3. Proofs

3.1. Proof of Lemma 1

We show the implications (i)$\implies$(ii), (ii)$\implies$(iii), and (iii)$\implies$(i).

3.1.1. (i)$\implies$(ii).

Let $\delta>0$ and let $f\,:\,\mathbb{R}^{M}\rightarrow\mathbb{R}$ be a continuous and bounded function with the properties

\begin{align*}f\geq0,\quad f(0)=0,\quad\mathrm{and}\quad f(x)=1,x\in\mathbb{R}^{M}\backslash\prod_{\lambda=1}^{M}\left[-\delta,\delta\right].\end{align*}

The weak convergence of $\left(\mu_{n}\right)_{n}$ to $\delta_{0}$ implies

\begin{align*}\int_{\mathbb{R}^{M}}f\,\mathrm{d}\mu_{n}\xrightarrow[n\rightarrow\infty]{}\int_{\mathbb{R}^{M}}f\,\mathrm{d}\delta_{0}=f(0)=0.\end{align*}

Since $f\geq0$ and f equals 1 on $\mathbb{R}^{M}\backslash\prod_{\lambda=1}^{M}\left[-\delta,\delta\right]$, we have

\begin{align*}\mu_{n}\left(\mathbb{R}^{M}\backslash\prod_{\lambda=1}^{M}\left[-\delta,\delta\right]\right)\xrightarrow[n\rightarrow\infty]{}0.\end{align*}

3.1.2. $(ii)\implies(iii)$.

By (ii), for all $m\in\mathbb{N}$ we have

(4) \begin{equation}\mu_{n}\left(\mathbb{R}^{M}\backslash\prod_{\lambda=1}^{M}\left[-\frac{1}{m},\frac{1}{m}\right]\right)\xrightarrow[n\rightarrow\infty]{}0.\end{equation}

We define the natural numbers $n_{1},n_{2},\ldots$ as follows:

\begin{align*}n_{m}\,:\!=\,\min\left\{ k\in\mathbb{N}\,\left|\,\mu_{n}\left(\prod_{\lambda=1}^{M}\left[-\frac{1}{m},\frac{1}{m}\right]\right)\geq1-\frac{1}{m},n\geq k\right.\right\} ,\quad m\in\mathbb{N}.\end{align*}

Note that the sets in the display are non-empty for each m by (4). Set

\begin{align*}\varepsilon_{n,\lambda}\,:\!=\,\frac{1}{m},\quad\lambda=1,\ldots,M,\quad n_{m}\leq n\lt n_{m+1},\quad m\in\mathbb{N}.\end{align*}

By construction, $\left\Vert \varepsilon_{n}\right\Vert _{\infty}\xrightarrow[n\rightarrow\infty]{}0$ and $\mu_{n}\left(\left[-\varepsilon_{n},\varepsilon_{n}\right]\right)\xrightarrow[n\rightarrow\infty]{}1$.

3.1.3. $(iii)\implies(i)$.

Let $f\,:\,\mathbb{R}^{M}\rightarrow\mathbb{R}$ be a continuous and bounded function. Let $\eta>0$. Since f is continuous, there is a $\delta>0$ such that, for all $x\in\mathbb{R}^{M}$,

\begin{align*}\left\Vert x\right\Vert _{\infty}<\delta\quad\mathrm{implies}\quad\left|f(x)-f(0)\right|<\eta.\end{align*}

Then

(5) \begin{equation}\left|\int_{\mathbb{R}^{M}}f\,\mathrm{d}\mu_{n}-\int_{\mathbb{R}^{M}}f\,\mathrm{d}\delta_{0}\right|\leq\int_{\mathbb{R}^{M}}\left|f(x)-f(0)\right|\,\mu_{n}\left(\mathrm{d}x\right).\end{equation}

Let $n_{0}\in\mathbb{N}$ be large enough that

\begin{align*}\left\Vert \varepsilon_{n}\right\Vert _{\infty}<\delta\quad\mathrm{and}\quad\mu_{n}\left(\left[-\varepsilon_{n},\varepsilon_{n}\right]\right)>1-\eta,\quad n\geq n_{0}.\end{align*}

We continue with our calculation:

\begin{align*} ({5}) & =\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}\left|f(x)-f(0)\right|\,\mu_{n}\left(\mathrm{d}x\right)+\int_{\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]}\left|f(x)-f(0)\right|\,\mu_{n}\left(\mathrm{d}x\right)\\ & \leq\eta\mu_{n}\left(\left[-\varepsilon_{n},\varepsilon_{n}\right]\right)+2\left\Vert f\right\Vert _{\infty}\eta,\end{align*}

which proves the weak convergence of $\left(\mu_{n}\right)_{n}$ to $\delta_{0}$.

3.2. Proof of Theorem 4

The proofs of the theorems in Section 2 use characteristic functions. Thus, we have to show pointwise convergence of the sequence of characteristic functions of the normalised sums to the characteristic function of the limiting distribution. In the case of Theorem 4, we define $\varphi_{n}$ to be the characteristic function of the distribution of

\begin{align*}\boldsymbol{S}_{n}=\left(\frac{S_{n,1}}{\sqrt{n_{1}}},\ldots,\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\end{align*}

and $\varphi_{\mathcal{N}\left(0,I_{M}\right)}$ to be the characteristic function of $\mathcal{N}\left(0,I_{M}\right)$. Then our task is to show $\varphi_{n}(t)\xrightarrow[n\rightarrow\infty]{}\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)$ for all $t\in\mathbb{R}^{M}$. We will only prove the result for sequences of de Finetti measures on $\left[-1,1\right]^{M}$. The results for de Finetti sequences on $\mathbb{R}^{M}$ can be shown analogously.

We will use the following notation for ‘asymptotic equivalence’.

Notation 9. Complex-valued sequences $\left(f_{n}\right)_{n},\left(g_{n}\right)_{n}$ are called asymptotically equivalent (as $n\to\infty$), and we write $f_{n}\approx g_{n}$, if

\begin{align*}\lim_{n\rightarrow\infty}\frac{f_{n}}{g_{n}}=1.\end{align*}

Recall Definition 2 and Notation 1. By Definition 3, $\varphi_{n}(t)$ can be expressed as

(6) \begin{align}\mathbb{E}\exp\left(it\cdot\boldsymbol{S}_{n}\right) & =\mathbb{E}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\sqrt{n_{1}}}+\cdots+t_{M}\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\right)\nonumber \\ & =\int_{\left[-1,1\right]^{M}}E_{m}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\sqrt{n_{1}}}+\cdots+t_{M}\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\right)\mu_{n}\left(\mathrm{d}m\right).\end{align}

We first note that from the boundedness of the integrand in (6) and the concentration property

$$\mu_{n}\left[-\varepsilon_{n},\varepsilon_{n}\right]\xrightarrow[n\rightarrow\infty]{}1,$$

we obtain

(7) \begin{equation} ({6}) \approx\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}E_{m}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\sqrt{n_{1}}}+\cdots+t_{M}\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\right)\mu_{n}\left(\mathrm{d}m\right).\end{equation}

This step follows from the next lemma, provided that (7) converges to a non-zero limit, which we will show in (14).

Lemma 2. Let $f_{n}\,:\,\left[-1,1\right]^{M}\rightarrow\mathbb{C}$ be measurable functions with $\left\Vert f_{n}\right\Vert _{\infty}\leq1$, $n\in\mathbb{N}$, and $\left(\mu_{n}\right)_{n}$ the de Finetti sequence in the statement of Theorem 4. Then

\begin{align*}\left|\int_{\left[-1,1\right]^{M}}f_{n}\,\mathrm{d}\mu_{n}-\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}f_{n}\,\mathrm{d}\mu_{n}\right|\xrightarrow[n\rightarrow\infty]{}0.\end{align*}

Proof. We estimate

\begin{align*}\left|\int_{\left[-1,1\right]^{M}}f_{n}\,\mathrm{d}\mu_{n}-\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}f_{n}\,\mathrm{d}\mu_{n}\right| & =\left|\int_{\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]}f_{n}\,\mathrm{d}\mu_{n}\right|\leq\int_{\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]}\left|f_{n}\right|\,\mathrm{d}\mu_{n}\\ & \leq\mu_{n}\left(\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]\right)\xrightarrow[n\rightarrow\infty]{}0.\end{align*}

Next we have to calculate the expectation

$$E_{m}\exp\left(i\left(t_{1}S_{n,1}/\sqrt{n_{1}}+\cdots+t_{M}S_{n,M}/\sqrt{n_{M}}\right)\right).$$

Under $P_{m}^{\otimes n}$, all random variables $X_{\lambda i}$, $\lambda=1,\ldots,M$, $i=1,\ldots,n_{\lambda}$, are independent. Within each group $\lambda$, the $X_{\lambda i}$ are even independent and identically distributed. We subtract the expected value $m_{\lambda}$ of each $X_{\lambda i}$ under $P_{m}^{\otimes n}$. Note that for $t_{\lambda}=0$,

\begin{align*}E_{m}\exp\left(i\left(t_{\lambda}\frac{\sum_{j}\left(X_{\lambda j}-m_{\lambda}\right)}{\sqrt{n_{\lambda}}}\right)\right)=1.\end{align*}

