds for the original Pólya urn, where L is a constant function. If L diverges, then the process exhibits weak monopoly resp. deterministic limits for finite N (see Section 6), which is again not captured by Theorem 7.1 as it takes more than O(N ) steps to reach the long-time limit.
Moreover, the assumptions of Theorem 7.1 are not fulfilled for exponential feedback, since G is not continuous. Nevertheless, the dynamics in the limit N → ∞ are already described by Corollary 4.8, which states that all steps are won by the same agents as long as χ(0) is not on the boundary between the attraction domains. Note that this is consistent with Theorem 7.1, i.e. (41) still holds.
Since F i only depends on X i and not X j , j = i there are no limit cycles and the dynamics tends to a fixed point, as opposed to models discussed in [11].

A Functional Central Limit Theorem for the dynamics
In Section 7 we derived a law of large numbers for the process of market shares for large initial values. For that we decomposed the processes (χ (N ) (n)) n into a predictable part H (N ) and a random part M (N ) (defined in (43)). We showed that the random part vanishes for large N , whereas the predictable part converges to a deterministic function Z. This section derives a central limit theorem for the martingales M A ). For simplicity we will at first only consider one fixed agent (without loss of generality agent 1) while keeping A ≥ 2 general. We will use the notation introduced in Section 7. Note that the convergence of p(k, (·)) is equivalent to the convergence of G k in Theorem 7.1. Alternatively, the inhomogeneous Markov-process M 1 is characterized as the solution of the stochastic differential equation where B denotes a standard Brownian motion. Thus, M 1 is a time-changed Brownian motion. To be more precise, is the quadratic variation process of M 1 . Note that M 1 (t) is deterministic and monotone increasing in t, and thus M 1 (t) converges almost surely for t → ∞ and the limit has a centered Gaussian distribution with variance lim t→∞ M 1 (t).
For the proof of Theorem 8.1, we first show tightness of the sequence √ N M Proof. According to a version of the Aldous criterion in [39,Lemma 3.11], the following two properties are sufficient for the tightness. 1. Stochastic Boundedness: For C, T > 0 we have by Doob's inequality and (42) Similarly, we get for 0 < t ≤ T and 0 < u ≤ δ: By the definition of tightness and Theorem 7.1, we also get tightness of the joint sequence (Z (N ) , N 1− β 2 M (N ) 1 ( N β (·) )) N . Before we turn to the proof of Theorem 8.1, we add another helpful lemma.
Proof. Taylor-expansion of f with Lagrange's remainder yields: Now we are well prepared for the proof of Theorem 8.1.
Proof. We show that for any limit (Z, M 1 ) of a convergent subsequence of (Z (N ) , √ N M (N ) 1 ( N (·) )) N , M 1 is a Markov process with generator (L s ) s>0 . For simplicity of notation, assume that the sequence is convergent itself.
Take a smooth test-function f : R → R with compact support. Then for each N f √ N M Convergence for N → ∞ holds almost surely on an appropriate probability space by Skorochod's representation theorem, which implies weak convergence. Summing up, we have that (47) converges to for N → ∞.
As f and f are bounded, the sequence in (47) is obviously uniformly integrable in N . Thus, [39,Theorem 5.3] implies that (48) is a martingale as well. Moreover, the solution of the martingale problem (48) is unique as a time-changed Brownian motion is always the unique solution if its corresponding martingale problem. Hence, M 1 is a time-inhomogeneous Markov-process with generator (L s ) s≥0 .
and suppose that χ i (0)α i > χ j (0)α j for an i ∈ [A] and all j = i. Then M 1 (t) = 0 almost surely for all t ≥ 0, since p(x) = e (i) for x ∈ D i , in particular on the path of Z. This complies with the idea of a total monopoly described in Section 4.

If
for all t ≥ 0. Note that in this case the martingale part and β > 0. Since we start in a stable or unstable equilibrium point, we have Z(t) ≡ χ(0) and hence M 1 (t) = A−1 1+t for all t ≥ 0. In particular, M 1 does not depend on β.
