Silviase/jgov_v1.4_full_ann huggingface dataset:
## 3 Propagation of Chaos

In this section we present the result of propagation of chaos for a $ \Lambda $ -individual system and discuss restrictions on the result. The heart of propagation of chaos is that a countably infinite particle system, in which particles are described by a common stochastic process and are stochastically independent, obtained as the natural limit of a finite particle system, in which particles are symmetrically interacted and are stochastically dependent. The common stochastic process is called a McKean process. The formal treatments of stochastic processes and the proofs are given in the Appendix.

### 3.1 Result

Let W be the set of all S-valued right continuous step functions with left limits $ {}^{11} $ , and define $ \sigma $ -field of W by $ \mathcal{F}_{t} = \sigma(w(s); s \geq t) $ and $ \mathcal{F} = \vee_{t \geq 0} \mathcal{F}_{t} $ for coordinate process $ w(t), w \in W $ . Denote by $ \mathsf{C}_{b}(W) $ the set of all bounded continuous functions on W. Further we set $ W^{\Lambda} = W \times \cdots \times W $ and $ \mathcal{F}^{\Lambda} = \mathcal{F} \times \cdots \times \mathcal{F} $ and denote a general element of $ W^{\Lambda} $ by $ w^{(\Lambda)} = (w_{1}, \ldots, w_{\Lambda}) $ .

Let $ X^{(\Lambda)} $ and X be the Markov process on $ S^{\otimes\Lambda} $ that describes a $ \Lambda $ -individual system and the corresponding McKean process on S. Then we denote by $ P^{(\Lambda)} $ and $ \nu $ the probability measures induced by $ X^{(\Lambda)} $ on $ (W^{\Lambda},\mathcal{F}^{\Lambda}) $ and X on $ (W,\mathcal{F}) $ respectively. We begin with the definition about the propagation of chaos. Although propagation of chaos is usually defined for a sequence of symmetric probability measures (see Sznitman (1984) and Tanaka (1982)), we give the definition for a sequence of non-symmetric probability measures, since our result covers models in which $ P^{(\Lambda)} $ is not necessarily symmetric.

Definition. A sequence of probability measures $ \{P^{(\Lambda)},\Lambda>1\} $ is said to be v-chaotic if for any M>0 and any $ \varphi_{\kappa}\in\mathsf{C}_{b}(W) $

$$ \lim_{\Lambda\rightarrow\infty}E^{(\Lambda)}[1\otimes\cdots\otimes\varphi_{1}\otimes\cdots\otimes\varphi_{2}\otimes\cdots\otimes\varphi_{M}\otimes\cdots\otimes1]=\prod_{\kappa=1}^{M}E^{\nu}[\varphi_{\lambda}], $$

where $ E^{(\Lambda)}(E^{\nu}) $ stands for the expectation by $ P^{(\Lambda)} $ (resp. $ P^{\nu} $ ), and $ \varphi_{\kappa}(1 \leq \kappa \leq M) $
may appear at any position in the integrand of the lefthand.

(3.1) implies that any marginal distribution of $ P^{(\Lambda)} $ converges weakly to a product measure of independent copies of the McKean process. The following proposition is proved by an application of a standard argument (Shiga and Tanaka (1985), Sznitman (1984) and Tanaka (1982)) to our multiple population case.

Proposition. Let us be given a $ \Lambda $ -individual system in the modified formulation with review rates $ \bar{r}_{\lambda}^{i}(X^{(\Lambda)}(t)) = r_{X_{\lambda}^{i}(t)}^{i}(V(X^{(\Lambda)}(t))) + D r_{\lambda}^{i}(X^{(\Lambda)}(t)) $ and choice probabilities $ \bar{p}_{\lambda}^{i}(X^{(\Lambda)}(t)) = p_{X_{\lambda}^{i}(t)}^{i}(V(X^{(\Lambda)}(t))) + D p_{\lambda}^{i}(X^{(\Lambda)}(t)), i \in I, \lambda \in \bar{\Lambda} $ . Then the following statements hold.