Now assume $t_{\lambda}\neq0$. Using a Taylor expansion yields

\begin{align*}E_{m}\exp\left(i\left(t_{\lambda}\frac{\sum_{j}\left(X_{\lambda j}-m_{\lambda}\right)}{\sqrt{n_{\lambda}}}\right)\right) & =\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}+O\left(\frac{1}{n_{\lambda}^{3/2}}\right)\right)^{n_{\lambda}}.\end{align*}

Thus, the conditional expectation in the integral in (7) can be written as

\begin{align*}\prod_{\lambda\,:\,t_{\lambda}\neq0}\exp\left(im_{\lambda}t_{\lambda}\sqrt{n_{\lambda}}\right)\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}+O\left(\frac{1}{n_{\lambda}^{3/2}}\right)\right)^{n_{\lambda}}.\end{align*}

Set

\begin{align*}a_{n,\lambda} & \,:\!=\,\exp\left(im_{\lambda}t_{\lambda}\sqrt{n_{\lambda}}\right),\\b_{n,\lambda} & \,:\!=\,\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}+O\left(\frac{1}{n_{\lambda}^{3/2}}\right)\right)^{n_{\lambda}},\\c_{n,\lambda} & \,:\!=\,a_{n,\lambda}b_{n,\lambda}\end{align*}

for each n and each $\lambda$.

First we analyse the asymptotic behaviour of $\left(a_{n,\lambda}\right)_{n}$.

Lemma 3. For all $n\in\mathbb{N}$ and all $\lambda=1,\ldots,M$ we have

\begin{align*}a_{n,\lambda} & \in A_{n}\,:\!=\,\left\{ \exp\left(i\omega\right)\,\big|\,\omega\in\left[-\left|t_{\lambda}\right|\varepsilon_{n,\lambda}\sqrt{n_{\lambda}},\left|t_{\lambda}\right|\varepsilon_{n,\lambda}\sqrt{n_{\lambda}}\right]\right\} ,\end{align*}

and

\begin{align*}A_{n}\searrow\left\{ 1\right\} \quad\mathrm{as }n\rightarrow\infty.\end{align*}

Proof. Since $t_{\lambda}\neq0$ and m takes values on the interval $\left[-\varepsilon_{n},\varepsilon_{n}\right]$, $a_{n,\lambda}$ lies on the unit circle in the complex plane, specifically in the section defined by $A_{n}$. The convergence speed $\varepsilon_{n,\lambda}=o\left(\frac{1}{\sqrt{n_{\lambda}}}\right)$ implies $\pm\left|t_{\lambda}\right|\varepsilon_{n,\lambda}\sqrt{n_{\lambda}}\xrightarrow[n\rightarrow\infty]{}0$ and therefore $A_{n}\searrow\left\{ 1\right\} $ as $n\rightarrow\infty$.

Next we deal with the sequence $\left(b_{n,\lambda}\right)_{n}$. We estimate

(8) \begin{align}\left|b_{n,\lambda}-\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\right| &\leq\left|b_{n,\lambda}-\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}\right| \nonumber\\&\quad+\left|\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}-\exp\left(-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}\right)\right|.\end{align}

We will find upper bounds for each of the summands in (8).

Lemma 4. There is a constant $C>0$ such that for all $n\in\mathbb{N}$ we have

\begin{align*}\left|b_{n,\lambda}-\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}\right|\leq\frac{C}{n_{\lambda}^{3/2}}.\end{align*}

Proof. We define $r_{n_{\lambda}}\,:\!=\,\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}$ and $s_{n_{\lambda}}$ to be the $O\left(\frac{1}{n_{\lambda}^{3/2}}\right)$ term in the definition of $b_{n,\lambda}$. Then we have

\begin{align*}b_{n,\lambda}=\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}+s_{n_{\lambda}}\right)^{n_{\lambda}}=\sum_{k=0}^{n_{\lambda}}\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{n_{\lambda}-k}s_{n_{\lambda}}^{k}=\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{n_{\lambda}}\sum_{k=0}^{n_{\lambda}}\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{k}.\end{align*}

We calculate

\begin{align*}\left|b_{n,\lambda}-\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}\right| & =\left|\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{n_{\lambda}}\sum_{k=0}^{n_{\lambda}}\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{k}-\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{n_{\lambda}}\right|\\ & =\left|1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right|^{n_{\lambda}}\left|\sum_{k=0}^{n_{\lambda}}\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{k}-1\right|\\ & =\left|1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right|^{n_{\lambda}}\left|\frac{1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}+1}}{1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}}-1\right|,\end{align*}

where in the last step we applied the formula for the value of a geometric series with complex summands. We note that $r_{n_{\lambda}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}\frac{t_{\lambda}^{2}}{2}$ and hence

\begin{align*}\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}0\quad\mathrm{and}\quad\left|1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right|^{n_{\lambda}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}\exp\left(-\frac{t_{\lambda}^{2}}{2}\right).\end{align*}

There is a constant $C_{1}>0$ such that

\begin{align*}\left|1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right|^{n_{\lambda}}\leq C_{1},\quad n_{\lambda}\in\mathbb{N}.\end{align*}

We continue with

\begin{align*}\left|\frac{1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}+1}}{1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}}-1\right| & =\left|\frac{1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}+1}-1+\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}}{1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}}\right|\\ & =\left|\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right|\left|\frac{1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}}}{1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}}\right|.\end{align*}

Now we note that

\begin{align*}1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}1,\quad1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}1,\quad\mathrm{and}\quad1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}1.\end{align*}

The above implies

\begin{align*}1-\frac{r_{n_{\lambda}}}{n_{\lambda}},1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}},1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}}=\Theta(1)\end{align*}

and the existence of a constant $C_{2}>0$ such that

\begin{align*}\left|\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right|\left|\frac{1-\left(\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right)^{n_{\lambda}}}{1-\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}}\right|\leq C_{2}\left|\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right|,\quad n_{\lambda}\in\mathbb{N}.\end{align*}

Since $1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}1$, there is a constant $C_{3}>0$ such that

\begin{align*}\left|\frac{s_{n_{\lambda}}}{1-\frac{r_{n_{\lambda}}}{n_{\lambda}}}\right|\leq C_{3}\left|s_{n_{\lambda}}\right|,\quad n_{\lambda}\in\mathbb{N}.\end{align*}

Finally, since $\left|s_{n_{\lambda}}\right|=O\left(\frac{1}{n_{\lambda}^{3/2}}\right)$, there is a constant $C_{4}>0$ such that

\begin{align*}\left|s_{n_{\lambda}}\right|\leq\frac{C_{4}}{n_{\lambda}^{3/2}},\quad n_{\lambda}\in\mathbb{N}.\end{align*}

The claim follows from setting $C\,:\!=\,C_{1}C_{2}C_{3}C_{4}>0$.

Next we will find an upper bound for the second summand in (8).

Lemma 5. There is a constant $D>0$ such that for all $n\in\mathbb{N}$ we have

\begin{align*}\left|\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}-\exp\left(-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}\right)\right|\leq\frac{D}{n_{\lambda}}.\end{align*}

Proof. We again define $r_{n_{\lambda}}\,:\!=\,\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}$. Recall that $r_{n_{\lambda}}\xrightarrow[n_{\lambda}\rightarrow\infty]{}\frac{t_{\lambda}^{2}}{2}$. Choose $m_{0}\in\mathbb{N}$ large enough that

(9) \begin{equation}\left|\frac{r_{n_{\lambda}}}{n_{\lambda}}\right|<\frac{1}{2}\quad\mathrm{and}\quad\left|r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right|+2\frac{r_{n_{\lambda}}^{2}}{n_{\lambda}}<1,\quad n_{\lambda}\geq m_{0}.\end{equation}

We use Taylor expansions for the functions $\exp$ and $\ln$:

\begin{align*}\exp z & =1+z+O\left(z^{2}\right),\quad z\in\mathbb{R},\ \left|z\right|\leq1,\\\ln\left(1+z\right) & =z+O\left(z^{2}\right),\quad z\in\mathbb{R},\ \left|z\right|\leq\frac{1}{2},\end{align*}

More precisely, we have the bounds

(10) \begin{align}\left|\exp z-\left(1+z\right)\right| & \leq\frac{e}{2}z^{2},\quad z\in\mathbb{R},\ \left|z\right|\leq1,\end{align}

(11) \begin{align}\left|\ln\left(1+z\right)-z\right| & \leq2z^{2},\quad z\in\mathbb{R},\ \left|z\right|\leq\frac{1}{2},\end{align}

on the remainder terms of order 2, as is easy to see from the Lagrange form of said remainder terms. The constant $m_{0}$ in (9) is chosen to be such that the value of z in the Taylor expansions will be small enough in our calculations.

