Proof. Partition $\mathbb {R}$ into Borel sets $\mathbb {R} = U_n \sqcup V_n \sqcup W_n$ such that $\mu _n|_{U_n} \sim \nu _n|_{U_n}$ , $\nu _n(V_n) = 0$ and $\mu _n(W_n) = 0$ . From equation (3.2), it follows in particular that

(3.3) $$ \begin{align} \sum_{n=1}^{\infty} \mu_n(V_n) < + \infty \quad\text{and}\quad \sum_{n =1}^{\infty} \nu_n(W_n) < + \infty \;. \end{align} $$

Define $(\Omega ,\mu )$ and $(\Omega _n,\widetilde {\mu }_n)$ as in equation (3.1) so that the tail boundary A of $(\mu _n)_{n \in \mathbb {N}}$ is realized as the intersection of $A_n = (\pi _n)_*(L^{\infty }(\mathbb {R} \times \Omega _n))$ inside $B = L^{\infty }(\mathbb {R} \times \Omega )$ . Similarly define $(\Omega ',\nu )$ , $(\Omega ^{\prime }_n,\widetilde {\nu }_n)$ and $\pi ^{\prime }_n$ so that the tail boundary C of $(\nu _n)_{n \in \mathbb {N}}$ is realized as the intersection of $C_n = (\pi ^{\prime }_n)_*(L^{\infty }(\mathbb {R} \times \Omega ^{\prime }_n))$ inside $D = L^{\infty }(\mathbb {R} \times \Omega ')$ .

Define $X_n \subset \mathbb {R} \times \Omega _n$ by $(t,x) \in X_n$ iff $x_m \in U_m$ for all $m \geq n+1$ . Similarly, define $X^{\prime }_n \subset \mathbb {R} \times \Omega ^{\prime }_n$ . Denote $p_n = (\pi _n)_*(1_{X_n})$ and $p^{\prime }_n = (\pi ^{\prime }_n)_*(1_{X^{\prime }_n})$ . The Kakutani criterion for the equivalence of product measures implies that the identity map $X_n \to X^{\prime }_n$ is a nonsingular isomorphism, inducing a $*$ -isomorphism $\theta _n : A_n p_n \to C_n p^{\prime }_n$ . By equation (3.3), the projections $p_n \in A$ and $p^{\prime }_n \in D$ are increasing to $1$ . The $*$ -isomorphisms $\theta _n$ are compatible so that there is a unique $*$ -isomorphism $\theta : A \to C$ satisfying $\theta (a)p^{\prime }_n = \theta _n(a p_n)$ for all $a \in A$ and $n \in \mathbb {N}$ . By construction, $\theta $ conjugates the tail boundary flows of $(\mu _n)_{n \in \mathbb {N}}$ and $(\nu _n)_{n \in \mathbb {N}}$ .

Proof. Fix $\kappa> 0$ , and choose couplings $\zeta _n \in \Gamma (\mu _n,\nu _n)$ such that

$$ \begin{align*}\int_{\mathbb{R}^2} T_{\kappa}(x-y)^2 \, d\zeta_n(x,y) \leq 2^{-n} + W_{2,\kappa}(\mu_n,\nu_n)^2 \; .\end{align*} $$

So we get that

$$ \begin{align*}\sum_{n=1}^{\infty} \int_{\mathbb{R}^2} T_{\kappa}(x-y)^2 \, d\zeta_n(x,y) < +\infty \; .\end{align*} $$

Define $\psi : \mathbb {R}^2 \to \mathbb {R}^2 : \psi (x,y) = (x-y,y)$ , and put $\eta _n = \psi _*(\zeta _n)$ . Fix a probability measure $\lambda $ on $\mathbb {R}$ that is equivalent with the Lebesgue measure. Consider

$$ \begin{align*}(X,\mu) = \prod_{n=1}^{\infty} (\mathbb{R},\mu_n), \quad (Y,\nu) = \prod_{n=1}^{\infty} (\mathbb{R},\nu_n) \quad\text{and}\quad (\Omega,\eta) = \prod_{n=1}^{\infty} (\mathbb{R}^2,\eta_n) \; .\end{align*} $$

Whenever $x,y \in \mathbb {R}^{\mathbb {N}}$ , we denote by $(x,y) \in \Omega $ the element $(x_1,y_1,x_2,y_2,\ldots )$ . As in the definition of the tail boundary flow, define

$$ \begin{align*} & (\Omega_m,\widetilde{\eta}_m) = \prod_{n=m+1}^{\infty} (\mathbb{R}^2,\eta_n) \quad\text{and}\\ & S_m : \mathbb{R} \times \Omega \to \mathbb{R} \times \Omega_m : S_m(t,x,y) = \left(t + \sum_{i=1}^m (x_i+y_i), x_{m+1},y_{m+1},x_{m+2},y_{m+2},\ldots\right) \;. \end{align*} $$

Note that the maps $S_m$ are nonsingular factor maps. Denote $B = L^{\infty }(\mathbb {R} \times \Omega ,\lambda \times \eta )$ , and define $A_m \subset B$ by $A_m = (S_m)_*(L^{\infty }(\mathbb {R} \times \Omega _m))$ . Define $A = \bigcap _{m=1}^{\infty } A_m$ . We let $\mathbb {R}$ act by translation in the first variable and obtain in this way an ergodic action $\mathbb {R} \curvearrowright A$ .

We identify the ergodic action $\mathbb {R} \curvearrowright A$ with both the tail boundary flow of $(\mu _n)_{n \in \mathbb {N}}$ and the tail boundary flow of $(\nu _n)_{n \in \mathbb {N}}$ .

The first identification can be easily proved as follows and basically holds by definition. Writing

$$ \begin{align*} & (X_m,\widetilde{\mu}_m) = \prod_{n=m+1}^{\infty} (\mathbb{R},\mu_n) \quad\text{and}\\ & s_m : \mathbb{R} \times X \to \mathbb{R} \times X_m : s_m(t,x) = \left(t + \sum_{i=1}^m x_i, x_{m+1},x_{m+2},\ldots\right) , \end{align*} $$

the tail boundary of $(\mu _n)$ is defined by the intersection C of $C_m = (s_m)_*(L^{\infty }(\mathbb {R} \times X_m))$ inside $D = L^{\infty }(\mathbb {R} \times X)$ . Write $S : \mathbb {R}^2 \to \mathbb {R} : S(x,y) = x+y$ . By construction, $S_*(\eta _n) = \mu _n$ , and we get the measure preserving factor map

$$ \begin{align*}P : \mathbb{R} \times \Omega \to \mathbb{R} \times X : P(t,x,y) = (t, x_1 + y_1, x_2 + y_2 , \ldots ) \; .\end{align*} $$

By independence, we have that $A \subset P_*(D)$ and $A_m \cap P_*(D) = P_*(C_m)$ . Therefore, $A = P_*(C)$ .

The second identification requires more work. We disintegrate the probability measures $\eta _n$ w.r.t. the second variable. We thus find probability measures $\eta _{n,y}$ on $\mathbb {R}$ such that

$$ \begin{align*}\int_{\mathbb{R}^2} F \, d\eta_n = \int_{\mathbb{R}} d\nu_n(y) \, \int_{\mathbb{R}} d\eta_{n,y}(x) \, F(x,y) \end{align*} $$

for all bounded Borel functions $F : \mathbb {R}^2 \to \mathbb {R}$ . Writing, for $y \in Y$ ,

$$ \begin{align*}(\Omega_y,\eta_y) = \prod_{n=1}^{\infty} (\mathbb{R},\eta_{n,y_n}) ,\end{align*} $$

we can view $(\Omega _y,\eta _y)_{y \in Y}$ as the disintegration of $(\Omega ,\eta )$ w.r.t. the measure preserving factor map $R : \Omega \to Y : R(x,y) = y$ .