For non-linear, polynomial feedback functions and general initial market shares, the expressions for Z are lengthy or even not explicit. Figure 5 shows some realisations of the process M 1 . It can be seen that the convergence of M 1 (t) for t → ∞ is faster the faster the feedback functions grow. In the monopoly case, the variation of M 1 is small if χ(0) is already close to zero or one.
So far in this section, we only considered one fixed agent. Nevertheless, one can obtain an extension of Theorem 8.1 for all agents by a completely analogous, but lengthy argument, which we leave to the reader.
The specific form of the generator is due to the conditioned covariance matrix of the increments  ξ (N ) , which is for j = i: Alternatively, the A-dimensional generatorL s can be rewritten as ∂x i ∂x j is the second derivative along the diagonal x i = x j . From this form of the generator it is easy to see (e.g. by a coordinate transformation) that M solves the system of stochastic differential equations where B i,j is a standard Brownian motion, which is independent of B k,l if {i, j} = {k, l} and B j,i = −B i,j for i = j. It follows immediately that A i=1 dM i (t) = 0 for all t > 0. Hence, the sum A i=1 M i (t) = 0 is a conserved quantity. Consequently, the state space of M is the tagent space of the simplex ∆ A−1 . This allows the following interpretation of the limit process M : Each pair of agents exchanges mass according to a time-changed Brownian motion and the exchange of several distinct pairs of agents is independent. Figure 5 shows two simulations of the process M with polynomial feedback. From a stochastical point of view, the first steps of a generalized Pólya urn are of special interest because the randomness plays a significant role. In the later stages of the process, the market shares and thus the probability of winning in a certain step remain almost invariant, such that the sequence of winners (X(n + 1) − X(n)) n is almost independent and identically distributed for large n. Even in the Central Limit Theorem 8.1 the limiting process M becomes virtually constant for large t. In order to particularly focus on the early stages of the process, we finally analyse the process for large initial market size N and β ∈ (0, 1). Theorem 8.6. Suppose that the assumptions of Theorem 8.1 are fulfilled and denote by (B t ) t≥0 a standard Brownian motion. Then for any β ∈ (0, 1) we have convergence to a rescaled Brownian motion Proof. We will only sketch the proof as it is quite analogous to the proof of Theorem 8.1. We use the tightness given by Lemma 8.2 and assume that the sequence converges to a processM 1 . Then we take a smooth test-function f : R → R with compact support and consider the martingales Then we know that f N 1− β 2 M (N ) 1 ( N β t ) converges to f M 1 (t) and via Lemma 8.3 we get: where we have used β < 1 and k/N → 0 in the last step. This implies thatM 1 is a Markov process with generator p 1 (Z(0)) (1 − p 1 (Z(0)) f 2 . Hence,M 1 is the desired rescaled Brownian motion.
Note that Theorem 8.6 is consistent with Theorem 8.1 for small t. Again, a straight forward extension to higher dimensions is possible. Theorem 8.7. Suppose that the assumptions of Theorem 8.1 are fulfilled and let β ∈ (0, 1). Then the sequence of processes N 1− β 2 M (N ) ( N β t ) t≥0 converges for N → ∞ to a time-homogeneous Markov process with generator weakly on D([0, ∞), T ∆ A−1 ).
As before, the limit process can be interpreted as independent exchanges of mass between pairs of agents according to a rescaled standard Brownian motion.
We already know from Theorem 7.1 that χ (N ) ( N β t ) converges to χ(0) for N → ∞, when β < 1. Moreover, Theorem 8.6 states, that the process M (N ) ( N β t ) t≥0 converges to zero at rate In addition, it follows from that H (N ) ( N β t ) t≥0 converges to (G(χ(0))t) t≥0 at rate N 1−β , which immediately implies the following law of large numbers. Further functional central limit theorems in the context of Pólya urns have recently been studied in [8] and [12].