Then, for all $n_{\lambda}\geq m_{0}$, we have

\begin{align*}\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}} & =\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{n_{\lambda}}=\exp\left[n_{\lambda}\ln\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)\right]\\ & =\exp\left[n_{\lambda}\left(-\frac{r_{n_{\lambda}}}{n_{\lambda}}+s_{n_{\lambda}}\right)\right]=\exp\left[-r_{n_{\lambda}}+n_{\lambda}s_{n_{\lambda}}\right],\end{align*}

where the remainder term $s_{n_{\lambda}}$ satisfies $\left|s_{n_{\lambda}}\right|\leq2\left(\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{2}=O\left(\frac{1}{n_{\lambda}^{2}}\right)$ by (11). We continue with our calculation:

\begin{align*}\exp\left(-r_{n_{\lambda}}+n_{\lambda}s_{n_{\lambda}}\right) & =\exp\left[-\frac{t_{\lambda}^{2}}{2}-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}\right]\\ & =\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\exp\left[-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}\right]\\ & =\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\left[1-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}+u_{n_{\lambda}}\right],\end{align*}

where the remainder term $u_{n_{\lambda}}$ satisfies

(12) \begin{align}\left|u_{n_{\lambda}}\right| & \leq\frac{e}{2}\left(\left(-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}\right)^{2}\right)\nonumber \\ & =O\left(\left(-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}\right)^{2}\right)\nonumber \\ & =O\left(\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)^{2}\right)+O\left(\left(n_{\lambda}s_{n_{\lambda}}\right)^{2}\right)=O\left(\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)^{2}\right)+O\left(\frac{1}{n_{\lambda}^{2}}\right),\end{align}

by (10). We used $O\left(\left(n_{\lambda}s_{n_{\lambda}}\right)^{2}\right)=O\left(\frac{n_{\lambda}^{2}}{n_{\lambda}^{4}}\right)=O\left(\frac{1}{n_{\lambda}^{2}}\right)$ in the last step above. Now we estimate the difference

\begin{align*}& \quad \left|\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}-\exp\left(-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}\right)\right| \\& =\left|\left(1-\frac{r_{n_{\lambda}}}{n_{\lambda}}\right)^{n_{\lambda}}-\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\right|\\ & =\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\left|1-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}+u_{n_{\lambda}}-1\right|\\ & =\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\left|-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}+u_{n_{\lambda}}\right|.\end{align*}

Taking into account $n_{\lambda}s_{n_{\lambda}}=O\left(\frac{1}{n_{\lambda}}\right)$ and (12), we have

(13) \begin{align}&\quad \exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\left|-\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+n_{\lambda}s_{n_{\lambda}}+u_{n_{\lambda}}\right| \nonumber \\& \leq\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\left[\left|r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right|+\left|n_{\lambda}s_{n_{\lambda}}\right|+\left|u_{n_{\lambda}}\right|\right]\nonumber \\ & =O\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+O\left(\frac{1}{n_{\lambda}}\right) \nonumber \\ &\quad +O\left(\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)^{2}\right)+O\left(\frac{1}{n_{\lambda}^{2}}\right)\nonumber \\ & =O\left(r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right)+O\left(\frac{1}{n_{\lambda}}\right).\end{align}

Using the definition of $r_{n_{\lambda}}$ and the assumption $\varepsilon_{n,\lambda}=o\left(\frac{1}{\sqrt{n_{\lambda}}}\right)$, we have

\begin{align*}\left|r_{n_{\lambda}}-\frac{t_{\lambda}^{2}}{2}\right| & =m_{\lambda}^{2}\frac{t_{\lambda}^{2}}{2}\leq\varepsilon_{n,\lambda}^{2}\frac{t_{\lambda}^{2}}{2}=o\left(\frac{1}{n_{\lambda}}\right).\end{align*}

It follows from the above and (13) that

\begin{align*}\left|\left(1-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}\right)^{n_{\lambda}}-\exp\left(-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}\right)\right|=O\left(\frac{1}{n_{\lambda}}\right),\end{align*}

and hence the statement of the lemma is proved.

Lemmas 4 and 5 imply the following corollary.

Corollary 3. The range of $b_{n,\lambda}$ over all possible values of $m_{\lambda}\in\left[-\varepsilon_{n,\lambda},\varepsilon_{n,\lambda}\right]$ lies in a $\frac{D}{n_{\lambda}}$-neighbourhood $B_{n}$ of the set $\left[\exp\left(-\frac{t_{\lambda}^{2}}{2}\right),\exp\left(-\left(1-\varepsilon_{n,\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2}\right)\right]$ in $\mathbb{C}$. The set sequence $B_{n}$ converges:

\begin{align*}B_{n}\searrow\left\{ \exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\right\} \quad\mathrm{as }n\rightarrow\infty.\end{align*}

Since $\mu_{n}$ is symmetric, once we integrate $c_{n,\lambda}$ with respect to $\mu_{n}$, only the integral of the real part $\mathrm{Re}\left(c_{n,\lambda}\right)$ remains. This real part is the radius of the circle on which all values of $c_{n,\lambda}$ lie, and these radii are the values that comprise the set $B_{n}$. As each $\mu_{n}$ is a probability measure and we assumed $\mu_{n}\left[-\varepsilon_{n},\varepsilon_{n}\right]\xrightarrow[n\rightarrow\infty]{}1$, we have

(14) \begin{align} ({7}) & \xrightarrow[n\rightarrow\infty]{}\prod_{\lambda=1}^{M}\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)=\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t).\end{align}

As we see, (7) converges to a non-zero limit as $n\rightarrow\infty$, thereby proving the asymptotic equivalence of (7) to $\varphi_{n}(t)$.

This concludes the proof of Theorem 4.

3.3. Proof of Theorem 6

We now show the statements of Theorem 6. The first statement is a corollary of Theorem 4. To prove the second statement, we once again have to calculate $\varphi_{n}(t)=\mathbb{E}\exp\left(it\cdot\boldsymbol{S}_{n}\right)$. The normalisation $\gamma_{n,\lambda}=\sqrt{n_{\lambda}}$ is the same as in Theorem 4. Since $\mu_{n}$ is the contraction of $\mu$, which has support in $\left[-1,1\right]^{M}$, the support of $\mu_{n}$ is a subset of $\left[-\varepsilon_{n},\varepsilon_{n}\right]$ for each $n\in\mathbb{N}$, and hence

(15) \begin{equation}\varphi_{n}(t)=\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}E_{m}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\sqrt{n_{1}}}+\cdots+t_{M}\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\right)\mu_{n}\left(\mathrm{d}m\right).\end{equation}

At this point, we change variables by setting $s\,:\!=\,\left(\frac{1}{\varepsilon_{n,1}},\ldots,\frac{1}{\varepsilon_{n,M}}\right)\circ m$. Then

(16) \begin{align} ({15}) & =\int_{\left[-1,1\right]^{M}}E_{\varepsilon_{n}\circ s}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\sqrt{n_{1}}}+\cdots+t_{M}\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\right)\mu_{n}\left(\mathrm{d}\left(\varepsilon_{n}\circ s\right)\right)\nonumber \\ & =\int_{\left[-1,1\right]^{M}}E_{\varepsilon_{n}\circ s}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\sqrt{n_{1}}}+\cdots+t_{M}\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\right)\mu\left(\mathrm{d}s\right),\end{align}

where we used that, by definition, $\mu_{n}\left(\varepsilon_{n}\circ A\right)=\mu\left(A\right)$ for all measurable sets A. The conditional expectations $E_{\varepsilon_{n}\circ s}\exp\left(it_{\lambda}\frac{S_{n,\lambda}}{\sqrt{n_{\lambda}}}\right)$ can be expressed via a Taylor expansion as

\begin{align*}E_{\varepsilon_{n}\circ s}\exp\left(it_{\lambda}\frac{S_{n,\lambda}}{\sqrt{n_{\lambda}}}\right) & =\exp\left(i\varepsilon_{n,\lambda}s_{\lambda}t_{\lambda}\sqrt{n_{\lambda}}\right)\left(1-\left(1-\varepsilon_{n,\lambda}^{2}s_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2n_{\lambda}}+O\left(\frac{1}{n_{\lambda}^{3/2}}\right)\right)^{n_{\lambda}}\\ & \xrightarrow[n\rightarrow\infty]{}\exp\left(it_{\lambda}h_{\lambda}s_{\lambda}\right)\exp\left(-\frac{t_{\lambda}^{2}}{2}\right).\end{align*}

By dominated convergence, we obtain

\begin{align*} ({16}) & \xrightarrow[n\rightarrow\infty]{}\int_{\left[-1,1\right]^{M}}\prod_{\lambda=1}^{M}\exp\left(it_{\lambda}h_{\lambda}s_{\lambda}\right)\exp\left(-\frac{t_{\lambda}^{2}}{2}\right)\mu\left(\mathrm{d}s\right)=\varphi_{\mu}\left(h\circ t\right)\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t),\end{align*}

where $\varphi_{\mu}$ is the characteristic function of $\mu$.