(i) For any $ \bar{\nu} \in \Theta $ there uniquely exists a corresponding McKean process X with initial distribution $ \bar{\nu} $ such that $ v(t) $ , the distribution of $ X(t) $ , is the unique solution of (2.1).

(ii) If $ \{(X_{\lambda}^{1}(0),\ldots,X_{\lambda}^{n}(0)),\lambda\in\bar{\Lambda}\} $ are independently and identically distributed by $ \bar{\nu}\in\Theta $ , then $ P^{(\Lambda)} $ is v-chaotic, i.e., (3.1) holds for $ P^{(\Lambda)} $ , and

$$ \lim_{\Lambda\rightarrow\infty}P(\sup_{0\leq t\leq T}\|V(X^{(\Lambda)}(t))-\nu(t)\|>\varepsilon)=0 $$

holds for any $\varepsilon>0$ and any finite $T>0$.

That $ P^{(\Lambda)} $ is $ \mu $ -chaotic implies two things. One is that if the strategies of individuals are independent and identically distributed at time t = 0, the stochastic independence among individuals appears as the number of individuals goes to infinity. The other is that the behavior of each individual can be described by the McKean process in the limit. Then, as (3.2) shows, the law of large numbers holds for the $ \Lambda $ -individual system, and the limit of the average frequencies in the populations at time t are given by the dynamics (2.1) of Björnersted and Weibull.

The Proposition justifies not only that the dynamics of average frequencies of large populations in the modified formation of Björnersted and Weibull is approximated by $ (2.1) $ in a given finite time interval, but also that the stochastic dynamics of individuals themselves are approximated by independent copies of the McKean process, more fundamentally.
### 3.2 Discussion

We here clarify assumptions needed for our result and discuss restrictions on it. Our result is established based on two crucial assumptions on dynamics.

One is that dynamics is essentially determined by average frequencies. We have shown that this assumption can be typically satisfied for models in which individuals revise their strategies based on interactions with randomly grouped and randomly sampled individuals. Roughly speaking, the propagation of stochastic independence among individuals arises due to that the stochastic process of average frequencies, which describes interactions among individuals, is replaced by deterministic process. Then, the law of large numbers holds and the dynamics of Björnerstedt and Weibull is derived. However, as for models in which the decisions of individuals are made in interactions with particular individuals, say, the first individual in each population, not only is our result not applicable, but also the stochastic process of average frequencies cannot be approximated by the dynamics of Björnerstedt and Weibull. This is because the stochastic process is described not by the average frequencies but by the strategies of particular individuals in the models.

The other is on the regularity of the functions describing dynamics. We have assumed the Lipschitz continuity of the review rate and the choice probability. Lipschitz continuity is a most standard assumption in studying continuous dynamics. But the Lipschitz continuity of the review rate is actually violated, for example, by "switch below average payoff" dynamics in Dixon (2000), in which individuals revise their strategies only when they obtain payoffs below average.

Since the proof of the unique existence of the McKean process is based on the Lipschitz continuity, our result cannot be applied to this dynamics. However, if some noise is added to average payoff, the Lipschitz continuity of the review rate is obtained so that our result is applicable, as seen in the example of Section 2.4. Further, another possibility is shown by some studies on propagation of chaos such as Kotani and Osada (1985) and Sznitman (1986). These studies establish the propagation of chaos for dynamics that is represented by using the $ \delta $ -function. Therefore they suggest that the restriction on the regularity in our case might also be overcome and the approximation by the dynamics of Björnerstedt and Weibull.
might be justified by a more sophisticated approach.

## 4 Concluding Remarks

In this paper we rigorously derived the deterministic dynamics for aggregate imitational behavior given by Björnerstedt and Weibull. We first clarified that models of finite populations in which individuals are independently given reviewing opportunities and revise the strategies by interactions based on random grouping and random sampling, fit the modified formulation of Björnerstedt and Weibull.