Define the Borel functions

$$ \begin{align*} & \varphi_n : \mathbb{R} \to [-\kappa,\kappa] : \varphi_n(z) = \int_{\mathbb{R}} T_{\kappa}(x) \, d\eta_{n,z}(x) \quad\text{and}\\ & F : Y \to [0,+\infty] : F(y) = \sum_{n=1}^{\infty} \int_{\mathbb{R}} T_{\kappa}(x)^2 \, d\eta_{n,y_n}(x) \;. \end{align*} $$

Since

$$ \begin{align*}\int_Y F(y) \, d\nu(y) = \sum_{n=1}^{\infty} \int_{\mathbb{R}^2} T_{\kappa}(x)^2 \, d\eta_n(x,y) < +\infty ,\end{align*} $$

we get that $F(y) < +\infty $ for $\nu $ -a.e. $y \in Y$ . Also note that

$$ \begin{align*}\varphi_n(y)^2 \leq \int_{\mathbb{R}} T_{\kappa}(x)^2 \, d\eta_{n,y}(x) \quad\text{so that}\quad \sum_{n=1}^{\infty} \int_{\mathbb{R}} \varphi_n(y)^2 \, d\nu_n(y) < +\infty \; .\end{align*} $$

Write $s_n = \int _{\mathbb {R}} \varphi _n(y) \, d\nu _n(y)$ . Define the Borel sets $\mathcal {U} \subset Y$ and $\mathcal {V}_y \subset \Omega _y$ by

$$ \begin{align*} y \in \mathcal{U} &\quad\text{iff}\quad \varphi(y) = \lim_{n \to +\infty} \sum_{k=1}^n (\varphi_k(y_k) - s_k) \;\;\text{exists,}\\ x \in \mathcal{V}_y &\quad\text{iff}\quad \theta_y(x) = \lim_{n \to +\infty} \sum_{k=1}^n (x_k - \varphi_k(y_k)) \;\;\text{exists.} \end{align*} $$

Also define the Borel set

$$ \begin{align*}\mathcal{V} = \{(x,y) \in \Omega \mid F(y) < +\infty , \;\; y \in \mathcal{U} \;\;\text{and}\;\; x \in \mathcal{V}_y \} \; .\end{align*} $$

We already proved that $F(y) <+\infty $ for $\nu $ -a.e. $y \in Y$ . By van Kampen’s version of Kolmogorov’s three series theorem (see, e.g., [Reference Shiryaev47, Theorem 3 in Section 4.2]), we have that $\nu (Y \setminus \mathcal {U}) = 0$ and that for all $y \in Y$ with $F(y) < +\infty $ , also $\eta _y(\Omega _y \setminus \mathcal {V}_y) = 0$ . Since $F(y) < +\infty $ for $\nu $ -a.e. $y \in Y$ , it follows that $\eta (\Omega \setminus \mathcal {V}) = 0$ . Write

(3.5) $$ \begin{align} \pi : \mathcal{V} \to \mathbb{R} : \pi(x,y) = \theta_y(x) + \varphi(y) = \lim_{n \to +\infty} \sum_{k=1}^n (x_k - s_k), \end{align} $$

and define the nonsingular factor map $Q : \mathbb {R} \times \Omega \to \mathbb {R} \times Y : Q(t,x,y) = (t+\pi (x,y),y)$ . We view the tail boundary of $(\nu _n)_{n \in \mathbb {N}}$ as a von Neumann subalgebra $N \subset L^{\infty }(\mathbb {R} \times Y)$ . We prove that $A = Q_*(N)$ .

More precisely, we write

$$ \begin{align*} & (Y_m,\widetilde{\nu}_m) = \prod_{n=m+1}^{\infty} (\mathbb{R},\nu_n) \quad\text{and}\\ & r_m : \mathbb{R} \times Y \to \mathbb{R} \times Y_m : r_m(t,y) = \left(t + \sum_{i=1}^m y_i, y_{m+1},y_{m+2},\ldots\right) \;. \end{align*} $$

Defining $N_m = (r_m)_*(L^{\infty }(\mathbb {R} \times Y_m))$ , we have $N = \bigcap _{m=1}^{\infty } N_m$ .

It immediately follows from (3.5) that $Q_*(N_m) \subset A_m$ , so that $Q_*(N) \subset A$ . To prove the converse, fix $F \in A$ . It follows that for $\nu $ -a.e. $y \in Y$ , we may view the function $F(\cdot ,y)$ as an element of the tail boundary for the measures $(\eta _{n,y})_{n \in \mathbb {N}}$ . By e.g. [Reference Berendschot and Vaes8, Proposition 2.1], this tail boundary is given by the translation action $\mathbb {R} \curvearrowright \mathbb {R}$ and we find a unique $K_y \in L^{\infty }(\mathbb {R})$ such that $F(t,x,y) = K_y(t+\theta _y(x))$ for a.e. $(t,x) \in \mathbb {R} \times \Omega _y$ . Defining $G_y(t) = K_y(t-\varphi (y))$ , we get that $F(t,x,y) = G_y(t+\pi (x,y))$ for a.e. $(t,x) \in \mathbb {R} \times \Omega _y$ . Writing $G(t,y) = G_y(t)$ , we have found $G \in L^{\infty }(\mathbb {R} \times Y)$ such that $Q_*(G) = F$ .

It also follows from equation (3.5) that $Q_*(L^{\infty }(\mathbb {R} \times Y)) \cap A_m = Q_*(N_m)$ . Therefore, $G \in N_m$ for all $m \in \mathbb {N}$ so that $G \in N$ . This concludes the proof of the proposition.

Proof. Note that it suffices to prove the proposition assuming that all $\beta _n \neq 0$ . Indeed, it then follows that for $I = \{n \in \mathbb {N} \mid \beta _n \neq 0\}$ , the tail boundary flow of $(\mu _n)_{n \in I}$ is periodic, from which it follows that, a fortiori, the tail boundary flow of $(\mu _n)_{n \in \mathbb {N}}$ is periodic.

Assume that the sum in equation (3.7) is infinite. We prove that the tail boundary flow of $(\mu _n)_{n \in \mathbb {R}}$ is periodic.

Define the finite positive measures $\alpha _n$ such that $\mu _n = \alpha _n + \beta _n$ . Define $X_0 = \mathbb {R} \sqcup \mathbb {R}^2$ , and equip $X_0$ with the probability measures $\zeta _n = \alpha _n \sqcup \beta _n(\mathbb {R})^{-1}(\beta _n \times \beta _n)$ . Define the map $T : X_0 \to \mathbb {R}$ by $T(x) = 0$ if $x \in \mathbb {R}$ and $T(x,y) = x-y$ if $(x,y) \in \mathbb {R}^2$ . Define $(X,\zeta ) = \prod _{n \in \mathbb {N}} (X_0,\zeta _n)$ and the independent random variables $T_n : X \to \mathbb {R} : T_n(z) = T(z_n)$ . Note that $|T_n| \leq C$ for all n, $E(T_n) = 0$ and

$$ \begin{align*}E(T_n^2) = 2 \beta_n(\mathbb{R}) \, \operatorname{Var}(\beta_n(\mathbb{R})^{-1} \beta_n) \; .\end{align*} $$

Since $\sum _{n=1}^{\infty } E(T_n^2) =+\infty $ and $E(T_n^2) \leq C^2$ for every $n \in \mathbb {N}$ , we can choose $0 \leq n_1 < n_2 < \cdots $ such that

$$ \begin{align*}(40 C)^2 \leq \sum_{n=n_{k}+1}^{n_{k+1}} E(T_n^2) \leq (50 C)^2\end{align*} $$

for all $k \in \mathbb {N}$ . Since $|T_n| \leq C$ for all n, we have that $E(|T_n|^3) \leq C \, E(T_n^2)$ for all $n \in \mathbb {N}$ . Define $\sigma _k \in [40 C,50 C]$ by

$$ \begin{align*}\sigma_k^2 = \sum_{n=n_{k}+1}^{n_{k+1}} E(T_n^2) \; .\end{align*} $$

Write

$$ \begin{align*}S_k = \sum_{n=n_{k}+1}^{n_{k+1}} T_n \quad\text{and}\quad F_k(t) = \zeta\bigl(\{x \in X \mid S_k(x) \leq \sigma_k \, t \, \}\bigr) \; .\end{align*} $$

Denote by G the distribution function of a standard Gaussian random variable. By the Berry–Esseen theorem (see, e.g., [Reference Stroock48, Theorem 2.2.17]), we get that

$$ \begin{align*}|F_k(t) - G(t)| \leq \frac{10 C}{\sigma_k} \leq \frac{1}{4}\end{align*} $$

for all $k \in \mathbb {N}$ and all $t \in \mathbb {R}$ . Take $\delta < 0$ such that $G(\delta ) = 1/3$ . Then, $F_k(\delta ) \geq 1/12$ . Since $\sigma _k \geq 40 C$ and $\delta < 0$ , we conclude that

(3.8) $$ \begin{align} \zeta\bigl(\{x \in X \mid S_k(x) \leq 40 C \delta \}\bigr) \geq \frac{1}{12} \quad\text{for all } k \in \mathbb{N}. \end{align} $$

Define $(\Omega ,\mu ) = \prod _{n \in \mathbb {N}} (\mathbb {R},\mu _n)$ . Denote by $\mu _0$ the probability measure on $\mathbb {R}$ given by $d\mu _0(t) = (\pi (1+t^2))^{-1} \, dt$ . Consider the Hilbert spaces

$$ \begin{align*}H = L^2(\mathbb{R} \times \Omega, \mu_0 \times \mu) \quad\text{and}\quad K = L^2(\mathbb{R} \times X,\mu_0 \times \zeta),\end{align*} $$

and define the subspaces $H_N \subset H$ by $H_N = L^2((\mathbb {R},\mu ) \times \prod _{n=1}^N (\mathbb {R},\mu _n))$ .