Since $\varphi_{\mu}\left(h\circ t\right)=\varphi_{h\circ\mu}\left(t\right)$, this concludes the proof of the second statement of Theorem 6.

In the statement (iii), we have the normalisation sequences $\gamma_{n,\lambda}\,:\!=\,\varepsilon_{n,\lambda}n_{\lambda}$. Let $\varphi_{n}$ be the characteristic function of the distribution of

\begin{align*}\boldsymbol{S}_{n}=\left(\frac{S_{n,1}}{\gamma_{n,1}},\ldots,\frac{S_{n,M}}{\gamma_{n,M}}\right).\end{align*}

The proof of (iii) proceeds as outlined above and yields

(17) \begin{equation}\varphi_{n}(t)=\int_{\left[-1,1\right]^{M}}E_{\varepsilon_{n}\circ s}\exp\left(i\left(t_{1}\frac{S_{n,1}}{\gamma_{n,1}}+\cdots+t_{M}\frac{S_{n,M}}{\gamma_{n,M}}\right)\right)\mu\left(\mathrm{d}s\right).\end{equation}

We once again expand the conditional expectation in the integral above:

\begin{align*}E_{\varepsilon_{n}\circ s}\exp\left(it_{\lambda}\frac{S_{n,\lambda}}{\gamma_{n,\lambda}}\right) & =\exp\left(i\varepsilon_{n,\lambda}s_{\lambda}t_{\lambda}\frac{n_{\lambda}}{\gamma_{n,\lambda}}\right)\left(1-\left(1-\varepsilon_{n,\lambda}^{2}s_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{2\gamma_{n,\lambda}^{2}}+O\left(\frac{1}{\gamma_{n,\lambda}^{3}}\right)\right)^{n_{\lambda}}.\end{align*}

Recall that we assumed $1/\sqrt{n_{\lambda}}=o\left(\varepsilon_{n,\lambda}\right)$. It follows that $1/n_{\lambda}=o\left(\varepsilon_{n,\lambda}^{2}\right)$ and $n_{\lambda}\varepsilon_{n,\lambda}^{2}\xrightarrow[n\rightarrow\infty]{}\infty$. Hence, $n_{\lambda}=o\left(\varepsilon_{n,\lambda}^{2}n_{\lambda}^{2}\right)$, and since $\gamma_{n,\lambda}^{2}=\varepsilon_{n,\lambda}^{2}n_{\lambda}^{2}$ we have

\begin{align*}E_{\varepsilon_{n}\circ s}\exp\left(it_{\lambda}\frac{S_{n,\lambda}}{\gamma_{n,\lambda}}\right)\approx\exp\left(i\varepsilon_{n,\lambda}s_{\lambda}t_{\lambda}\frac{n_{\lambda}}{\gamma_{n,\lambda}}\right)=\exp\left(is_{\lambda}t_{\lambda}\right).\end{align*}

By dominated convergence,

\begin{align*}\varphi_{n}(t) & \xrightarrow[n\rightarrow\infty]{}\int_{\left[-1,1\right]^{M}}\prod_{\lambda=1}^{M}\exp\left(is_{\lambda}t_{\lambda}\right)\mu\left(\mathrm{d}s\right)=\varphi_{\mu}\left(t\right),\end{align*}

and our proof is complete.

We omit the proof of Theorem 7 as it amounts to a repetition of the arguments presented above, carefully keeping track of the coordinates belonging to each of the three clusters. Corollaries 1 and 2 follow immediately from Theorem 7.

3.4. Proof of Proposition 1

We calculate the correlation $\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)$ using Definition 3:

\begin{align*}\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right) & =\int_{\mathbb{R}^{M}}E_{\bar{m}}\left(X_{\lambda1}X_{\nu2}\right)\,\mu_{n}\left(\mathrm{d}m\right) \\& =\int_{\mathbb{R}^{M}}\left(P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}=X_{\nu2}\right)-P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}\neq X_{\nu2}\right)\right)\,\mu_{n}\left(\mathrm{d}m\right)\\ & =2\int_{\mathbb{R}^{M}}P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}=X_{\nu2}\right)\,\mu_{n}\left(\mathrm{d}m\right)-1.\end{align*}

We express $P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}=X_{\nu2}\right)$ in terms of $\bar{m}$:

\begin{align*}P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}=X_{\nu2}\right) & =P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}=X_{\nu2}=1\right)+P_{\bar{m}}^{\otimes n}\left(X_{\lambda1}=X_{\nu2}=-1\right)\\ & =\frac{1+\bar{m}_{\lambda}}{2}\frac{1+\bar{m}_{\nu}}{2}+\frac{1-\bar{m}_{\lambda}}{2}\frac{1-\bar{m}_{\nu}}{2}=\frac{1+\bar{m}_{\lambda}\bar{m}_{\nu}}{2},\end{align*}

where in the second step we used that all $X_{\kappa i}$ are independent random variables under $P_{\bar{m}}^{\otimes n}$. Hence,

\begin{align*}\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)=\int_{\mathbb{R}^{M}}\left(1+\bar{m}_{\lambda}\bar{m}_{\nu}\right)\,\mu_{n}\left(\mathrm{d}m\right)-1=\int_{\mathbb{R}^{M}}\bar{m}_{\lambda}\bar{m}_{\nu}\,\mu_{n}\left(\mathrm{d}m\right).\end{align*}

We have

\begin{align*}\int_{\mathbb{R}^{M}}\bar{m}_{\lambda}\bar{m}_{\nu}\,\mu_{n}\left(\mathrm{d}m\right) & =\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}\bar{m}_{\lambda}\bar{m}_{\nu}\,\mu_{n}\left(\mathrm{d}m\right)+\int_{\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]}\bar{m}_{\lambda}\bar{m}_{\nu}\,\mu_{n}\left(\mathrm{d}m\right),\end{align*}

and we upper-bound each summand in turn:

\begin{align*}\left|\int_{\left[-\varepsilon_{n},\varepsilon_{n}\right]}\bar{m}_{\lambda}\bar{m}_{\nu}\,\mu_{n}\left(\mathrm{d}m\right)\right|\leq\sup\nolimits_{m\in\left[-\varepsilon_{n},\varepsilon_{n}\right]}\left|\bar{m}_{\lambda}\bar{m}_{\nu}\right|=O\left(\varepsilon_{n,\lambda}\varepsilon_{n,\nu}\right)\xrightarrow[n\rightarrow\infty]{}0,\end{align*}

where we used the fact that $\lim_{m\rightarrow0}\frac{m_{\kappa}}{\bar{m}_{\kappa}}=1$ for all $\kappa=1,\ldots,M$, and

\begin{align*}\left|\int_{\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]}\bar{m}_{\lambda}\bar{m}_{\nu}\,\mu_{n}\left(\mathrm{d}m\right)\right|\leq\mu_{n}\left(\mathbb{R}^{M}\backslash\left[-\varepsilon_{n},\varepsilon_{n}\right]\right)\xrightarrow[n\rightarrow\infty]{}0,\end{align*}

where we used that $\left\Vert \bar{m}_{\lambda}\bar{m}_{\nu}\right\Vert _{\infty}\leq1$.

3.5. Proof of Theorem 8

For our proof of the local limit theorem, Theorem 8, we will need the following auxiliary lemmas.

Lemma 6. Let $Y\,:\!=\,\left(Y_{1},\ldots,Y_{d}\right)$ be a random vector with values on the lattice $\prod_{\lambda=1}^{d}\left(v_{\lambda}+w_{\lambda}\mathbb{Z}\right)$, where $v_{\lambda}\in\mathbb{R}$ and $w_{\lambda}>0$, and let $\varphi$ be the characteristic function of the distribution of Y. Then for all $k_{1},\ldots,k_{d}\in\mathbb{Z}$ we have

\begin{align*}\left|\varphi\left(2\pi\left(\frac{k_{1}}{w_{1}},\cdots,\frac{k_{d}}{w_{d}}\right)\right)\right|=1,\end{align*}

and for all $t\in\mathbb{R}^{d}$ such that $0\lt t_{\lambda}\lt \frac{2\pi}{w_{\lambda}}$ for at least one component $\lambda$, we have

\begin{align*}\left|\varphi\left(t\right)\right|<1.\end{align*}

Proof. The lemma follows from the proof of Theorem 3.5.2 on page 140 in [Reference Durrett7].

Lemma 6 gives an upper bound for the characteristic function of a random vector on a lattice (such as $\boldsymbol{S}_{n}$), which we shall use in our calculations later on. We will use the following inversion formulas to recover distributions from their characteristic functions.