Then we showed that if the strategies of individuals are initially independent and identically distributed, the average frequencies of strategies in populations come to represent the deterministic dynamics of Björnerstedt and Weibull when the number of individuals goes to infinity. This establishment of law of large numbers is due to the fact that the propagation of chaos holds, i.e., the initial independence among individuals propagates in any finite time interval as the number of individuals goes to infinity.

Our result assures that the stochastic process of average frequencies in large populations can be approximated by the deterministic dynamics of Björnerstedt and Weibull in a given finite time interval. In particular, by application to the specific models of subsection 2.4, it gives examples of the approximation of aggregate imitational behavior in large populations by replicator dynamics in a different way from Börgers and Sarin (1997) and Schlag (1998).

## A Appendix

### A.1 Stochastic Processes for Interacting Individuals

Here we formally construct stochastic processes for interacting individuals. This is done in a martingale formulation $ {}^{12} $ , following Shiga and Tanaka (1985).

First we define some operators and the Markov processes generated by them in a martingale formulation. Given $ r^{i}, p^{i} $ , set $ m_{i} \times m_{i} $ -matrix valued function $ q^{i} $ on
Θ by

$$ q^{i}(h,h^{\prime};v)=\begin{array}{c c}{r_{h}^{i}(v)p_{h h^{\prime}}^{i}(v),}&{h^{\prime}\neq h}\\ {0,}&{h^{\prime}=h^{\prime}}\\ \end{array}. $$

Then define bounded linear operators $ Q(v), v \in \Theta $ on $ \mathsf{B}(S) $ by

$$ Q(v)\varphi(h^{1},\cdots,h^{n})=\sum_{i\in I}\sum_{l\in S_{i}}q^{i}(h^{i},l;v)\triangle_{l}^{i}\varphi(h^{1},\cdots,h^{n}),\varphi\in\mathsf{B}(S), $$

where $ \triangle_{l}^{i}\varphi(h^{1},\ldots,h^{n})=\varphi(h^{1},\ldots,l,\ldots,h^{n})-\varphi(h^{1},\ldots,h^{n}) $ , and $ \mathsf{B}(S) $ is the Banach space of all bounded functions on S equipped with the supremum norm $ \|\cdot\| $ .

For any $ \Theta $ -valued measurable function $ \nu_{1}(t) $ on $ [0,\infty] $ and $ \bar{\nu}\in\Theta $ , a Markov process $ X(t) $ on S with initial distribution $ \bar{\nu} $ is uniquely generated by $ \{Q(v_{1}(t))\} $ . Then the probability measure $ P^{X} $ on $ (W,\mathcal{F}) $ induced by $ X(t) $ is a unique solution of the following martingale problem (Theorem 7.3 in page 223 of Ethier and Kurtz (1985)). For a probability measure P on W, we denote by $ \mathcal{L}(w(t):P) $ the probability distribution of $ w(t) $ under P.

Definition A1. For $ \Theta $ -valued measurable function $ v_{1}(t) $ and $ \bar{v} \in \Theta $ , a probability measure P on $ (W, \mathcal{F}) $ is a solution of the martingale problem for $ (\{Q(v_{1}(t))\}, \bar{v}) $ if P satisfies

$$ \begin{align*}\varphi(w(t))-\int_{0}^{t}Q(v_{1}(s))\varphi(w(s))ds is a P-martingale for any\varphi\in\mathsf{B}(S),\\\mathcal{L}(w(0):P)=\bar{v}.\end{align*} $$

Now we give the definition of a McKean process. It is a Markov process that moves under the influence of its distribution at each time, and therefore it represents a stochastic phenomenon with interactions.

Definition A2. For $ \bar{\nu}\in\Theta $ , a probability measure $ \nu $ on $ (W,\mathcal{F}) $ is called a McKean measure corresponding to (abbreviated as c.t.) $ \{Q(\nu); \nu\in\Theta\} $ with $ \mathcal{L}(w(0):\nu)=\bar{\nu} $ if $ \nu $ satisfies

$$ \varphi(w(t))-\int_{0}^{t}Q(v(s))\varphi(w(s))ds is a\nu-martingale for any\varphi\in\mathsf{B}(S), $$

$$ \mathcal{L}(w(t):\nu)=\nu(t)and\mathcal{L}(w(0):\nu)=\bar{\nu}. $$
Then a Markov process realized by the McKean measure is called a McKean process c.t. $ \{Q(v); v \in \Theta\} $ with initial distribution $ \bar{v} $ .