Define the maps

$$ \begin{align*} &\theta_1 : X_0 \to \mathbb{R} : \theta_1(x) = x \;\;\text{if } x \in \mathbb{R},\;\; \theta_1(x,y) = x \;\;\text{if } (x,y) \in \mathbb{R}^2,\\ &\theta_2 : X_0 \to \mathbb{R} : \theta_2(x) = x \;\;\text{if } x \in \mathbb{R},\;\; \theta_2(x,y) = y \;\;\text{if } (x,y) \in \mathbb{R}^2. \end{align*} $$

Note that $T(z) = \theta _1(z) - \theta _2(z)$ for all $z \in X_0$ . We consider the associated measure preserving factor maps

$$ \begin{align*}\pi : X \to \Omega : (\pi(z))_n = \theta_1(z_n) \;\text{and}\; \rho_k : X \to \Omega : (\rho_k(z))_n = \begin{cases} \theta_2(z_n) &\text{if } n_k + 1 \leq n \leq n_{k+1},\\ \theta_1(z_n) &\text{otherwise.}\end{cases}\end{align*} $$

Define the isometries $V : H \to K$ and $W_k : H \to K$ by

$$ \begin{align*}V(\xi)(t,z) = \xi(t,\pi(z)) \quad\text{and}\quad W_k(\xi)(t,z) = \xi(t,\rho_k(z)) \; .\end{align*} $$

When $\xi \in H_N$ , we have $W_k(\xi ) = V(\xi )$ for all k large enough. By density, we get that $W_k \to V$ strongly.

Let $F \in L^{\infty }(\mathbb {R} \times \Omega )$ be a function that generates the tail boundary of $(\mu _n)_{n \in \mathbb {N}}$ . Since F belongs to the tail boundary algebra, we have

(3.9) $$ \begin{align} (W_k(F))(t,z) = (V(F))(t-S_k(z),z) \quad\text{for all } k \in \mathbb{N} \text{ and a.e.}\ (t,z) \in \mathbb{R} \times X. \end{align} $$

Since

(3.10) $$ \begin{align} \frac{1+(t+s)^2}{1+t^2} \leq 2(1+s^2) \quad\text{for all } s,t \in \mathbb{R}, \end{align} $$

and since each $S_k$ is a bounded function, we can define the bounded linear operators

$$ \begin{align*}R_k : K \to K : (R_k(\xi))(t,z) = \xi(t-S_k(z),z) \; .\end{align*} $$

Write $D = \sqrt {2(1+(50C)^2)}$ . We claim that

(3.11) $$ \begin{align} \|R_k(\xi)\|_2^2 \leq D \, \|\xi\|_{\infty} \, \|\xi\|_2 \quad\text{for all } \xi \in L^{\infty}(\mathbb{R} \times X) \subset K. \end{align} $$

To prove this claim, note that the left-hand side can be estimated, using the Cauchy–Schwartz inequality and equation (3.10), by

$$ \begin{align*} \frac{1}{\pi} \int_{X} d\zeta(z) \, &\int_{\mathbb{R}} dt \; \frac{1}{(1 + (t+S_k(z))^2)} \, |\xi(t,z)|^2 \\ & \leq \|\xi\|_{\infty} \, \int_{X} d\zeta(z) \, \int_{\mathbb{R}} dt \; \frac{1}{\sqrt{\pi (1+ t^2)}} \, |\xi(t,z)| \, \frac{\sqrt{1+t^2}}{\sqrt{\pi} (1+(t+S_k(z))^2)} \\ & \leq \|\xi\|_{\infty} \, \|\xi\|_2 \, \Bigl( \int_{X} d\zeta(z) \, \int_{\mathbb{R}} dt \; \frac{1 + t^2}{\pi (1+(t+S_k(z))^2)^2}\Bigr)^{1/2} \\ & = \|\xi\|_{\infty} \, \|\xi\|_2 \, \Bigl( \int_{X} d\zeta(z) \, \int_{\mathbb{R}} dt \; \frac{1 + (t-S_k(z))^2}{\pi (1+t^2)^2}\Bigr)^{1/2} \\ & \leq \|\xi\|_{\infty} \, \|\xi\|_2 \, \Bigl( \int_{X} d\zeta(z) \, \int_{\mathbb{R}} d\mu_0(t) \; 2 (1+S_k(z)^2)\Bigr)^{1/2} = D \, \|\xi\|_{\infty} \, \|\xi\|_2 \;. \end{align*} $$

So, the claim is proven.

Define the probability measure $\eta _k = (S_k)_*(\zeta )$ . Then

$$ \begin{align*}\int_{\mathbb{R}} t^2 \, d\eta_k(t) = E(S_k^2) \leq (50 C)^2 \quad\text{for all } k \in \mathbb{N} \end{align*} $$

so that $\eta _k$ is a tight family of probability measures. Choose a sequence $k_j \to \infty $ such that $\eta _{k_j}$ converges weakly to a probability measure $\eta $ on $\mathbb {R}$ with $\int _{\mathbb {R}} t^2 \, d\eta (t) \leq (50 C)^2 < +\infty $ . By equation (3.8), we have that $\eta _k((-\infty ,40 C \delta ]) \geq 1/12$ for all $k \in \mathbb {N}$ . Therefore, $\eta \neq \delta _0$ .

We can then define the bounded convolution operator

$$ \begin{align*}K \to K : \xi \mapsto \eta * \xi : (\eta * \xi)(t,z) = \int_{\mathbb{R}} \xi(t-s,z) \, d\eta(s) \; .\end{align*} $$

Note that by the Cauchy–Schwarz inequality and equation (3.10),

$$ \begin{align*} \|\eta * \xi\|_2^2 &\leq \int_X d\zeta(z) \, \int_{\mathbb{R}} d\eta(s) \, \int_{\mathbb{R}} d\mu_0(t) \, |\xi(t-s,z)|^2 \\ &\leq \int_X d\zeta(z) \, \int_{\mathbb{R}} d\eta(s) \, \int_{\mathbb{R}} d\mu_0(t) \, 2(1+s^2) \, |\xi(t,z)|^2 \; \end{align*} $$

so that $\|\eta * \xi \|_2 \leq D \, \|\xi \|_2$ for all $\xi \in K$ . We have a similar convolution operator $\xi \mapsto \eta * \xi $ on H. Note that $V(\eta * \xi ) = \eta * V(\xi )$ .

We finally prove that, for every $\xi \in L^{\infty }(\mathbb {R} \times X) \subset K$ and every $\xi ' \in K$ , we have that

(3.12) $$ \begin{align} \lim_j \langle R_{k_j}(\xi),\xi' \rangle = \langle \eta * \xi , \xi' \rangle \;. \end{align} $$

Define the closed subspaces $K_N = L^2(\mathbb {R} \times \prod _{n=1}^N X_0)$ of K. Fix $\xi \in L^{\infty }(\mathbb {R} \times X) \subset K$ and $\xi ' \in K$ . The orthogonal projection $E_N : K \to K_N$ corresponds to the conditional expectation so that $\|E_N(\xi )\|_{\infty } \leq \|\xi \|_{\infty }$ for all $N \in \mathbb {N}$ . By equation (3.11), the sequence $\|R_{k_j}(\xi )\|_2$ is bounded. To prove equation (3.12), we may thus assume that $\xi ' \in K_N$ for some N. By (3.11), we also have that

$$ \begin{align*} \|R_{k_j}(\xi) - R_{k_j}(E_N(\xi))\|_2^2 &= \|R_{k_j}(\xi - E_N(\xi))\|_2^2 \leq D \, \|\xi - E_N(\xi)\|_{\infty} \, \|\xi - E_N(\xi)\|_2 \\ & \leq 2 D \, \|\xi\|_{\infty} \, \|\xi - E_N(\xi)\|_2 , \end{align*} $$

which tends to zero as $N \to \infty $ , uniformly in j. To prove (3.12), we may thus also assume that $\xi \in K_N$ . When $\xi ,\xi ' \in K_N$ and $n_{k_j}> N$ , we have

$$ \begin{align*}\langle R_{k_j}(\xi),\xi' \rangle = \langle \eta_{k_j} * \xi , \xi' \rangle\end{align*} $$

for all j so that equation (3.12) follows.

We now return to our element F generating the tail boundary. By equation (3.9), we get that $W_k(F) = R_k(V(F))$ . Since $W_k \to V$ strongly, we get that $\|W_k(F) - V(F)\|_2 \to 0$ . By equation (3.12), we have that $R_k(V(F)) \to \eta * V(F) = V(\eta * F)$ weakly. We thus conclude that $F = \eta * F$ . Since $\eta \neq \delta _0$ , it follows from the Choquet–Dény theorem (see [Reference Choquet and Deny9]) that F is periodic in the first variable. So, the tail boundary flow is periodic.