Lemma 7. Let $\left(Y_{1},\ldots,Y_{d}\right)$ be a random vector as in Lemma 6 with characteristic function $\varphi$. Then, for all $x\in\prod_{\lambda=1}^{d}\left(v_{\lambda}+w_{\lambda}\mathbb{Z}\right)$,

\begin{align*}\mathbb{P}\left(\left(Y_{1},\ldots,Y_{d}\right)=x\right)=\frac{\prod_{\lambda=1}^{d}w_{\lambda}}{\left(2\pi\right)^{d}}\int_{\prod_{\lambda}\left[-\frac{\pi}{w_{\lambda}},\frac{\pi}{w_{\lambda}}\right]}e^{-it\cdot x}\,\varphi(t)\,\mathrm{d}t.\end{align*}

Proof. See e.g. Section 3.10 in [Reference Durrett7].

Lemma 8. Let $\varphi$ be the characteristic function of some probability distribution on $\mathbb{R}^{d}$ such that $\varphi$ is Lebesgue integrable. Then

\begin{align*}f(x)=\frac{1}{\left(2\pi\right)^{d}}\int_{\mathbb{R}^{d}}e^{-it\cdot x}\,\varphi(t)\,\mathrm{d}t\end{align*}

for all $x\in\mathbb{R}^{d}$ defines a continuous Lebesgue density function f for said probability distribution.

Proof. This is Theorem 5.5 in [Reference Wengenroth36] (alternatively, see Theorem 3.3.14 in [Reference Durrett7]).

Now we start the proof proper of Theorem 8. Let $\mathbb{1}_{A}$ be the indicator function of a measurable set A. The lattice on which $\boldsymbol{S}_{n}$ lives is $\mathcal{L}_{n}\,:\!=\,\prod_{\lambda=1}^{M}\left(\sqrt{n_{\lambda}}+\frac{2}{\sqrt{n_{\lambda}}}\mathbb{Z}\right)$. Let $x\in\mathcal{L}_{n}$. By Lemmas 7 and 8, we have

(18) \begin{align}& \quad \left|\frac{\prod_{\lambda=1}^{M}\sqrt{n_{\lambda}}}{2^{M}}\mathbb{P}\left(\boldsymbol{S}_{n}=x\right)-\phi(x)\right| \nonumber \\& =\frac{1}{\left(2\pi\right)^{M}}\left|\int_{\prod_{\lambda}\left[-\frac{\pi\sqrt{n_{\lambda}}}{2},\frac{\pi\sqrt{n_{\lambda}}}{2}\right]}e^{-it\cdot x}\varphi_{n}(t)\,\mathrm{d}t-\int_{\mathbb{R}^{M}}e^{-it\cdot x}\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)\,\mathrm{d}t\right|\nonumber \\ & \leq\frac{1}{\left(2\pi\right)^{M}}\int_{\mathbb{R}^{M}}\mathbb{1}_{\prod_{\lambda}\left[-\frac{\pi\sqrt{n_{\lambda}}}{2},\frac{\pi\sqrt{n_{\lambda}}}{2}\right]}(t)\left|\varphi_{n}(t)-\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)\right|\mathrm{d}t \end{align}

(19) \begin{align} \quad+\frac{1}{\left(2\pi\right)^{M}}\int_{\mathbb{R}^{M}\backslash\prod_{\lambda}\left[-\frac{\pi\sqrt{n_{\lambda}}}{2},\frac{\pi\sqrt{n_{\lambda}}}{2}\right]}\left|\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)\right|\mathrm{d}t.\qquad\qquad\qquad\end{align}

Note that both (18) and (19) are independent of the point $x\in\mathcal{L}_{n}$. The term (19) converges to 0 as $n\rightarrow\infty$, since $\left|\varphi_{\mathcal{N}\left(0,I_{M}\right)}\right|$ is integrable. Thus, our remaining task is to show that (18) converges to 0. By Theorem 4, which we have already shown, $\varphi_{n}(t)\xrightarrow[n\rightarrow\infty]{}\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)$ holds for all $t\in\mathbb{R}^{M}$. Hence, we can prove that (18) converges to 0 by finding an appropriate integrable majorant and applying the theorem of dominated convergence.

We pick some $0<\delta<\pi/2$ and partition the set $\prod_{\lambda}\left[-\frac{\pi\sqrt{n_{\lambda}}}{2},\frac{\pi\sqrt{n_{\lambda}}}{2}\right]$ into the disjoint sets

$A_{n}\,:\!=\,\prod_{\lambda}\left[-\delta\sqrt{n_{\lambda}},\delta\sqrt{n_{\lambda}}\right]$ and $B_{n}\,:\!=\,\prod_{\lambda}\left[-\frac{\pi\sqrt{n_{\lambda}}}{2},\frac{\pi\sqrt{n_{\lambda}}}{2}\right]\backslash A_{n}$ for each $n\in\mathbb{N}$. For all $s\in\left[-1,1\right]$, let $\varphi_{\mathcal{R}(s)}$ be the characteristic function of the Rademacher distribution on $\left\{ -1,1\right\} $ (see Definition 2).

We construct an integrable majorant first over $A_{n}$ and then $B_{n}$. The following upper bound holds over $A_{n}$:

(20) \begin{equation}\mathbb{1}_{A_{n}}(t)\left|\varphi_{n}(t)\right|\leq\mathbb{1}_{A_{n}}(t)\int_{\left[-1,1\right]^{M}}\prod_{\lambda}\left|\varphi_{\mathcal{R}(m_{\lambda})}\left(\frac{t_{\lambda}}{\sqrt{n_{\lambda}}}\right)\right|^{n_{\lambda}}\mu_{n}(\mathrm{d}m).\end{equation}

The next lemma will provide an upper bound for the characteristic function of the Rademacher distribution.

Lemma 9. For any $x\in\mathbb{R}$ and any $n\in\mathbb{N}$, the upper bound

\begin{align*}\left|\exp\left(ix\right)-\sum_{k=0}^{n}\frac{\left(ix\right)^{k}}{k!}\right|\leq\min\left\{ \frac{\left|x\right|^{n+1}}{\left(n+1\right)!},\frac{2\left|x\right|^{n}}{n!}\right\}\end{align*}

holds.

Proof. We use Taylor expansions with the remainder term in Legendre form. First we expand to order n,

(21) \begin{equation}\exp\left(ix\right)-\sum_{k=0}^{n}\frac{\left(ix\right)^{k}}{k!}=\frac{\left(ix\right)^{n+1}\exp\left(i\xi_{n+1}\right)}{\left(n+1\right)!},\end{equation}

and then to order $n-1$,

\begin{align*}\exp\left(ix\right)-\sum_{k=0}^{n-1}\frac{\left(ix\right)^{k}}{k!}=\frac{\left(ix\right)^{n}\exp\left(i\xi_{n}\right)}{n!}.\end{align*}

The variables $\xi_{n}$ and $\xi_{n+1}$ each lie between 0 and x. Combining the two displays above yields

(22) \begin{align}\exp\left(ix\right)-\sum_{k=0}^{n}\frac{\left(ix\right)^{k}}{k!} & =\exp\left(ix\right)-\sum_{k=0}^{n-1}\frac{\left(ix\right)^{k}}{k!}-\frac{\left(ix\right)^{n}}{n!}\nonumber \\ & =\frac{\left(ix\right)^{n}\exp\left(i\xi_{n}\right)}{n!}-\frac{\left(ix\right)^{n}}{n!}.\end{align}

Taking into account $\left|i^{n}\right|=\left|i^{n+1}\right|=\left|\exp\left(i\xi_{n}\right)\right|=1$, we obtain the desired result by applying upper bounds to the absolute value of (21) and (22).

Applying the lemma to $x=X_{\lambda1}-s$ and $n=2$ and taking the expectation $E_{s}$, we obtain

(23) \begin{align}&\quad \left|E_{s}\exp\left(iu\left(X_{\lambda1}-s\right)\right)-E_{s}\left(\sum_{k=0}^{2}\frac{\left(iu\right)^{k}\left(X_{\lambda1}-s\right)^{k}}{k!}\right)\right| \nonumber \\& \leq u^{2}E_{s}\min\left\{ \frac{\left|u\right|\left|X_{\lambda1}-s\right|^{3}}{3!},\left|X_{\lambda1}-s\right|^{2}\right\} \nonumber \\ & \leq u^{2}\min\left\{ \frac{\left|u\right|E_{s}\left|X_{\lambda1}-s\right|^{3}}{3!},E_{s}\left|X_{\lambda1}-s\right|^{2}\right\} .\end{align}

We calculate the expectations in the upper bound above:

\begin{align*}E_{s}\left|X_{\lambda1}-s\right|^{2}=1-s^{2}\quad\mathrm{and}\quad E_{s}\left|X_{\lambda1}-s\right|^{3}=2\left|s\right|\left(1-s^{2}\right).\end{align*}

We calculate an upper bound for the Rademacher characteristic function:

\begin{align*}\left|\varphi_{\mathcal{R}(s)}(u)\right| & =\left|E_{s}\exp\left(iu\left(X_{\lambda1}-s\right)\right)\right|\left|\exp\left(ius\right)\right|\\ & \leq\left|1-\left(1-s^{2}\right)\frac{u^{2}}{2}\right|+u^{2}\left(1-s^{2}\right)\min\left\{ \left|u\right|\left|s\right|,1\right\} \\ & \leq1-\left(1-s^{2}\right)\frac{u^{2}}{2}+\left(1-s^{2}\right)\frac{u^{2}}{4}\leq\exp\left(-\left(1-s^{2}\right)\frac{u^{2}}{4}\right).\end{align*}

Above, in the first inequality, we used a Taylor expansion of order 2 with remainder term upper-bounded by (23). The second inequality holds for all $u\in\mathbb{R}$ such that $\left|u\right|\left|s\right|\leq1/4$, for which we have

\begin{align*}\left|u\right|^{3}\left(1-s^{2}\right)\left|s\right|\leq\left(1-s^{2}\right)\frac{u^{2}}{4}.\end{align*}

The third inequality follows from $1-x\leq\exp\left(-x\right)$ for all $x\in\mathbb{R}$.