For $ h \in S^{i} $ let us denote by $ 1_{h} $ the indicator function of h on $ S^{i} $ . We identify $ 1_{h} $ with function $ 1 \ldots 1 \otimes 1_{h} \otimes 1 \ldots 1 $ on S, i.e., $ 1_{h}(H) = 1 $ for $ H = (h^{1}, \ldots, h^{n}) \in S $ if and only if $ h^{i} = h $ . If we substitute $ \varphi = 1_{h}, h \in S^{i} $ in (A.3) and take the expectation, we have

$$ \begin{aligned}&v_{h}^{i}(t)-\bar{v}_{h}^{i}\\ &=\int_{0}^{t}[\sum_{l}r_{l}^{i}(v(s))p_{lh}^{i}(v(s))v_{l}^{i}(s)-r_{h}^{i}(v(s))v_{h}^{i}(s)]ds,h\in S^{i},i\in I.\\ \end{aligned} $$

By the continuity of $ r^{i} $ and $ p^{i} $ , (A.4) is equivalent to (2.1). The unique existence of a solution for (2.1) is guaranteed by the Lipschitz continuity of $ r^{i}, p^{i}, i \in I $ . So the marginal distribution v of the McKean process is the unique solution of the imitation dynamics (2.1).

Next we turn to a Markov process $ X^{(\Lambda)}(t) = (X_{1}(t), \ldots, X_{\Lambda}(t)) $ on $ S^{\otimes\Lambda} $ that a $ \Lambda $ -individual system represents. Based on the identification of pure strategies in $ S^{i} $ with unit vectors in $ \triangle^{i} $ , for $ H^{(\Lambda)} = (h_{1}^{1}, \ldots, h_{1}^{n}, \ldots, h_{\Lambda}^{1}, \ldots, h_{\Lambda}^{n}) \in S^{\otimes\Lambda} $ set an empirical distribution on $ S^{i} $ and S by $ V^{i}(H^{(\Lambda)}) = \frac{1}{\Lambda} \sum_{\lambda=1}^{\Lambda} h_{\lambda}^{i} \in \triangle^{i} $ and $ V(H^{(\Lambda)}) = (V^{1}(H^{(\Lambda)}), \ldots, V^{n}(H^{(\Lambda)})) \in \Theta $ respectively, as before.

As (A.1) and (A.2), for $ i \in I $ and $ \lambda \in \bar{\Lambda} $ set $ R_{+}^{m_{i}} $ - valued function $ \bar{q}_{\lambda}^{i} $ on $ S^{\otimes\Lambda} $ by

$$ \bar{q}_{\lambda}^{i}(h_{\lambda}^{i},h^{\prime};H^{(\Lambda)})=\begin{array}{c c}\bar{r}_{\lambda}^{i}(H^{(\Lambda)})\bar{p}_{\lambda h^{\prime}}^{i}(H^{(\Lambda)}),&h^{\prime}\neq h_{\lambda}^{i}\\ 0,&h^{\prime}=h_{\lambda}^{i}\end{array} $$

and define bounded linear operators $ \bar{Q}(H^{(\Lambda)}), H^{(\Lambda)} \in S^{\otimes\Lambda} $ on $ \mathsf{B}(S) $ . Then $ X^{(\Lambda)}(t) $ is generated by the following $ \bar{Q}^{(\Lambda)} $ based on $ \bar{Q}(H^{(\Lambda)}) $ :

$$ \bar{Q}^{(\Lambda)}\Phi(H^{(\Lambda)})=\sum_{\lambda=1}^{\Lambda}\bar{Q}_{\lambda}(H^{(\Lambda)})\Phi(H^{(\Lambda)}),\Phi\in\mathsf{B}(S^{\otimes\Lambda}), $$

where $ \bar{Q}_{\lambda}(H^{(\Lambda)}) $ stands for the operation of $ \bar{Q}(H^{(\Lambda)}) $ with respect to $ H_{\lambda} $