Proof. For a fixed $k \in \mathbb {N}$ , we have that the measures $(p_k \delta _0 + q_k \beta _n)_{n \in J_k}$ have a uniformly bounded support in $\{0\} \cup I_k$ . It thus follows from Proposition 3.3 that

$$ \begin{align*}(p_k + q_k) \sum_{n \in J_k} \operatorname{Var}((p_k+q_k)^{-1} (p_k \delta_0 + q_k \beta_n)) < +\infty \; .\end{align*} $$

Since $I_k \subset \mathbb {R} \setminus (-1,1)$ and $\beta _n$ is supported on $I_k$ , by equation (3.13), the left-hand side is larger or equal than $p_k q_k (p_k+q_k)^{-1} |J_k|$ so that $J_k$ is a finite set.

We can also apply Proposition 3.3 to all the finite measures $q_k \beta _n \leq \mu _n$ , with $k \in \mathbb {N}$ and $n \in J_k$ , because they have uniformly bounded width C. Defining for any $k \in \mathbb {N}$ and $n \in J_k$ , the point $s_n \in I_k$ by

$$ \begin{align*}s_n = \int_{\mathbb{R}} x \, d\beta_n(x) ,\end{align*} $$

we get that

(3.16) $$ \begin{align} \sum_{k=1}^{\infty} \sum_{n \in J_k} q_k \int_{\mathbb{R}} (x-s_n)^2 \, d\beta_n(x) < +\infty \;. \end{align} $$

For every $k \in \mathbb {N}$ , we consider the finitely many points $(s_n)_{n \in J_k}$ in the interval $I_k$ . We denote by $t_k$ a ‘middle point’. More precisely, if $|J_k|$ is odd, we write $J_k = A_k \sqcup B_k \sqcup \{j_k\}$ with $|A_k| = |B_k|$ such that $s_n \leq s_{j_k} \leq s_m$ for all $n \in A_k$ and $m \in B_k$ . We put $t_k = s_{j_k}$ . If $|J_k|$ is even, we write $J_k = A_k \sqcup B_k$ with $|A_k| = |B_k|$ such that for some $t_k \in I_k$ , we again have that $s_n \leq t_k \leq s_m$ for all $n \in A_k$ and $m \in B_k$ . We choose a bijection $\alpha _k : A_k \to B_k$ .

Since the tail boundary flow of $(\mu _n)_{n \in \mathbb {N}}$ is aperiodic, also the measures $\mu _n * \mu _{\alpha _k(n)}$ with $k \in \mathbb {N}$ and $n \in A_k$ have an aperiodic tail boundary. By construction,

$$ \begin{align*} p_k q_k (\beta_n + \beta_{\alpha_k(n)}) \leq \mu_n * \mu_{\alpha_k(n)} \quad\text{for all } k \in \mathbb{N}, n \in A_k. \end{align*} $$

Moreover, the measures on the left have their support in $I_k$ , with $|I_k| \leq C$ for all $k \in \mathbb {N}$ . Again applying Proposition 3.3, we conclude that

$$ \begin{align*}\sum_{k=1}^{\infty} \sum_{n \in A_k} p_k q_k \, \operatorname{Var}((\beta_n + \beta_{\alpha_k(n)})/2) < + \infty \; .\end{align*} $$

The mean value of $(\beta _n + \beta _{\alpha _k(n)})/2$ is $(s_n + s_{\alpha _k(n)})/2$ , and we conclude that

(3.17) $$ \begin{align} \sum_{k=1}^{\infty} \sum_{n \in A_k} p_k q_k \int_{\mathbb{R}} \left(x - \frac{s_n + s_{\alpha_k(n)}}{2}\right)^2 \, d\beta_n(x) < + \infty \;. \end{align} $$

A direct computation gives that, for $n \in A_k$ ,

$$ \begin{align*}\int_{\mathbb{R}} \left(x - \frac{s_n + s_{\alpha_k(n)}}{2}\right)^2 \, d\beta_n(x) &= \operatorname{Var}(\beta_n) + \frac{1}{4}(s_{\alpha_k(n)}-s_n)^2 \\ &\geq \frac{1}{4}\bigl((s_{\alpha_k(n)}-t_k)^2 + (t_k - s_n)^2\bigr)\end{align*} $$

because $s_n \leq t_k \leq s_{\alpha _k(n)}$ . It thus follows from equation (3.17) that

$$ \begin{align*}\sum_{k = 1}^{\infty} \sum_{n \in J_k} p_k q_k \, (s_n - t_k)^2 = \sum_{k=1}^{\infty} \sum_{n \in A_k} p_k q_k \, \bigl((s_{\alpha_k(n)}-t_k)^2 + (t_k - s_n)^2\bigr) < +\infty \; .\end{align*} $$

Since $(x-t_k)^2 \leq 2(x-s_n)^2 + 2(s_n -t_k)^2$ and since $p_k \leq 1$ , in combination with equation (3.16), we have proven that equation (3.15) holds.

Proof. By induction, it suffices to prove that, for every integer $N \geq 2$ , every ITPFI $_{N+1}$ factor of type III $_0$ is isomorphic with the tensor product of an ITPFI $_N$ factor and an ITPFI $_2$ factor.

We denote in this proof by $\delta (a)$ the Dirac measure in $a \in \mathbb {R}$ . For every $a \geq 0$ , we define the probability measure

(3.18) $$ \begin{align} \gamma(a) = (1+\exp(-a))^{-1} \, \left(\delta(0) + \exp(-a) \delta(a)\right) \;. \end{align} $$

For every $a \in \mathbb {R}_{\geq 0}^N$ , we define the probability measure

$$ \begin{align*}\rho(a) = \left(1 + \sum_{i=1}^N \exp(-a_i)\right)^{-1} \, \left(\delta(0) + \sum_{i=1}^N \exp(-a_i) \delta(a_i)\right)\end{align*} $$

and the state $\psi _a$ on $M_{N+1}(\mathbb {C})$ by

$$ \begin{align*}\psi_a(A) = \left(1 + \sum_{i=1}^N \exp(-a_i)\right)^{-1} \, \left(A_{00} + \sum_{i=1}^N \exp(-a_i) A_{ii} \right) .\end{align*} $$

By diagonalizing states, any ITPFI $_{N+1}$ factor can be written as the tensor product of a sequence $(M_{N+1}(\mathbb {C}),\psi _{a_n})$ with $a_n \in \mathbb {R}_{\geq 0}^N$ . By [Reference Connes and Woods17, Theorem 3.1], its flow of weights is precisely the tail boundary flow of the sequence $(\rho (a_n))_{n \in \mathbb {N}}$ .

We thus fix such a sequence $a_n \in \mathbb {R}_{\geq 0}^N$ , we assume that the tail boundary flow of

$$ \begin{align*}\mu_n = \rho(a_n) , \;\; n \in \mathbb{N} ,\end{align*} $$

is aperiodic and we prove that it is isomorphic with the tail boundary flow of a family of probability measures of the form $\rho (b)$ , $b \in \mathbb {R}_{\geq 0}^{N-1}$ , and $\gamma (c)$ , $c \geq 0$ .

For every $k \in \mathbb {N}$ and $1 \leq i \leq N$ , we denote $J_{k,i} = \{n \in \mathbb {N} \mid a_{n,i} \in [k-1,k)\}$ . We put $p = (1+N)^{-1}$ and $q_k = p \, \exp (-k)$ . Fix $i \in \{1,\ldots ,N\}$ .

For every $n \in J_{1,i}$ , we have that

$$ \begin{align*}p \, \delta(0) + p \exp(-a_{n,i}) \, \delta(a_{n,i}) \leq \mu_n\end{align*} $$

and that the measures $\delta (a_{n,i})$ are supported on the interval $[0,1]$ . It follows from Proposition 3.3 and equation (3.13) that

(3.19) $$ \begin{align} \sum_{n \in J_{1,i}} \exp(-a_{n,i}) \, a_{n,i}^2 < +\infty \;. \end{align} $$

For every $k \geq 2$ and $n \in J_{k,i}$ , we have that

$$ \begin{align*}p \, \delta(0) + q_k \, \delta(a_{n,i}) \leq \mu_n\end{align*} $$

and that the measures at the left are supported on the interval $[k-1,k)$ . For $n \in J_{k,i}$ , we have that $\exp (-a_{n,i}) \leq \exp (1) \, (N+1) \, q_k$ . By Proposition 3.4, we thus find $b_{k,i} \in [k-1,k)$ such that

(3.20) $$ \begin{align} \sum_{k=2}^{\infty} \sum_{n \in J_{k,i}} \exp(-a_{n,i}) \, (b_{k,i} - a_{n,i})^2 < +\infty \;. \end{align} $$

Defining $b_{1,i}=0$ for all $i \in \{1,\ldots ,N\}$ and summing over $i \in \{1,\ldots ,N\}$ , it follows from equations (3.19) and (3.20) that

(3.21) $$ \begin{align} \sum_{k=1}^{\infty} \sum_{i=1}^N \sum_{n \in J_{k,i}} \exp(-a_{n,i}) \, (b_{k,i} - a_{n,i})^2 < +\infty \;. \end{align} $$

Define the maps $\alpha _n : \{1,\ldots ,N\} \to \mathbb {N}$ by $\alpha _n(i) = k$ if and only if $n \in J_{k,i}$ . Define the probability measures

$$ \begin{align*}\mu^{\prime}_n = \left(1 + \sum_{i=1}^N \exp(-a_{n,i})\right)^{-1} \, \left(\delta(0) + \sum_{i=1}^N \exp(-a_{n,i}) \delta(b_{\alpha_n(i),i})\right).\end{align*} $$

Using the Wasserstein $2$ -distance, it follows from equations (3.21) and (3.6) that $\sum _{n=1}^{\infty } W_2(\mu _n,\mu ^{\prime }_n)^2 < +\infty $ . It thus follows from Proposition 3.2 that $(\mu _n)_{n \in \mathbb {N}}$ and $(\mu ^{\prime }_n)_{n \in \mathbb {N}}$ have isomorphic tail boundary flows.