Therefore,

\begin{align*}\left|\varphi_{\mathcal{R}\left(m_{\lambda}\right)}\left(\frac{t_{\lambda}}{\sqrt{n_{\lambda}}}\right)\right|\leq\exp\left(-\left(1-m_{\lambda}^{2}\right)\frac{t_{\lambda}^{2}}{4n_{\lambda}}\right)\end{align*}

holds for all $n\in\mathbb{N}$ large enough.

By assumption, there is some $\tau\in\left(0,1\right)^{M}$ with the property (3). We continue with our calculation:

(24) \begin{align} ({20}) & \leq\mathbb{1}_{A_{n}}(t)\int_{\left[-1,1\right]^{M}}\exp\left(-\frac{1}{4}\sum_{\lambda}\left(1-m_{\lambda}^{2}\right)t_{\lambda}^{2}\right)\mu_{n}(\mathrm{d}m)\nonumber \\ & \leq\int_{\left[-\tau,\tau\right]}\exp\left(-\frac{1}{4}\sum_{\lambda}\left(1-m_{\lambda}^{2}\right)t_{\lambda}^{2}\right)\mu_{n}(\mathrm{d}m)+\mathbb{1}_{A_{n}}(t)\,\mu_{n}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right)\nonumber \\ & \leq\exp\left(-\frac{1}{4}\sum_{\lambda}\left(1-\tau_{\lambda}^{2}\right)t_{\lambda}^{2}\right)+\mathbb{1}_{A_{n}}(t)\,\mu_{n}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right).\end{align}

It is clear that the first summand in (24) is integrable. For the second summand, we have

\begin{align*}\mathbb{1}_{A_{n}}(t)\,\mu_{n}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right) & \leq\mathbb{1}_{A_{1}}\left(t\right)\mu_{1}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right) \\&\quad +\sum_{k=1}^{\infty}\mathbb{1}_{A_{k+1}\backslash A_{k}}\left(t\right)\,\mu_{k}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right)\\ & =\!:\,f(t).\end{align*}

Let $\boldsymbol{\lambda}^{M}$ be the Lebesgue measure on $\mathbb{R}^{M}$. We show that the function f on the right-hand side is an integrable majorant for all $\mathbb{1}_{A_{n}}(t)\,\mu_{n}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right)$, $n\in\mathbb{N}$:

\begin{align*}\int_{\mathbb{R}^{M}}f(t)\mathrm{d}t & =\boldsymbol{\lambda}^{M}\left(A_{1}\right)\mu_{1}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right)+\sum_{k=1}^{\infty}\boldsymbol{\lambda}^{M}\left(A_{k+1}\backslash A_{k}\right)\mu_{k}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right).\end{align*}

Each summand in the series above can be bounded above by

\begin{align*}& \quad \boldsymbol{\lambda}^{M}\left(A_{k+1}\backslash A_{k}\right)\mu_{k}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right) \\& \leq O\left(\left(\sqrt{k+1}-\sqrt{k}\right)\left(\sqrt{k+1}\right)^{M-1}\right)\mu_{k}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right),\end{align*}

which is summable in k by (3).

As $\left|\varphi_{\mathcal{N}\left(0,I_{M}\right)}\right|$ is integrable as well, we have thus found an integrable majorant for $\mathbb{1}_{A_{n}}(t)\left|\varphi_{n}(t)-\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)\right|$. By Theorem 4, $\left|\varphi_{n}(t)-\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)\right|\xrightarrow[n\rightarrow\infty]{}0$ pointwise, so we conclude that the integral of

$\mathbb{1}_{A_{n}}(t)\left|\varphi_{n}(t)-\varphi_{\mathcal{N}\left(0,I_{M}\right)}(t)\right|$ over $\mathbb{R}^{M}$ converges to 0 as $n\rightarrow\infty$.

We proceed with the integrand over the set $B_{n}$:

(25) \begin{align}\mathbb{1}_{B_{n}}(t)\left|\varphi_{n}(t)\right| & \leq\mathbb{1}_{B_{n}}(t)\int_{\left[-1,1\right]^{M}}\prod_{\lambda}\left|\varphi_{\mathcal{R}\left(m_{\lambda}\right)}\left(\frac{t}{\sqrt{n_{\lambda}}}\right)\right|^{n_{\lambda}}\mu_{n}(\mathrm{d}m)\nonumber \\ & \leq\mathbb{1}_{B_{n}}(t)\int_{\left[-1,1\right]^{M}}\left(\theta(m)\right)^{n}\mu_{n}(\mathrm{d}m),\end{align}

where the existence of

\begin{align*}\theta(m)=\max_{t\in B_{n},\lambda=1,\ldots,M}\left|\varphi_{\mathcal{R}\left(m_{\lambda}\right)}\left(t_{\lambda}\right)\right|<1\end{align*}

is a consequence of Lemma 6. We continue with the calculation of an upper bound using the constant $\tau$ from (3):

\begin{align*} ({25}) \leq\mathbb{1}_{B_{n}}(t)\int_{\left[-\tau,\tau\right]}\left(\theta(m)\right)^{n}\,\mu_{n}(\mathrm{d}m)+\mathbb{1}_{B_{n}}(t)\,\mu_{n}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right).\end{align*}

On the interval $\left[-\tau,\tau\right]$, $\theta$ is bounded away from 1:

\begin{align*}s\,:\!=\,\sup\nolimits_{m\in\left[-\tau,\tau\right]}\theta(m)<1.\end{align*}

This leads us to our final upper bound

\begin{align*}\mathbb{1}_{B_{n}}(t)\left|\varphi_{n}(t)\right| & \leq\mathbb{1}_{B_{n}}(t)\left(s^{n}+\mu_{n}\left(\left[-1,1\right]^{M}\backslash\left[-\tau,\tau\right]\right)\right).\end{align*}

For the last expression, we can construct an integrable majorant in the same manner as for the second summand in (24). This concludes the proof of Theorem 8.

4. Examples of de Finetti voting models

We discuss two classes of voting models featured prominently in the literature that are de Finetti voting models according to Definition 3. As mentioned in the introduction, models of voting behaviour are often adapted from statistical physics. Two categories are the collective bias model (CBM) and the Curie-Weiss model (CWM). The single-group CBM first appeared in Straffin’s work on voting systems [Reference Straffin, Brams, Lucas, Stran and Springer32] with a uniform de Finetti measure. It was later generalised in the works [Reference Kirsch14, Reference Kirsch and Langner15]. The CWM in its single-group version has a long history in statistical physics and was first defined by Husimi [Reference Husimi13] and Temperley [Reference Temperley33]. It has seen a multitude of applications to the social sciences (see [Reference Brock and Durlauf2] for the first instance in which the CWM was used in such a context). The multi-group version of the CWM was defined independently in [Reference Berthet, Rigollet and Srivastava1, Reference Contucci and Gallo4]. Subsequently, the model has received considerable attention from other authors (see [Reference Collet3, Reference Fedele8, Reference Fedele and Contucci9, Reference Kirsch and Toth17, Reference Kirsch and Toth18, Reference Kirsch and Toth21–Reference Löwe and Schubert23).

In the CWM, the voters tend to align with each other and there is no external influence on them. In the CBM, the voters do not care about others’ opinions, but there is some central influence (such as a cultural or religious institution) that induces correlation between the voters. In this sense, the two models are opposites. However, mathematically, they have more in common than a cursory glance would suggest. A multi-group CBM is defined by the voting measure

(26) \begin{equation}\mathbb{P}_{\mathrm{CBM}}\left(X_{11}=x_{11},\ldots,X_{Mn_{M}}=x_{Mn_{M}}\right)\,:\!=\,\int_{\left[-1,1\right]^{M}}P_{m}^{\otimes n}\left(x_{11},\ldots,x_{Mn_{M}}\right)\mu\left(\mathrm{d}m\right)\end{equation}

for all voting configurations $\left(x_{11},\ldots,x_{Mn_{M}}\right)$. In the integral above, $\mu$ is a symmetric probability measure on $\left[-1,1\right]^{M}$; also, recall the probability measure $P_{m}^{\otimes n}$ from Notation 1. By (26), the CBM is a de Finetti voting model according to Definition 3 with a constant sequence of de Finetti measures $\mu_{n}\,:\!=\,\mu$, $n\in\mathbb{N}$.