Further we define $ Q^{(\Lambda)} $ based on $ Q(v) $ in (A.2) and $ Q_{\lambda}(V(H^{(\Lambda)})) $ as above. Then we note that $ \bar{Q}^{(\Lambda)} $ coincides with $ Q^{(\Lambda)} $ in the case of the original formulation of Björnerstedt and Weibull, where $ Dr_{\lambda}^{i} = 0 $ and $ Dp_{\lambda}^{i} = 0 $ for $ i \in I, \lambda \in \bar{\Lambda} $ .
Let us denote by $ P^{(\Lambda)} $ the probability measure induced by the Markov process $ X^{(\Lambda)}(t) $ with initial distribution $ \bar{\nu}^{\otimes\Lambda} $ on $ (W^{\Lambda}, \mathcal{F}^{\Lambda}) $ . Then $ P^{(\Lambda)} $ is a unique solution of the martingale problem for $ (Q^{(\Lambda)}, \bar{\nu}^{\otimes\Lambda}) $ as $ P^{X} $ in the above. We denote by $ E^{(\Lambda)} $ the expectation with respect to $ P^{(\Lambda)} $ .

### A.2 Proofs for Proposition

Proof of Proposition (i). Based on the assumption of the Lipschitz continuity of the review rate and the choice probability and the finiteness of S, Proposition (i) is proved by a simple extension of the proof of lemma 2.3 in Shiga and Tanaka (1985) to the multiple population case, although our model is not contained in their formulation. We prove it with three steps.

1st step. In the first step we note that for any $ \Theta $ -valued measurable function $ v_{1}(t) $ on $ [0,\infty] $ and $ \bar{v} $ , the martingale problem for $ (\{Q(v_{1}(t))\},\bar{v}) $ has a unique solution as already mentioned.

2nd step. We show the following equation has a unique $ \Theta $ -valued solution for any $ \bar{\nu} \in \Theta $ .

$$ -<\bar{v},\varphi>=\int_{0}^{t}ds,\varphi\in\mathsf{B}(S), $$

where $ \langle v,\varphi\rangle $ stands for the integration of $ \varphi $ by a probability measure v. This follows from that by linearity (A.7) is equivalent to

$$ \begin{aligned}&v(t)_{k^{1}\ldots k^{n}}-v(0)_{k^{1}\ldots k^{n}}\\ &=\int_{0}^{t}\sum_{i\in I}\sum_{l\neq k^{i}}[q^{i}(l,k_{i};v(s))v(s)_{k^{1},\ldots,l,\ldots,k^{n}}-q^{i}(k_{i},l;v(s))v(s)_{k_{1},\ldots,k_{i},\ldots,k_{n}}]d s,\\ \end{aligned} $$

where $ v_{k^{1}\ldots k^{n}}(t)=v(t)(\{(k^{1},\ldots,k^{n})\}) $ for $ (k^{1},\ldots,k^{n})\in S $ .

Because (A.8) is a $ \prod_{i} m^{i} $ -dimensional integral equation for $ v(t)_{k_{1}\ldots k_{n}}, (k_{1}, \ldots, k_{n}) \in S $ and has a unique solution, since $ q^{i}(h, k; \cdot), h, k \in S^{i}, i \in I $ are Lipschitz continuous. Further it is easily verified that $ v(t)_{k_{1}\ldots k_{n}}, k_{1} \in S^{1}, \ldots, k_{n} \in S^{n} $ constitute a probability.
Similarly, for any $ \Theta $ -valued measurable function $ v_{1}(t) $ and $ \bar{v} \in \Theta $ , the following equation has a unique $ \Theta $ -valued solution.