When $s, t \in \mathbb {R}$ satisfy $|s-t| \leq 1$ , we have that

$$ \begin{align*}\bigl(\exp(-s/2) - \exp(-t/2)\bigr)^2 = \exp(-s) \, \bigl(1-\exp((s-t)/2)\bigr)^2 \leq \exp(-s) \, (s-t)^2 \; .\end{align*} $$

It follows that for all $n \in J_{k,i}$ ,

(3.22) $$ \begin{align} \bigl(\exp(-a_{n,i}/2) - \exp(-b_{k,i}/2)\bigr)^2 \leq \exp(-a_{n,i}) \, (b_{k,i} - a_{n,i})^2 \;. \end{align} $$

Defining $\mu ^{\prime \prime }_n = \rho (b_{\alpha _n(1),1},\ldots ,b_{\alpha _n(N),N})$ , it thus follows from equations (3.21) and (3.22) that

$$ \begin{align*}\sum_{n=1}^{\infty} H^2(\mu^{\prime}_n,\mu^{\prime\prime}_n) < + \infty \; .\end{align*} $$

By Lemma 3.1, also $(\mu ^{\prime }_n)_{n \in \mathbb {N}}$ and $(\mu ^{\prime \prime }_n)_{n \in \mathbb {N}}$ have isomorphic tail boundary flows.

We order the elements $(b_{k,i})_{i = 1,\ldots ,N}$ in $[k-1,k)$ from smaller to larger, remove duplicates and relabel so as to find an increasing sequence $0 \leq b_1 < b_2 < \cdots $ with $b_m \geq m/N - 1$ for all $m \in \mathbb {N}$ and with all $b_{k,i}$ appearing in this sequence. Permuting elements $(d_1,\ldots ,d_N)$ does not change the probability measure $\rho (d_1,\ldots ,d_N)$ . So every $\mu ^{\prime \prime }_n$ is of the form

$$ \begin{align*}\mu^{\prime\prime}_n = \rho(b_{\theta_n(1)},\ldots,b_{\theta_n(N)})\end{align*} $$

for a nondecreasing function $\theta _n : \{1,\ldots ,N\} \to \mathbb {N}$ .

For every $1 \leq M \leq N$ , denote by $\mathcal {F}_M$ the set of nondecreasing functions from $\{1,\ldots ,M\}$ to $\mathbb {N}$ . For every $\theta \in \mathcal {F}_M$ , denote

$$ \begin{align*}\zeta(\theta) = \rho(b_{\theta(1)},\ldots,b_{\theta(M)}) \; .\end{align*} $$

Define the subset $\mathcal {J} \subset \mathcal {F}_N$ as the set of all $\theta \in \mathcal {F}_N$ that are of the form $\theta _n$ for some $n \in \mathbb {N}$ . Define $L_{\theta } \in \mathbb {N}$ as the cardinality of $\{ n \in \mathbb {N} \mid \theta _n = \theta \}$ . As explained in the beginning of Section 3.1, the tail boundary flow does not depend on the order of the transition probability measures. So, by construction, the tail boundary flow of the family $(\mu ^{\prime \prime }_n)_{n \in \mathbb {N}}$ , and hence also the tail boundary flow of $(\mu _n)_{n \in \mathbb {N}}$ , is isomorphic with the tail boundary flow of the family of measures $\zeta (\theta )^{* L_{\theta }}$ , $\theta \in \mathcal {J}$ .

For every $\theta \in \mathcal {F}_N$ , we denote by $\widehat {\theta }$ the restriction of $\theta $ to $\{1,\ldots ,N-1\}$ . For every $\theta \in \mathcal {J}$ , we apply Lemma 3.6 below to

$$ \begin{align*} & \alpha = \sum_{i=1}^{N-1} \exp(-b_{\theta(i)}) \geq \beta = \exp(-b_{\theta(N)}) \quad\text{and}\\ & P = \alpha^{-1} \sum_{i=1}^{N-1} \exp(-b_{\theta(i)}) \delta(b_{\theta(i)}) , \;\; Q = \delta(b_{\theta(N)}) \;. \end{align*} $$

By Lemma 3.6, we thus find $K_{\theta }, M_{\theta } \in \mathbb {N}$ such that with the notation of equation (3.18),

(3.23) $$ \begin{align} d_{\operatorname{TV}}\bigl(\zeta(\theta)^{*L_{\theta}} , \zeta(\widehat{\theta})^{* K_{\theta}} * \gamma(b_{\theta(N)})^{* M_{\theta}}\bigr) \leq 4 \exp(-b_{\theta(N)}/2) \;. \end{align} $$

Given $k \in \mathbb {N}$ , the number of elements $\theta \in \mathcal {F}_N$ with $\theta (N) = k$ is smaller than $k^{N-1}$ . Also recall that $b_k \geq k/N - 1$ . Thus,

$$ \begin{align*}\sum_{\theta \in \mathcal{F}_N} \exp(-b_{\theta(N)}) \leq \sum_{k=1}^{\infty} k^{N-1} \exp(-b_k) \leq \sum_{k=1}^{\infty} k^{N-1} \, \exp(1) \, \exp(-k/N) < +\infty . \end{align*} $$

It then follows from equation (3.23) and Lemma 3.1 that the tail boundary flow of $(\mu _n)_{n \in \mathbb {N}}$ is isomorphic with the tail boundary flow of the family of measures

$$ \begin{align*}\{\zeta(\widehat{\theta})^{* K_{\theta}} \mid \theta \in \mathcal{J}\} \cup \{\gamma(b_{\theta(N)})^{* M_{\theta}} \mid \theta \in \mathcal{J} \} \; .\end{align*} $$

This concludes the proof of the theorem.

Proof. For integers $k,\; m \geq 0$ with $k+m \leq L$ , we write

$$ \begin{align*}\mu(k,m) = \frac{L!}{(L-k-m)! \, k! \, m!} \, \frac{\alpha^k \, \beta^m}{(1+\alpha+\beta)^L} \; .\end{align*} $$

For all other integers $k,m \geq 0$ , we write $\mu (k,m) = 0$ . Then,

$$ \begin{align*}\bigl((1+\alpha+\beta)^{-1}(\delta_0 + \alpha P + \beta Q)\bigr)^{*L} = \sum_{k,m \geq 0} \mu(k,m) \, P^{*k} \, Q^{*m} \; .\end{align*} $$

We similarly write for all integers $0 \leq k \leq K$ and $0 \leq m \leq M$ ,

$$ \begin{align*}\mu'(k,m) = \frac{K!}{(K-k)! \, k!} \, \frac{\alpha^k}{(1+\alpha)^K} \, \frac{M!}{(M-m)! \, m!} \, \frac{\beta^m}{(1+\beta)^M}\end{align*} $$

and $\mu '(k,m) = 0$ for all other integers $k,m \geq 0$ so that

$$ \begin{align*}\bigl((1+\alpha)^{-1}(\delta_0+\alpha P)\bigr)^{* K} \; * \; \bigl((1+\beta)^{-1} (\delta_0 + \beta Q)\bigr)^{* M} = \sum_{k,m \geq 0} \mu'(k,m) \, P^{*k} \, Q^{*m} \; .\end{align*} $$

Parts (a) and (b) of the proof of [Reference Giordano and Skandalis26, Lemma 2.2] are saying that $H^2(\mu ,\mu ') \leq 2 \beta $ . Then also $d_{\operatorname {TV}}(\mu ,\mu ') \leq 2 \sqrt {\beta }$ .

Proof of Theorem 3.8

The cases of a trivial flow or a periodic flow are straightforward. By, e.g., [Reference Berendschot and Vaes8, Proposition 2.1], taking $a> 0$ and $\mu _n = \mathcal {E}(\delta _a)$ for all $n \in \mathbb {N}$ , the tail boundary flow is the periodic flow $\mathbb {R} \curvearrowright \mathbb {R}/a\mathbb {Z}$ . Similarly, taking $a> 0$ irrational and $\mu _n = \mathcal {E}(\delta _1 + \delta _a)$ for all $n \in \mathbb {N}$ , the tail boundary flow is trivial.