The variable m in the integral (26) measures the prevalent bias in the population. A positive bias means the voters are more likely to be in favour of the proposal. For a fixed bias m, the voters are conditionally independent in their decisions. In a CBM,

(27) \begin{equation}\left(\frac{S_{n,1}}{n_{1}},\ldots,\frac{S_{n,M}}{n_{M}}\right)\xrightarrow[n\rightarrow\infty]{\mathrm{d}}\mu.\end{equation}

The above limit theorem provides useful information on the large-population behaviour of the CBM with the single exception of the (degenerate) case $\mu=\delta_{0}$, in which all voters are independent and we have the central limit theorem

(28) \begin{equation}\left(\frac{S_{n,1}}{\sqrt{n_{1}}},\ldots,\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\xrightarrow[n\rightarrow\infty]{\mathrm{d}}\mathcal{N}\left(0,I_{M}\right).\end{equation}

See [Reference Kirsch and Toth20] for these results.

While a CBM is a de Finetti voting model, it does not satisfy the assumption of decreasing social cohesion embodied by the weak convergence of $\mu_{n}$ to $\delta_{0}$, except when $\mu=\delta_{0}$. For more on single-group CBMs in the context of voting theory, see [Reference Kirsch14], and for an in-depth discussion of multi-group versions of the CBM, see [Reference Kirsch and Toth20].

The single-group CWM is defined for all voting configurations $\left(x_{1},\ldots,x_{n}\right)$ by

\begin{align*}\mathbb{P}_{\mathrm{CWM}}\left(X_{1}=x_{1},\ldots,X_{n}=x_{n}\right)\,:\!=\,Z_{\beta,n}^{-1}\exp\left(\frac{\beta}{2n}\left(\sum_{i=1}^{n}x_{i}\right)^{2}\right),\end{align*}

where $Z_{\beta,n}$ is a normalisation constant which depends on $\beta$ and n. The parameter $\beta\geq0$ is the inverse temperature in the physical context of the CWM as a model of ferromagnetism. As a voting model, $\beta$ measures the degree of influence the voters exert over each other. As we see, the most probable voting configurations are those with unanimous votes in favour of or against the proposal. However, there are only two of these extreme configurations, whereas there is a multitude of low-probability configurations with roughly equal numbers of votes for and against. This is the ‘conflict between energy and entropy’. Which one of these pseudo-forces dominates depends on the regime the model is in, determined by the value of the parameter $\beta\geq0$. Using the Hubbard–Stratonovich transformation, the CWM can also be expressed in the following way (see e.g. Chapter 2 of [Reference Friedli and Velenik10] or Theorem 5.6 in the monograph [Reference Kirsch16]):

(29) \begin{align}& \quad \mathbb{P}_{\mathrm{CWM}}\left(X_{1}=x_{1},\ldots,X_{n}=x_{n}\right) \nonumber \\ & =\left(Z^{\prime}_{\beta,n}\right)^{-1}\int_{\left[-1,1\right]}P_{m}^{\otimes n}\left(x_{11},\ldots,x_{Mn_{M}}\right)\frac{\exp\left(-nF\left(m\right)\right)}{1-m^{2}}\,\mathrm{d}m,\end{align}

for all voting configurations $\left(x_{1},\ldots,x_{n}\right)$. Here

\begin{align*}Z^{\prime}_{\beta,n}=\int_{\left[-1,1\right]}\frac{\exp\left(-nF\left(m\right)\right)}{1-m^{2}}\,\mathrm{d}m\end{align*}

is a normalisation constant and the function F above is given by

\begin{equation*}F\left(m\right)\,:\!=\,\frac{1}{2}\left(\frac{1}{\beta}\left(\frac{1}{2}\ln\frac{1+m}{1-m}\right)^{2}+\ln\left(1-m^{2}\right)\right), \qquad m\in\left[-1,1\right].\end{equation*}

The representation (29) of the CWM is called the ‘de Finetti representation’. We note that for fixed n we can define

(30) \begin{equation}\mu_{n}\,:\!=\,\left(Z^{\prime}_{\beta,n}\right)^{-1}\frac{\exp\left(-nF\left(m\right)\right)}{1-m^{2}}\,\mathbb{1}_{\left[-1,1\right]}\,\boldsymbol{\lambda},\end{equation}

where $\boldsymbol{\lambda}$ is the Lebesgue measure, and the CWM is a CBM. However, this is only true for fixed n. Whereas in (26) the de Finetti measure $\mu$ is static, i.e. independent of n, in (29) we see that there is a different de Finetti measure $\mu_{n}$ for each model of size n. Hence, taken as a class of models with any $n\in\mathbb{N}$, the CWM is not a special case of the CBM.

The CWM has three distinct ‘regimes’ in which the voting margin $S_{n}$ behaves differently. For $\beta<1$, the so-called ‘high-temperature regime’, $S_{n}/\sqrt{n}$ converges in distribution to a centred normal distribution. The sequence of de Finetti measures defined in (30) converges weakly to the Dirac measure at 0. Moreover, it has a concentration property: for any $\delta>0$, there are constants $C,D>0$ such that

(31) \begin{equation}\mu_{n}\left(\left[-1,1\right]\backslash\left[-\delta,\delta\right]\right)\lt C\exp\left(-Dn\right),\quad n\in\mathbb{N}.\end{equation}

This exponential decay is key to analysing the asymptotic behaviour of the CWM. In the ‘critical regime’, $\beta=1$, a limit theorem holds for $S_{n}/n^{3/4}$ with a limiting distribution which is not normal. Instead, the distribution has a density proportional to $\exp\left(-x^{4}/12\right)$, $x\in\mathbb{R}$. In the ‘low-temperature regime’, $\beta>1$, $S_{n}/n$ converges to the convex combination of two Dirac measures, $1/2\left(\delta_{-m(\beta)}+\delta_{m(\beta)}\right)$, where $0\lt m(\beta)<1$ is the positive solution of the Curie-Weiss equation $x=\tanh(\beta x)$.

The multi-group version of the CWM defined in (29) is given by a positive semi-definite coupling matrix $J=\left(J_{\lambda\nu}\right)_{\lambda,\nu=1,\ldots,M}\in\mathbb{R}^{M\times M}$ and the Gibbs measure

(32) \begin{equation}\mathbb{P}_{\mathrm{CWM}}\left(X_{11}=x_{11},\ldots,X_{Mn_{M}}=x_{Mn_{M}}\right)\,:\!=\,Z_{J,n}^{-1}\exp\left(\frac{1}{2}\sum_{\lambda,\nu=1}^{M}\frac{J_{\lambda\nu}}{\sqrt{n_{\lambda}n_{\nu}}}\sum_{i=1}^{n_{\lambda}}\sum_{j=1}^{n_{\nu}}x_{\lambda i}x_{\nu j}\right)\end{equation}

for all voting configurations. The normalisation constant $Z_{J,n}$ depends on the group sizes $n_{\lambda}=n_{\lambda}(n)$ and the coupling matrix J. The entries $J_{\lambda\nu}$ of the coupling matrix give the strength of interaction between any pair of voters, one belonging to group $\lambda$ and the other to group $\nu$. For this probability measure, there is a de Finetti representation given by

\begin{align*}\mathbb{P}_{\mathrm{CWM}}\left(X_{11}=x_{11},\ldots,X_{Mn_{M}}=x_{Mn_{M}}\right) & =\int_{\mathbb{R}^{M}}P_{\bar{m}}^{\otimes n}\left(x_{11},\ldots,x_{Mn_{M}}\right)\exp\left(-nF_{J,n}\left(x\right)\right)\mathrm{{d}}x.\end{align*}

Above, the mapping $m\mapsto\bar{m}$ is $\bar{m}\,:\!=\,\left(\tanh m_{\lambda}\right)_{\lambda=1,\ldots,M}$. The CWM has a sequence of de Finetti measures given by

(33) \begin{equation}\mu_{n}\,:\!=\,\exp\left(-nF_{J,n}\right)\boldsymbol{\lambda}^{M},\quad n\in\mathbb{N},\end{equation}

with $\boldsymbol{\lambda}^{M}$ being the Lebesgue measure on $\mathbb{R}^{M}$. See [Reference Kirsch and Toth21, display (12)] for the form of $F_{J,n}$ and [Reference Kirsch and Toth21, Section 2] for the three regimes of the model. An alternative representation with compactly supported de Finetti measures can be obtained by a change of variables in the integral in (32), setting $t\,:\!=\,\left(\tanh x_{\lambda}\right)_{\lambda=1,\ldots,M}$, which yields de Finetti measures $\mu^{\prime}_{n}$ whose support belongs to $\left[-1,1\right]^{M}$. The multi-group CWM thus fits into the framework outlined in Definition 3.