Convergence of the predictable part
In Section 7 we derived a law of large numbers for the evolution of market shares. In order to gain a better understanding of the convergence of Z (N ) (t) = χ (N ) ( N t ) to the deterministic process Z, we now want to derive a corresponding central limit theorem by examining the sequence of processes √ N (Z (N ) − Z) of deviations. We already know from Section 8 that in the decomposition (43) the martingale part √ N M (N ) ( N t ) converges to a system of time-changed Brownian motions, characterizing the random part of the deviations. Thus, it remains to determine the asymptotics of √ N χ(0) + H (N ) ( N t ) − Z(t) , the systematic part of the deviations. For that, it is important to notice that H (N ) ( N t ) is deterministic when M (N ) ( N s ) is given for s ≤ t. Because of that, it is possible to express the limit process of √ N χ(0) + H (N ) ( N t ) − Z(t) for N → ∞ in terms of the limit M of √ N M (N ) . We use the notation introduced in Section 7.
Proof. Let L > 0 be a Lipschitz constant for G. We use (43) and calculate: In line 2, the second summand is of order 1/ √ N due to assumption (51). Now Grönwall's inequality yields: Repeating the same calculation with 0 in the place of s and s instead of t yields: Combining these two inequalities proves the claim.
We are now ready for the proof of Theorem 9.1.
from Lemma 9.2 and the stochastic boundedness of the sequence ( √ N M (N ) ( N t )) t≥0 (see proof of Lemma 8.2). Now we show that the limit of any convergent subsequence is as desired. For simplicity of notation, assume that the sequence is convergent itself. Since Theorem 8.1 applies we can take an appropriate probability space Ω, such that the convergence holds locally uniformly almost surely. Note that this already implies Z (N ) (ω) N →∞ − −−− → Z locally uniformly. Now, fix ω ∈ Ω. Using (43) and the mean value theorem, we get where m (N ) (s) is an intermediate value between Z(s) and χ (N ) ( N s ). In line 3, we used assumption (51) once again. The claim follows since (52) has a unique solution due to the Theorem of Picard-Lindelöf, and H(t) ∈ T ∆ A−1 since H (N ) (k) ∈ T ∆ A−1 for all N ≥ 1. Figure 6 shows a simulation of the process √ N χ(0) + H (N ) ( N t ) − Z(t) for large N and small t. The long time behaviour will be discussed later on. Note that the limit process (52) has continuously differentiable paths, their regularity is equivalent to that of integrated Brownian motion. Combining Theorem 8.5 and Theorem 9.1 yields the desired central limit theorem for the difference Corollary 9.3. In the situation of Theorem 9.1, we have whereZ is the solution of the system of stochastic differential equations Here, B i,j are Brownian motions as defined in (49). Again, the differential operator DG i (z) : T ∆ A → R for z ∈ ∆ o A−1 is the product with the gradient ∇G i (z), when G is defined on an open neighbourhood of T ∆ A−1 in R A . Figure 7 shows the process Z (N ) − Z for large N . We can observe that Z (N ) (t) − Z(t) is close to zero for large t. Indeed, this complies with formula (52). Proof. Similarly to Section 8, the generator ofZ is given by for a bounded function b(t). Since t → DG(Z(t)) is continuous and DG(Z(∞)) is negative definite, DG(Z(t)) is also negative definite for t ≥ t 0 , when t 0 > 0 is large enough. Thus, there is λ > 0 such that DG(Z(t))x, x ≤ −λ x 2 for all x ∈ R A and t ≥ t 0 . In summary, we get for t ≥ t 0 . Now, applying Dynkin's formula yields for t ≥ t 0 . Finally, the claim follows from Grönwall's inequality: For the almost sure convergence we fix a realisation ω ∈ Ω, such that m := lim t→∞ M (t)(ω) exists. Then we get from (52) and the Cauchy-Schwarz inequality that In generic examples one can show that DG(Z(∞)) is indeed negative definite, but it is also possible to find a counterexample.