So, it suffices to consider an almost periodic, aperiodic, ergodic flow. Such a flow is given by the translation action of $\mathbb {R}$ on a compact second countable abelian group L under a dense embedding $\pi : \mathbb {R} \to L$ . We have to realize this flow as the tail boundary flow of a sequence of compound Poisson distributions $\mathcal {E}(\mu _n)$ with $\mu _n$ supported on $\mathbb {R}_{\geq 0}$ . We start by reducing this problem to a similar question for $\mathbb {Z}$ -actions.

Fix a nontrivial character $\omega _0 \in \widehat {L}$ . Take $t_0 \in \mathbb {R} \setminus \{0\}$ such that $\omega _0(\pi (t)) = \exp (2\pi i t / t_0)$ for all $t \in \mathbb {R}$ . So, $\omega _0$ gives rise to a surjective, continuous group homomorphism $\theta : L \to \mathbb {R}/t_0 \mathbb {Z}$ such that $\theta \circ \pi : \mathbb {R} \to \mathbb {R} / t_0 \mathbb {Z}$ is the natural quotient map. Define $K = \operatorname {Ker} \theta $ as the kernel of $\theta $ . Note that the restriction of $\pi $ to $t_0 \mathbb {Z}$ is a dense embedding of $t_0 \mathbb {Z}$ into K. Note that the translation action $\mathbb {R} \curvearrowright L$ is the induction to $\mathbb {R}$ of the translation action $t_0 \mathbb {Z} \curvearrowright K$ . By the discussion preceding this proof, it thus suffices to prove that this translation action $t_0 \mathbb {Z} \curvearrowright K$ can be realized as the tail boundary action of $t_0 \mathbb {Z}$ associated with a sequence of compound Poisson distributions $\mathcal {E}(\mu _n)$ , where $\mu _n$ are finite positive measures on $t_0 \mathbb {N} \subset t_0 \mathbb {Z}$ . Since we can rescale everything with $t_0$ , we may assume that $t_0 = 1$ .

Combining [Reference Connes and Woods16, Corollary 2.8] and [Reference Connes and Woods17, Theorem 3.4], we can take finitely supported probability measures $(\eta _n)_{n \in \mathbb {N}}$ on $\mathbb {Z}$ such that the associated tail boundary flow is given by the translation $\mathbb {Z} \curvearrowright K$ . Denote by $\widetilde {\eta }_n$ the probability measure given by $\widetilde {\eta }_n(\mathcal {U}) = \eta _n(-\mathcal {U})$ . The tail boundary flow of $(\widetilde {\eta }_n)_{n \in \mathbb {N}}$ is given by the action of $\mathbb {Z}$ on K by $n \cdot k = -n + k$ . Through the isomorphism $k \mapsto -k$ , this flow is isomorphic with the original translation action $\mathbb {Z} \curvearrowright K$ . Since the joint flow of $\mathbb {Z} \curvearrowright K$ with itself is again $\mathbb {Z} \curvearrowright K$ , the tail boundary flow of the measures $(\eta _n * \widetilde {\eta }_n)_{n \in \mathbb {N}}$ is still given by $\mathbb {Z} \curvearrowright K$ .

For every $\mu \in \ell ^1(\mathbb {Z})$ , we consider the Fourier transform

$$ \begin{align*}\widehat{\mu} : \mathbb{T} \to \mathbb{C} : \widehat{\mu}(z) = \sum_{k \in \mathbb{Z}} \mu(k) z^k \; .\end{align*} $$

We view $\widehat {K}$ as a dense countable subgroup of $\mathbb {T}$ . The Fourier transform of $\eta _n * \widetilde {\eta }_n$ is a positive function. We replace $\eta _n$ by $\eta _n * \widetilde {\eta }_n$ , and we may thus assume that the probability measures are finitely supported, with $\widehat {\eta }_n(\omega ) \geq 0$ for all $n \in \mathbb {N}$ and $\omega \in \widehat {K}$ , and with associated tail boundary flow $\mathbb {Z} \curvearrowright K$ .

As each $\omega \in \widehat {K}$ is an eigenvalue of the translation action $\mathbb {Z}\curvearrowright K$ , we know from [Reference Connes and Woods17, Theorem 4.2] that

$$ \begin{align*} \lim_{n \to +\infty} \prod_{m=n}^{\infty}\widehat{\eta}_m(\omega) = 1 \;\;\text{for every } \omega\in \widehat{K}. \end{align*} $$

Choose an increasing sequence of finite subsets $\mathcal {F}_k \subset \widehat {K}$ such that $\widehat {K} = \bigcup _{k=1}^{\infty } \mathcal {F}_k$ . Then inductively choose $1 \leq n_1 < n_2 < \cdots $ such that

$$ \begin{align*} \prod_{m=1+n_k}^{\infty} \widehat{\eta}_m(\omega)> 1 - (k+1)^{-3} \quad\text{for all } \omega \in \mathcal{F}_{k+1}. \end{align*} $$

Then define the probability measures $\alpha _k$ on $\mathbb {Z}$ by

(3.25) $$ \begin{align} \alpha_1=\eta_1 * \eta_2 * \cdots * \eta_{n_1} \;\;\text{and}\;\; \alpha_k=\eta_{1+n_{k-1}} * \cdots * \eta_{n_k} \;\;\text{for } k \geq 2. \end{align} $$

Since $0 \leq \widehat {\eta }_m(\omega ) \leq 1$ for all $m \in \mathbb {N}$ , $\omega \in \widehat {K}$ , we thus have

(3.26) $$ \begin{align} \widehat{\alpha}_k(\omega) \geq 1 - k^{-3} \quad\text{for every } \omega \in \mathcal{F}_k. \end{align} $$

Also, the tail boundary flow of $(\alpha _k)_{k \in \mathbb {N}}$ is still given by $\mathbb {Z} \curvearrowright K$ .

For every $k \in \mathbb {N}$ ,

$$ \begin{align*} \mathcal{U}_k = \{g \in K \mid \, |\omega(g) - 1| < k^{-3} \;\;\text{for all } \omega \in \mathcal{F}_k\;\} \end{align*} $$

is a neighborhood of $0 \in K$ . Since $\mathbb {Z} \subset K$ is dense, there are arbitrarily large positive integers $n \in \mathcal {U}_k$ . Since the measures $\alpha _k$ have finite support, we can choose $m_k \in \mathbb {N}$ large enough such that $m_k \in \mathcal {U}_k$ and such that the translated measures defined by $\gamma _k(\mathcal {V}) = \alpha _k(\mathcal {V}-m_k)$ have their support in $\mathbb {N}$ . Since $m_k \in \mathcal {U}_k$ , using equation (3.26), we get that

(3.27) $$ \begin{align} |1 - \widehat{\gamma}_k(\omega)| \leq 2k^{-3} \quad\text{for every } \omega \in \mathcal{F}_k. \end{align} $$

The tail boundary flow of $(\gamma _k)_{k \in \mathbb {N}}$ is still given by $\mathbb {Z} \curvearrowright K$ .

Define the compound Poisson distributions $\beta _k = \mathcal {E}(k \gamma _k)$ , and denote by $\mathbb {Z} \curvearrowright X$ the tail boundary flow of $(\beta _k)_{k \in \mathbb {N}}$ . We prove that $\mathbb {Z} \curvearrowright X$ is isomorphic with $\mathbb {Z} \curvearrowright K$ .

Since $\widehat {\beta }_k(\omega ) = \exp (-k(1-\widehat {\gamma }_k(\omega )))$ , we get that $|\widehat {\beta }_k(\omega )| = \exp (-k(1-\operatorname {Re} \widehat {\gamma }_k(\omega )))$ . From equation (3.27) and the assumption that $\widehat {K} = \bigcup _{k=1}^{\infty } \mathcal {F}_k$ , it follows that

$$ \begin{align*}\prod_{k=1}^{\infty} |\widehat{\beta}_k(\omega)|> 0 \quad\text{for all } \omega \in \widehat{K}. \end{align*} $$

Again by [Reference Connes and Woods17, Theorem 4.2], it follows that each $\omega \in \widehat {K}$ is an eigenvalue of the tail boundary flow of $(\beta _k)_{k \in \mathbb {N}}$ .