The parameter space of the multi-group CWM is

\begin{align*}\Phi\,:\!=\,\left\{ \left.J\in\mathbb{R}^{M\times M}\,\right|\,J\mathrm{ is positive definite or }J=\left(\beta\right)_{\lambda,\nu=1,\ldots,M}\mathrm{ for some }\beta\geq0\right\} .\end{align*}

Analogously to the single-group model, the multi-group CWM has three regimes. If J is positive definite, the high-temperature regime is the region in $\Phi$ where $C\,:\!=\,\left(I_{M}-J\right)^{-1}$ is positive definite. If $J=\left(\beta\right)_{\lambda,\nu=1,\ldots,M}$, $\beta<1$ characterises the high-temperature regime. In both cases, this regime features weak interactions between voters. The sequence $\left(\mu_{n}\right)_{n}$ given in (33) exhibits a critical convergence speed according to Definition 5, and the limiting distribution of $\left(S_{n,1}/\sqrt{n_{1}},\ldots,S_{n,M}/\sqrt{n_{M}}\right)$ is $\mathcal{N}\left(0,C\right)$ with a (generally) non-diagonal covariance matrix C. As such, the limiting distribution of $\left(S_{n,1}/\sqrt{n_{1}},\ldots,S_{n,M}/\sqrt{n_{M}}\right)$ can be expressed as the convolution of two normal distributions:

\begin{align*}\left(\frac{S_{n,1}}{\sqrt{n_{1}}},\ldots,\frac{S_{n,M}}{\sqrt{n_{M}}}\right)\xrightarrow[n\rightarrow\infty]{\mathrm{d}}\mathcal{N}\left(0,I_{M}\right)\ast\mathcal{N}\left(0,\Sigma\right),\end{align*}

where $\Sigma=\left(J^{-1}-I_{M}\right)^{-1}$ if the coupling matrix J is positive definite, and $\Sigma=\left(\beta/\left(1-\beta\right)\right)_{\lambda,\nu=1,\ldots,M}$ is singular if J is a homogeneous matrix with all its entries equal to some $0\leq\beta<1$. The critical regime of the CWM fits into the subcritical regime according to Definition 5. Lastly, the low-temperature regime of the CWM features a sequence $\left(\mu_{n}\right)_{n}$ defined in (33) which does not converge weakly to $\delta_{0}$. See [Reference Kirsch and Toth21, Sections 3 and 5.2] for more details.

In the CWM, all groups must be in the same regime of convergence speeds given in Definition 5, unless there are clusters of groups which are already independent for finite n, because the coupling constants between groups belonging to different clusters are equal to 0. This setup is realised by choosing the coupling matrix J to be a block diagonal matrix. For CBMs, since the de Finetti sequence $\left(\mu_{n}\right)_{n}$ is constant and equal to $\mu$ given in (26), $\mu_{n}$ does not converge to $\delta_{0}$ at all, except in the degenerate case $\mu=\delta_{0}$ noted above.

No version of either the CBM or the CWM exhibits behaviour in line with the empirical evidence provided in [Reference Gelman, Katz and Bafumi11, Reference Gelman, Katz and Tuerlinckx12] and discussed in the introduction. Depending on the de Finetti measure $\mu$, the CBM produces expected per-capita absolute voting margins for each group $\lambda$, $\mathbb{E}\left(\left|S_{n,\lambda}\right|/n_{\lambda}\right)$, of order $1/\sqrt{n_{\lambda}}$ or 1, as follows from the limit theorems given in (27) and (28), with nothing in between. The CWM features three distinct regimes with $\mathbb{E}\left(\left|S_{n,\lambda}\right|/n_{\lambda}\right)$ of order $1/\sqrt{n_{\lambda}}$, $1/n_{\lambda}^{1/4}$, and 1, respectively. (See [Reference Kirsch and Toth21, Section 3] for these results.)

Table 1.

De Finetti regimes of the CBM and the CWM.

We sum up the convergence behaviour of the CBM and the CWM in Table 1. As for $J=0$ being the only case in which the CWM is in the fast-convergence regime, we note that since J is assumed to be either a positive definite or a homogeneous matrix, this is indeed the only J for which the CWM is in fast convergence. If we allowed slightly more general J, such as block diagonal matrices with one or more blocks equal to 0, then the groups $\lambda$ for which $J_{\lambda\nu}=0$ for all $\nu=1,\ldots,M$ are in fast convergence, whereas other groups might fall into other regimes.

We next take a look at the asymptotic behaviour of the correlations between two votes $\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)$. By Proposition 1, de Finetti voting models with an asymptotic loss of social cohesion, represented by the weak convergence of the de Finetti measures $\left(\mu_{n}\right)_{n}$ to $\delta_{0}$, feature decaying correlations, i.e. $\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)\xrightarrow[n\rightarrow\infty]{}0$. Given the de Finetti measure $\mu$ of the CBM defined in (26), we have the equality

\begin{align*}\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)=\int_{\left[-1,1\right]^{M}}m_{\lambda}m_{\nu}\,\mu\left(\mathrm{d}m\right),\quad\lambda,\nu=1,\ldots,M.\end{align*}

So correlations do not depend on the size of the population, and, except for some special cases such as $\mu=\delta_{0}$, correlations are non-zero in a CBM.

In the CWM, the asymptotic behaviour of the correlations $\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)$ depends on the regime the model is in. We have (cf. [Reference Kirsch and Toth21, Section 5.2])

\begin{align*}\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right) & =\begin{cases}\Theta\left(\frac{1}{n}\right) & \mathrm{in the high-temperature regime,}\\\Theta\left(\frac{1}{\sqrt{n}}\right) & \mathrm{in the critical regime,}\\\Theta\left(1\right) & \mathrm{in the low-temperature regime.}\end{cases}\end{align*}

So we see that in a CWM the loss of social cohesion in the high-temperature and the critical regime is reflected in the decay of correlations between pairs of votes. In the low-temperature regime, where social cohesion does not decay, neither do we have a decay of $\mathbb{E}\left(X_{\lambda1}X_{\nu2}\right)$.

Finally, we provide a concrete example of a de Finetti voting model in line with the cited empirical evidence.

Example 1. Let $0.1\leq\alpha_{1},\ldots,\alpha_{M}\leq0.2$. Let $\mu$ be a probability measure on the space $\left\{ -1,1\right\} ^{M}$ that is symmetric with respect to the origin. We set

\begin{align*}\mu_{n}\,:\!=\,\sum_{x\in\left\{ -1,1\right\} ^{M}}\mu\left\{ x\right\} \,\delta_{\left(\frac{1}{n_{1}^{\alpha_{1}}},\ldots,\frac{1}{n_{M}^{\alpha_{M}}}\right)\circ x},\quad n\in\mathbb{N},\end{align*}

and let $\left(\mathbb{P}_{n}\right)_{n}$ be the de Finetti voting model with de Finetti sequence $\left(\mu_{n}\right)_{n}$. This model falls into the contraction pattern defined in Theorem 6. Since $0.1\leq\alpha_{1},\ldots,\alpha_{M}\leq0.2$, its convergence speed is subcritical according to Definition 5. By Theorem 6, we have the limit theorem

\begin{align*}\left(\frac{S_{n,1}}{n_{1}^{1-\alpha_{1}}},\ldots,\frac{S_{n,M}}{n_{M}^{1-\alpha_{M}}}\right)\xrightarrow[n\rightarrow\infty]{\mathrm{d}}\mu.\end{align*}

Corollary 2 states that the expected per-capita voting margins exhibit the following behaviour:

\begin{align*}\mathbb{E}\left(\frac{\left|S_{n,\lambda}\right|}{n_{\lambda}}\right)=\Theta\left(n_{\lambda}^{-\alpha_{\lambda}}\right),\quad\lambda=1,\ldots,M.\end{align*}

Thus, choosing the parameters $\alpha_{\lambda}$, $\lambda=1,\ldots,M$, allows us to obtain a good fit of the model to the data in [Reference Gelman, Katz and Bafumi11, Reference Gelman, Katz and Tuerlinckx12]. Note that any possible correlation structure between the groups can be realised by choosing the probability measure $\mu$ appropriately. Hence, this choice can make the model fit the empirical correlations between the voting margins $S_{\lambda}$ and $S_{\nu}$ of different groups $\lambda$ and $\nu$. Because of both the discrete support and the symmetry of $\mu$ with respect to the origin, this is a fairly simple statistical model of binary voting. If desired, one can introduce further symmetries in $\mu$ to reduce the number of parameters. See [Reference Kirsch and Toth19] for a paper about optimal voting weights in a two-tier voting system under a CWM with a reduced set of parameters.