$$ -<\bar{v},\varphi>=\int_{0}^{t}ds,\varphi\in\mathsf{B}(S). $$

3rd step. Let $ v(t) $ be the unique solution of (A.7) for $ \bar{v} \in \Theta $ and v be the solution of MP for $ (\{Q(v(t))\}, \bar{v}) $ . Then $ \tilde{v}(t) = \mathcal{L}(w(t) : v) $ solves (A.9) with $ v_{1}(t) = v(t) $ . This means $ v(t) = \tilde{v}(t) $ and v is a McKean measure c.t. ( $ \{Q(v); v \in \Theta\} $ ) with $ \mathcal{L}(w(0) : v) = \bar{v} $ because of the uniqueness of the solution of (A.9). The uniqueness of the McKean measure is also proved from that of (A.9) and the martingale problem for $ (\{Q(v_{1}(t))\}, \bar{v}) $ .

Proof of Proposition (ii). We give a proof only for the case of the original formulation, where $ Dr_{\lambda}^{i}=0 $ and $ Dp_{\lambda}^{i}=0 $ for $ i\in I,\lambda\in\bar{\Lambda} $ . The proof for the general case is done by a slight modification based on the fact that $ |\bar{Q}^{(\Lambda)}\Phi(H^{(\Lambda)})-Q^{(\Lambda)}\Phi(H^{(\Lambda)})|=o(1) $ holds for $ H^{(\Lambda)}\in S^{\otimes\Lambda},\Phi\in\mathsf{B}(S^{\otimes\Lambda}) $ .

The proof proceeds in a standard way as in Sznitman (1984) and Tanaka (1983). Let $ W' $ be the space of $ \Theta $ -valued right continuous paths with left limits. Then $ V(w^{(\Lambda)}(t)) $ is a Markov process with sample paths in $ W' $ for a $ \Lambda $ -individual system. Denote by $ w'(t) $ the coordinate process for $ w' \in W' $ , by $ P'^{(\Lambda)} $ the probability measure on $ W' $ induced by $ V(w^{(\Lambda)}(t)) $ , and by $ E'^{(\Lambda)} $ the expectation with respect to $ P'^{(\Lambda)} $ respectively. We prove the law of large numbers (3.2), and then (3.1) easily follows. The proof also proceeds with three steps.

1st step. First, we show that $\{P'^{(\Lambda)}\}$ is tight. For $w_{\lambda}^{i}$ and $E\subset S^{i}$, set

$$ \begin{align*}N(t,E;w^{i}_{\lambda})&=\sum_{s\leq t}1(w^{i}_{\lambda}(s)\in E,w^{i}_{\lambda}(s)\neq w^{i}_{\lambda}(s-)),\\\tilde{N}(t,E;w^{i}_{\lambda})&=N(t,E;w^{i}_{\lambda})-\int_{0}^{t}\sum_{h\in E}q^{i}(w^{i}_{\lambda}(s),h;V(w^{(\Lambda)}(s)))ds,\end{align*} $$

where $ 1(w_{\lambda}^{i}(s) \in E, w_{\lambda}^{i}(s) \neq w_{\lambda}^{i}(s-)) $ is the indicator function such that it equals to 1 if and only if $ w_{\lambda}^{i}(s) \in E $ and $ w_{\lambda}^{i}(s) \neq w_{\lambda}^{i}(s-) $ .

Then it is easily shown as lemma 2.4 in Shiga and Tanaka (1985) that $ \tilde{N}(t, E; w_{\lambda}^{i}) $ is a $ P^{(\Lambda)} $ -martingale and

$$ <\tilde{N}(t,E_{1};w_{\lambda}^{i}),\tilde{N}(t,E_{2};w_{\kappa}^{i})>=\int_{0}^{t}\sum_{l\in E_{1}\cap E_{2}}q^{i}(w_{\lambda}^{i}(s),l;V(w^{(\Lambda)}(s)))ds, $$
where $ \langle\tilde{N}(t,E_{1};w_{\lambda}^{i}),\tilde{N}(t,E_{2};w_{\kappa}^{i})\rangle $ is the corresponding quadratic variational process.