Example 9.5. Let F i (k) = α i k β for α i > 0, β > 0, such that Since there is an obvious extension of G to R A , the operator DG(x) is negative definite if and only if the well-defined differential matrix ∂ ∂x j G i (x) i,j=1,...,A is negative definite.
2. In the monopoly case β > 1 assume that χ(0) is the unique unstable fixpoint of the vector field G. Then Z(∞) = χ(0) and DG(Z(∞)) is positive definite. Thus, E Z (t) 2 t→∞ − −− → ∞ follows by similar argumentation. 3. For β = 1, we have H(t) ≡ 0 since G(x) ≡ 0. In this caseZ(t) does not converge to zero for t → ∞. This is due to the fact that for β = 1 and large (but finite) N the time-limit lim n→∞ χ (N ) (n) is close to χ(0), but still random. For β = 1, the long-time limit can be predicted precisely for large N (at least with high probability).
Note that the time-change factor 1 1+t in (53) does not change the long-time limit of the dynamics, but slows down the rate of convergence. The Grönwall estimate in the proof of Proposition 9.4 implies thatZ(t) converges to zero at least at rate t −2λ . For the classical Pòlya urn we have λ = 0, such that there is no convergence to zero.

Summary
The purpose of this paper was a comprehensive analysis of the possible asymptotic behaviour in nonlinear generalized Pólya urn models with transition probabilities of the form (1). Various results often focusing on special cases of feedback functions have already been known in the literature, as summarized in detail in the introduction. Our approach provides a fairly complete study of different types of feedback functions, and the main novelties include a segmentation of the simplex into asymptotic attraction domains, an investigation of the difference between exponential and polynomial-type feedback (Section 4), and a detailed study of almost linear feedback functions, including a full characterization of the transition between strong monopoly and non-monopoly via weak monopoly (Section 6). Moreover, we provide a description of the dynamics in a functional LLN (Theorem 7.1) as well as a full characterization of fluctuations in a functional CLT ( Corollary 9.3) in the limit for large initial market sizes. The following table sums up the main results for the long-time behaviour of the model.

Examples
Assumption Main result lim sup 1 F i (k) 2 < ∞ and lim k→∞ Let us return to the competition of technologies described in the introduction. On the one hand, the results presented in the table imply, that in the case of increasing returns, i.e. superlinear feedback function, only one technology survives on the long run. The winner is random and basically determined by decisions made when the market is still young as the convergence in Theorem 4.7 and Proposition 4.19 can be considered as fast. Thus the winning technology does not need to be the best one, for generic polynomial feedback the winner is determined by a combination of quality and initial market share (see Example 4.15). In the words of Brian Arthur: the market gets locked-in by historically small events. On the other hand, in the case of decreasing returns, i.e. sublinear feedback functions, the market shares of the different technologies converge to a stable value, which is generically not zero and is not affected by events in the young market. This coincides with the ideas of the "conventional economic theory" as Arthur describes in [4]. In the case of (almost) linear feedback with different attractiveness, the best technology will dominate the market in the sense, that the corresponding market shares will get close to one, but nevertheless the other technologies survive as well.
So, does the generalizes Pólya urn suggest an answer to the question how the future car market is composed? One would have to decide whether the car market is subject to increasing or decreasing returns. Arthur explains in [4] that decreasing returns appear in resource based markets like agriculture, mining and bulk-goods production, whereas increasing returns are typical for knowledge-based markets like software, pharmaceuticals or automobiles, where high developing costs face low production costs. Hence, assuming that the car market is subject to increasing returns, the generalized non-linear Pólya urn model predicts that one technology will dominate in the long run and the winning technology is strongly influenced by decisions made at an early stage.
This means that the current support of battery-driven cars by politics, automobile concerns and financial investors will likely marginalize other technologies if the market is allowed to evolve freely in the future, regardless of whether battery driven cars are superior in ecological or economical terms.