Since $\exp (-k)$ is summable, by Lemma 3.1, the term $\exp (-k)\delta _0$ in the definition of $\beta _k = \mathcal {E}(k \gamma _k)$ is negligible so that $\mathbb {Z} \curvearrowright X$ is isomorphic with the tail boundary flow of

$$ \begin{align*}\rho_k = \gamma_k * \zeta_k \quad\text{where}\quad \zeta_k = (\exp(k)-1)^{-1} \left(k \delta_0 + \sum_{j=1}^{\infty} \frac{k^{j+1}}{(j+1)!} \gamma_k^{*j}\right). \end{align*} $$

Denoting by $\mathbb {Z} \curvearrowright Y$ the tail boundary flow of the sequence $(\zeta _k)_{k \in \mathbb {N}}$ , we conclude that $\mathbb {Z} \curvearrowright X$ is the joint flow of $\mathbb {Z} \curvearrowright K$ and $\mathbb {Z} \curvearrowright Y$ . Realizing this joint flow inside $L^{\infty }(K \times Y)$ with $\mathbb {Z}$ acting in the first variable, we conclude that the action $\mathbb {Z} \curvearrowright X$ can be continuously extended to an action of K. This means that $\mathbb {Z} \curvearrowright X$ is isomorphic with a factor of $\mathbb {Z} \curvearrowright K$ . We have already seen that each $\omega \in \widehat {K}$ is an eigenvalue of $\mathbb {Z} \curvearrowright X$ . So, $\mathbb {Z} \curvearrowright X$ must be isomorphic with $\mathbb {Z} \curvearrowright K$ .

Proof of Theorem 3.9

The trivial flow and every periodic flow arise as the flow of weights of an ITPFI $_2$ factor and also arise as a Poisson flow of positive type. We thus only consider the aperiodic case.

First assume that $\mathbb {R} \curvearrowright X$ is the flow of weights of an ITPFI $_2$ factor and that $\mathbb {R} \curvearrowright X$ is aperiodic. As we have seen in the proof of Theorem 3.5, we find for all $k \in \mathbb {N}$ , elements $b_k \in [k-1,k)$ and integers $L_k \geq 0$ such that $\mathbb {R} \curvearrowright X$ is isomorphic with the tail boundary flow of the sequence $\gamma (b_k)^{*L_k}$ , where $\gamma (b)$ is defined by equation (3.18). By equation (3.28), we get that

$$ \begin{align*}d_{\operatorname{TV}}\bigl(\gamma(b_k)^{*L_k} , \mathcal{E}(\lambda_k \delta_{b_k})\bigr) \leq \exp(-b_k) \quad\text{where}\;\; \lambda_k = \frac{L_k \exp(-b_k)}{1+\exp(-b_k)} \; .\end{align*} $$

Since $\exp (-b_k) \leq \exp (-k+1)$ is summable, it follows from Lemma 3.1 that $\mathbb {R} \curvearrowright X$ is a Poisson flow of positive type.

Conversely, assume that $\mathbb {R} \curvearrowright X$ is the Poisson flow of positive type defined by the sequence of probability measures $\mu _n = \mathcal {E}(\eta _n)$ , where each $\eta _n$ is a finite positive measure supported on $(0,+\infty )$ . We may assume that $\mathbb {R} \curvearrowright X$ is aperiodic. Taking integers $L_n \geq \eta _n(\mathbb {R})$ , we can replace $\mu _n$ by $\mathcal {E}(\eta _n / L_n)$ , each repeated $L_n$ times. We may thus assume that $\lambda _n := \eta _n(\mathbb {R}) \leq 1$ .

For every fixed $C> 0$ , we have that

$$ \begin{align*}\beta_n := \exp(-\lambda_n) (\delta_0 + \eta_n|_{(0,C]}) \leq \mu_n\end{align*} $$

for all $n \in \mathbb {N}$ and that all $\beta _n$ are supported on $[0,C]$ . It follows from Proposition 3.3 that

(3.29) $$ \begin{align} \sum_{n=1}^{\infty} \beta_n(\mathbb{R}) \, \operatorname{Var}(\beta_n(\mathbb{R})^{-1} \, \beta_n) < +\infty \;. \end{align} $$

Since $\exp (-\lambda _n) \geq \exp (-1)$ and $\eta _n((0,C]) \leq 1$ for all $n \in \mathbb {N}$ , it follows from equations (3.29) and (3.13) that for every $C> 0$ ,

(3.30) $$ \begin{align} \sum_{n=1}^{\infty} \int_{(0,C]} x^2 \, d\eta_n(x) < +\infty \;. \end{align} $$

Denote $\rho _n = \eta _n|_{(0,1]}$ and, for all $k \in \mathbb {N}$ , $\eta _{n,k} = \eta _n|_{(k,k+1]}$ . Since $\mathcal {E}(\eta _n)$ is the convolution of the measures $\mathcal {E}(\rho _n)$ and $\mathcal {E}(\eta _{n,k})$ , $n,k \in \mathbb {N}$ , we conclude that $\mathbb {R} \curvearrowright X$ also is the tail boundary flow of the union of the probability measures $\mathcal {E}(\rho _n)$ , $n \in \mathbb {N}$ , and $\mathcal {E}(\eta _{n,k})$ , $n,k \in \mathbb {N}$ .

By equations (3.30) and (3.24), we have $\sum _{n=1}^{\infty } \operatorname {Var}(\mathcal {E}(\rho _n)) < +\infty $ . By, e.g., [Reference Berendschot and Vaes8, Proposition 2.1], the tail boundary flow of $(\mathcal {E}(\rho _n))_{n \in \mathbb {N}}$ is given by the translation action $\mathbb {R} \curvearrowright \mathbb {R}$ , so that $\mathbb {R} \curvearrowright X$ is the tail boundary flow of the measures $\mathcal {E}(\eta _{n,k})$ , $n,k \in \mathbb {N}$ .

For every fixed $k \in \mathbb {N}$ , it follows from equation (3.30) that $\sum _{n =1}^{\infty } \eta _{n,k}(\mathbb {R}) < +\infty $ . We define $\zeta _k = \sum _{n=1}^{\infty } \eta _{n,k}$ and conclude that $\mathbb {R} \curvearrowright X$ is the tail boundary flow of $\mathcal {E}(\zeta _k)$ , $k \in I$ , where $\zeta _k$ is a finite positive measure supported on $(k,k+1]$ .

Write $\ell _k = \zeta _k(\mathbb {R})$ , and fix integers $L_k \geq \ell _k$ . Since $\mathbb {R} \curvearrowright X$ also is the tail boundary flow of the measures $\mathcal {E}(\zeta _k/L_k)$ , each repeated $L_k$ times, and since $\exp (-1) L_k^{-1} \zeta _k \leq \mathcal {E}(\zeta _k/L_k)$ , it follows from Proposition 3.3 that the points

$$ \begin{align*}b_k = \ell_k^{-1} \int_{\mathbb{R}} x \, d\zeta_k(x) \in (k,k+1] \quad\text{satisfy}\quad \sum_{k=1}^{\infty} \int_{\mathbb{R}} (x-b_k)^2 \, d\zeta_k(x) < +\infty \; .\end{align*} $$

Denote by $W_2$ the Wasserstein $2$ -distance. By equation (3.6), we have that

$$ \begin{align*}W_2(\mathcal{E}(\zeta_k), \mathcal{E}(\ell_k \delta_{b_k}))^2 \leq \exp(-\ell_k) \sum_{j=1}^{\infty} \frac{1}{j!} \int_{\mathbb{R}} (x - j b_k)^2 \, d\zeta_k^{* j}(x) \; .\end{align*} $$

For every $j \geq 1$ , we have that

$$ \begin{align*} \int_{\mathbb{R}} (x - j b_k)^2 \, d\zeta_k^{* j}(x) &= \int_{\mathbb{R}^j} \bigl( (x_1 - b_k) + \cdots + (x_j - b_k) \bigr)^2 \, d\zeta_k(x_1) \, \cdots \, d\zeta_k(x_j) \\ &= j \, \ell_k^{j-1} \, \int_{\mathbb{R}} (x-b_k)^2 \, d\zeta_k(x) \;. \end{align*} $$

We conclude that

$$ \begin{align*}W_2(\mathcal{E}(\zeta_k), \mathcal{E}(\ell_k \delta_{b_k}))^2 \leq \int_{\mathbb{R}} (x-b_k)^2 \, d\zeta_k(x) \quad\text{so that}\quad \sum_{k=1}^{\infty} W_2(\mathcal{E}(\zeta_k), \mathcal{E}(\ell_k \delta_{b_k}))^2 < + \infty \; .\end{align*} $$

By Proposition 3.2, $\mathbb {R} \curvearrowright X$ is also the tail boundary flow of the sequence $\mathcal {E}(\ell _k \delta _{b_k})$ , $k \in \mathbb {N}$ . Write $\ell ^{\prime }_k = \exp (-b_k)(1+\exp (-b_k))^{-1}$ . Take integers $M_k \geq 1$ such that $|M_k \ell ^{\prime }_k - \ell _k| \leq \ell ^{\prime }_k$ . For all $c,d> 0$ and $k \in \mathbb {N}$ , one has

$$ \begin{align*}|c^k - d^k| = |c-d| \, \bigl|\sum_{i=0}^{k-1} c^i d^{k-1-i}\bigr| \leq k \, |c-d| \, (\max\{c,d\})^{k-1} \; \end{align*} $$

so that $d_{\operatorname {TV}}(\mathcal {E}(c \delta _b) , \mathcal {E}(d \delta _b)) \leq |c-d|$ for all $c, d> 0$ . Therefore,

$$ \begin{align*}d_{\operatorname{TV}}\bigl(\mathcal{E}(M_k \ell^{\prime}_k \, \delta_{b_k}) , \mathcal{E}(\ell_k \, \delta_{b_k})\bigr) \leq \exp(-b_k) \quad\text{for all } k \in \mathbb{N}.\end{align*} $$

By equation (3.28), we have

$$ \begin{align*}d_{\operatorname{TV}}\bigl(\mathcal{E}(M_k \ell^{\prime}_k \, \delta_{b_k}) , \gamma(b_k)^{* M_k}\bigr) \leq \ell^{\prime}_k \leq \exp(-b_k) \; .\end{align*} $$

Since $b_k> k$ , the sequence $\exp (-b_k)$ is summable. It then follows from Lemma 3.1 that $\mathbb {R} \curvearrowright X$ is the tail boundary flow of the measures $(\gamma (b_k)^{* M_k})_{k \in \mathbb {N}}$ and, hence, is isomorphic with the flow of weights of an ITPFI $_2$ factor.