Moreover, since $ w_{\lambda}^{i} $ and $ w_{\kappa}^{j} $ have no common jumps unless $ i = j, \lambda = \kappa $ ,

$$ <\tilde{N}(t,E_{1};w_{\lambda}^{i}),\tilde{N}(t,E_{2};w_{\kappa}^{j})>=0. $$

Noting

$$ V(w^{(\Lambda)}(t))=(\frac{1}{\Lambda}\sum_{\lambda}1_{e_{1}^{1}}(w_{\lambda}^{1}(t)),\ldots,\frac{1}{\Lambda}\sum_{\lambda}1_{e_{h}^{i}}(w_{\lambda}^{i}(t)),\ldots,\frac{1}{\Lambda}\sum_{\lambda}1_{e_{m^{n}}^{n}}(w_{\lambda}^{n}(t))), $$

and

$$ 1_{e_{h}^{i}}(w_{\lambda}^{i}(t_{2}))-1_{e_{h}^{i}}(w_{\lambda}^{i}(t_{1}))=\int_{t_{1}}^{t_{2}}\sum_{l\in S^{i}}(1_{e_{h}^{i}}(l)-1_{e_{h}^{i}}(w_{\lambda}^{i}(s-)))N(d s,l;w_{\lambda}^{i}), $$

we have

$$ \begin{aligned}&E^{(\Lambda)}[\|V(w^{(\Lambda)}(t_{2}))-V(w^{(\Lambda)}(t_{1}))\|^{2}|\mathcal{F}_{t_{1}}^{\Lambda}]\\&=\frac{1}{\Lambda^{2}}E^{(\Lambda)}[\sum_{i}\sum_{h\in S^{i}}(\sum_{\lambda}\int_{t_{1}}^{t_{2}}\sum_{l\in S^{i}}(1_{e_{h}^{i}}(l)-1_{e_{h}^{i}}(w_{\lambda}^{i}(s-)))N(ds,l;w_{\lambda}^{i}))^{2}|\mathcal{F}_{t_{1}}^{\Lambda}]\\&\leq\frac{1}{\Lambda}E^{(\Lambda)}[\sum_{i}\sum_{h\in S^{i}}\sum_{\lambda}(\int_{t_{1}}^{t_{2}}N(ds,S^{i};w_{\lambda}^{i}))^{2}|\mathcal{F}_{t_{1}}^{\Lambda}]\\&\leq\frac{2}{\Lambda}E^{(\Lambda)}[\sum_{i}\sum_{h\in S^{i}}\sum_{\lambda}(\int_{t_{1}}^{t_{2}}\tilde{N}(ds,S^{i};w_{\lambda}^{i}))^{2}|\mathcal{F}_{t_{1}}^{\Lambda}]\\&+\frac{2}{\Lambda}E^{(\Lambda)}[\sum_{i}\sum_{h\in S^{i}}\sum_{\lambda}(\int_{t_{1}}^{t_{2}}\sum_{l\in S^{i}}q^{i}(w_{\lambda}^{i}(s),l;V^{(\Lambda)}(s)))ds)^{2}|\mathcal{F}_{t_{1}}^{\Lambda}]\\ \end{aligned} $$

Thus, using (A.10),

$$ E^{(\Lambda)}[\|V(w^{(\Lambda)}(t_{2}))-V(w^{(\Lambda)}(t_{1}))\|^{2}|\mathcal{F}_{t_{1}}^{\Lambda}]\leq C(T)|t_{2}-t_{1}|,0\leq t_{1}\leq t_{2}\leq T, $$

where $ C(T) $ is a positive constant depending on T.

From this, for $ 0 \leq t_{1} \leq t_{2} \leq t_{3} \leq T $ , we have

$$ E^{\prime(\Lambda)}[\|w^{\prime}(t_{1})-w^{\prime}(t_{2})\|^{2}\|w^{\prime}(t_{2})-w^{\prime}(t_{3})\|^{2}]\leq C^{2}(T)|t_{1}-t_{3}|^{2}, $$

which shows the tightness of $ \{P'^{(\Lambda)}\} $ (Theorem 15.6 in page 128 of Billingsley(1968)).
2nd step. Secondly for $ w' \in W' $ and $ \varphi \in \mathsf{B}(S) $ set

$$ F(w^{\prime})=--\int_{0}^{t}ds. $$

We show for any limit point $ P'^{(\infty)} $ of $ \{P'^{(\Lambda)}\} $ , $ F = 0 $ , $ P'^{(\infty)} - a.s. $ . Let $ \{P'^{(\Lambda_l)}\} $ be a subsequence converging to $ P'^{(\infty)} $ and set

$$ M_{\lambda}(t)=\varphi(w_{\lambda}(t))-\varphi(w_{\lambda}(0))-\int_{0}^{t}Q(V(w^{(\Lambda_{l})}(s)))\varphi(w_{\lambda}(s))ds,\\ \lambda\in\bar{\Lambda}_{l}. $$

Then $ M_{\lambda} $ is a $ P^{(\Lambda_{l})} $ -martingale, and $ =0, \lambda\neq\kappa $ by (A.11).

Hence we have

$$ E^{\prime(\infty)}F^{2}=\lim E^{(\Lambda_{l})}\frac{1}{\Lambda_{l}^{2}}|\sum_{\lambda}M_{\lambda}(t)|^{2}=\lim\frac{1}{\Lambda_{l}}E^{(\Lambda_{l})}|M_{1}(t)|^{2}=0, $$

which shows $ F = 0, P'(\infty) - a.s. $

Next we recall that $ v(t) $ is a unique $ \Theta $ -valued solution of (A.7). Moreover, $ w'(0) = \bar{v}, P'^{(\infty)} - a.s. $ by the law of large numbers. Therefore, from the uniqueness of solution of (A.7) we have $ w'(t) = v(t), 0 \leq t \leq T, P'^{(\infty)} - a.s., $ i.e., $ P'^{(\infty)} = \delta_v $ , which implies (3.2).

3rd step. Let $ P_{M}^{(\Lambda)}(M < \Lambda) $ be the marginal probability of $ P^{(\Lambda)} $ on the first $ W^{M} $ . For the propagation of chaos, it suffices to show $ P_{M}^{(\Lambda)} $ weakly converges to $ \nu^{\otimes M} $ because of the symmetry of $ P^{(\Lambda)} $ . We can first show the tightness of $ P_{M}^{(\Lambda)} $ as (A.12).

Next, for a set function $ \mu $ on the field of all subsets of $ S^{\otimes M} $ , let us denote by $ \|\mu\|_{var} $ the total variation norm of $ \mu $ , i.e., $ \|\mu\|_{var} = \sum_{H^{(M)} \in S^{\otimes M}} |\mu(\{H^{(M)}\})| $ . Then, by using (3.2) we can inductively establish that for any $ \varepsilon > 0 $ there exists $ \Lambda_n > 0 $ and $ C_{n-1}(T) > 0 $ such that for $ \Lambda > \Lambda_n $

$$ \begin{aligned}&|E_{M}^{(\Lambda)}[\Phi(w^{(M)}(t_{n}));A_{t_{1}}\cap\cdots\cap A_{t_{n-1}}]-E^{\vee^{\otimes M}}[\Phi(w^{(M)}(t_{n}));A_{t_{1}}\cap\cdots\cap A_{t_{n-1}}]|\\&\leq C_{n-1}(T)\|\Phi\|\varepsilon+2\|\Phi\|Mm\cdot\max\{r_{h}^{i}(v)\}\\&\cdot[\int_{t_{n-1}}^{t_{n}}\|P_{M}^{\Lambda}((w^{(M)}(s)\in\cdot)\cap A_{t_{1}}\cap\cdots\cap A_{t_{n-1}})\\&-v^{\otimes M}((w^{(M)}(s)\in\cdot)\cap A_{t_{1}}\cap\cdots\cap A_{t_{n-1}})\|_{var}ds],\\&A_{t_{1}}\in\sigma(w^{(M)}(t_{1})),\cdots,A_{t_{n-1}}\in\sigma(w^{(M)}(t_{n-1})),\Phi\in\mathsf{B}(S^{\otimes M}),\\ \end{aligned} $$