Proof. First, assume that $\mathbb {R} \curvearrowright Z$ is a Poisson flow. In most of the proof of Theorem 3.9, we did not use that the measures are supported on the positive real line. Writing $I = \mathbb {Z} \setminus \{-1,0\}$ , we find for every $k \in I$ , elements $b_k \in (k,k+1]$ and constants $\lambda _k> 0$ such that $\mathbb {R} \curvearrowright Z$ is isomorphic with the tail boundary flow of the family $(\mathcal {E}(\lambda _k \delta _{b_k}))_{k \in I}$ . Choose for every $k \in I$ an integer $M_k \geq 1$ such that $M_k^{-1} \lambda _k \leq 2^{-|k|}$ . Define for $k \in I$ , the probability measure

$$ \begin{align*}\eta_k = (1- M_k^{-1} \lambda_k)\delta_0 + M_k^{-1} \lambda_k \delta_{b_k} \; .\end{align*} $$

By equation (3.28), we have $d_{\operatorname {TV}}\bigl (\mathcal {E}(\lambda _k \delta _{b_k}) , \eta _k^{*M_k}\bigr ) \leq M_k^{-1} \lambda _k \leq 2^{-|k|}$ , which is summable. By Lemma 3.1, $\mathbb {R} \curvearrowright Z$ is isomorphic with the tail boundary flow of the probability measures $\eta _k$ repeated $M_k$ times, $k \in I$ . Since

$$ \begin{align*}\operatorname{Var} \eta_k \leq M_k^{-1} \lambda_k b_k^2 \leq 2^{-|k|} (|k|+1)^2 \to 0 ,\end{align*} $$

one implication of the proposition is proven.

Conversely, assume that $(\mu _n)_{n \in \mathbb {N}}$ is a sequence of probability measures whose support consists of two points and that satisfy $\operatorname {Var} \mu _n \leq C$ for all $n \in \mathbb {N}$ . Denote by $\mathbb {R} \curvearrowright Z$ their tail boundary flow. We have to prove that $\mathbb {R} \curvearrowright Z$ is a Poisson flow. We may thus assume that $\mathbb {R} \curvearrowright Z$ is aperiodic. Since translating the measures $\mu _n$ does not change the tail boundary flow, we may assume that $\mu _n(0) \geq 1/2$ for all $n \in \mathbb {N}$ , and we denote by $d_n \in \mathbb {R}$ the other atom of $\mu _n$ . Write $p_n = \mu _n(d_n)$ Since $\operatorname {Var} \mu _n \leq C$ , we get that $p_n d_n^2 \leq 2 C$ for all $n \in \mathbb {N}$ .

For every $k \in \mathbb {Z}$ , write $J_k = \{n \in \mathbb {N} \mid d_n \in [k,k+1)\}$ . Write $I = \mathbb {Z} \setminus \{-1,0,1\}$ . Arguing as in the proof of Theorem 3.9, it follows from Proposition 3.3 that

(3.31) $$ \begin{align} \sum_{n \in J_{-1} \cup J_0 \cup J_1} \operatorname{Var}(\mu_n) < + \infty , \end{align} $$

and it follows from Proposition 3.4 that we find for every $k \in I$ an element $b_k \in [k,k+1)$ such that

(3.32) $$ \begin{align} \sum_{k \in I} \sum_{n \in J_k} p_n (b_k - d_n)^2 < + \infty \;. \end{align} $$

By equation (3.31), the tail boundary flow of the measures $(\mu _n)_{n \in J_{-1} \cup J_0 \cup J_1}$ is given by the translation action $\mathbb {R} \curvearrowright \mathbb {R}$ so that these measures may be ignored. Defining for every $k \in I$ and $n \in J_k$ the probability measure $\eta _n = (1-p_n) \delta _0 + p_n \delta _{b_k}$ , note that equation (3.32) is saying that

$$ \begin{align*} \sum_{k \in I} \sum_{n \in J_k} W_2(\mu_n,\eta_n)^2 <+\infty \;. \end{align*} $$

It thus follows from Proposition 3.2 that $\mathbb {R} \curvearrowright Z$ is the tail boundary flow of the measures $\eta _n$ , $k \in I$ , $n \in J_k$ . Write $\lambda _k = \sum _{n \in J_k} p_n$ . By [Reference Barbour and Hall4, Theorem 1], we have for every $k \in I$ that

$$ \begin{align*} d_{\operatorname{TV}}\Bigl( \operatorname*{{\Large {\ast}}}_{n \in J_k} \eta_n \, , \, \mathcal{E}(\lambda_k \delta_{b_k})\Bigr) \leq \lambda_k^{-1} \sum_{n \in J_k} p_n^2 \leq \max_{n \in J_k} p_n \leq \max_{n \in J_k} 2 C d_n^{-2} \leq \frac{2C}{(|k|-1)^2} , \end{align*} $$

where we used that $d_n \in [k,k+1)$ if $n \in J_k$ . Since the right-hand side is summable, it follows from Lemma 3.1 that $\mathbb {R} \curvearrowright Z$ is also the tail boundary flow of the Poisson distributions $(\mathcal {E}(\lambda _k \delta _{b_k}))_{k \in I}$ and thus is a Poisson flow.

The positive case is proven entirely analogously.

Proof. If $\mathbb {R} \curvearrowright Z$ is the tail boundary flow of the sequence $(\mathcal {E}(\mu _n))_{n \in \mathbb {N}}$ , where $\mu _n$ is a sequence of finite positive measures on $\mathbb {R}$ , and if $L \geq 1$ is an integer, we can define $\mathbb {R} \curvearrowright Z_1$ as the tail boundary flow of the sequence $(\mathcal {E}(L^{-1} \mu _n))_{n \in \mathbb {N}}$ . By construction, $\mathbb {R} \curvearrowright Z$ is isomorphic with the joint flow of L copies of $\mathbb {R} \curvearrowright Z_1$ .

Since compound Poisson distributions $\mathcal {E}(\mu )$ are weakly dense in the set of infinitely divisible distributions (see, e.g., [Reference Stroock48, Theorem 3.2.7]), the second statement follows directly from Lemma 3.14 below.

Proof. Let $(\mu _n)_{n \in \mathbb {N}}$ be a sequence in $\overline {\mathcal {F}}$ . Denote by $\nu _n$ the uniform probability measure on the interval $[-1/n,1/n]$ so that $\operatorname {Var} \nu _n = n^{-2}/2$ is summable. By, e.g., [Reference Berendschot and Vaes8, Proposition 2.1], the tail boundary flow of the sequence $(\nu _n)_{n \in \mathbb {N}}$ is the translation action $\mathbb {R} \curvearrowright \mathbb {R}$ . Therefore, $(\mu _n)_{n \in \mathbb {N}}$ and $(\mu _n * \nu _n)_{n \in \mathbb {N}}$ give rise to isomorphic tail boundary flows.

Since $\mathcal {F}$ is weakly dense in $\overline {\mathcal {F}}$ and since $\nu _n$ is absolutely continuous, we can choose $\mu ^{\prime }_n \in \mathcal {F}$ such that $d_{\operatorname {TV}}\bigl (\mu ^{\prime }_n * \nu _n , \mu _n * \nu _n\bigr ) \leq n^{-2}$ . By Lemma 3.1, also $(\mu _n * \nu _n)_{n \in \mathbb {N}}$ and $(\mu ^{\prime }_n * \nu _n)_{n \in \mathbb {N}}$ give rise to isomorphic tail boundary flows, with the latter being isomorphic to the tail boundary flow of $(\mu ^{\prime }_n)_{n \in \mathbb {N}}$ .

Proof. The convolution products of measures of the form $\sigma _{\lambda ,a}$ are precisely the compound Poisson distributions $\mathcal {E}(\mu )$ where $\mu $ is a finite positive measure with finite support. These $\mathcal {E}(\mu )$ are weakly dense in the set of all compound Poisson distributions. The conclusion thus immediately follows from Lemma 3.14.