A New Characterization of Endogeny

Abstract

Aldous and Bandyopadhyay have shown that each solution to a recursive distributional equation (RDE) gives rise to a recursive tree process (RTP), which is a sort of Markov chain in which time has a tree-like structure and in which the state of each vertex is a random function of its descendants. If the state at the root is measurable with respect to the sigma field generated by the random functions attached to all vertices, then the RTP is said to be endogenous. For RTPs defined by continuous maps, Aldous and Bandyopadhyay showed that endogeny is equivalent to bivariate uniqueness, and they asked if the continuity hypothesis can be removed. We introduce a higher-level RDE that through its n-th moment measures contains all n-variate RDEs. We show that this higher-level RDE has minimal and maximal fixed points with respect to the convex order, and that these coincide if and only if the corresponding RTP is endogenous. As a side result, this allows us to answer the question of Aldous and Bandyopadhyay positively.

Appendices

Appendix A: The Convex Order

By definition, a G_δ-set is a set that is a countable intersection of open sets. By [4, §6 No. 1, Theorem. 1], for a metrizable space S, the following statements are equivalent.

• (i) S is Polish.

• (ii) There exists a metrizable compactification $\overline {S}$ of S s.t. S is a G_δ-subset of $\overline {S}$.

• (iii) For each metrizable compactification $\overline {S}$ of S, S is a G_δ-subset of $\overline {S}$.

Moreover, a subset S′ ⊂ S of a Polish space S is Polish in the induced topology if and only if S′ is a G_δ-subset of S.

Let S be a Polish space. Recall that $\mathcal {P}(S)$ denotes the space of probability measures on S, equipped with the topology of weak convergence. In what follows, we fix a metrizable compactification $\overline {S}$ of S. Then we can identify the space $\mathcal {P}(S)$ (including its topology) with the space of probability measures μ on $\overline {S}$ such that μ(S) = 1. By Prohorov’s theorem, $\mathcal {P}(\overline {S})$ is compact, so $\mathcal {P}(\overline {S})$ is a metrizable compactification of $\mathcal {P}(S)$. Recall the definition of $\mathcal {P}(\mathcal {P}(S))_{\mu }$ from (4.5).

Lemma 9 (Measures with given mean)

For any $\mu \in \mathcal {P}(S)$, the space $\mathcal {P}(\mathcal {P}(S))_{\mu }$ is compact.

Proof

Since any $\rho \in \mathcal {P}(\mathcal {P}(\overline {S}))$ whose first moment measure is μ must be concentrated on $\mathcal {P}(S)$, we can identify $\mathcal {P}(\mathcal {P}(S))_{\mu }$ with the space of probability measures on $\mathcal {P}(\overline {S})$ whose first moment measure is μ. From this we see that $\mathcal {P}(\mathcal {P}(S))_{\mu }$ is a closed subset of $\mathcal {P}(\mathcal {P}(\overline {S}))$ and hence compact. □

We let $\mathcal {C}(\overline {S})$ denote the space of all continuous real functions on $\overline {S}$, equipped with the supremumnorm, and we let $B(\overline {S})$ denote the space of bounded measurable real functions on $\overline {S}$. The following fact is well-known (see, e.g., [5, Corollary 12.11]).

Lemma 10 (Space of continuous functions)

$\mathcal {C}(\overline {S})$ is a separable Banach space.

For each $f\in \mathcal {C}(\overline {S})$, we define an affine function $l_{f}\in \mathcal {C}(\mathcal {P}(\overline {S}))$ by $l_{f}(\mu ):=\int f \mathrm {d}\mu$. The following lemma says that all continuous affine functions on $\mathcal {P}(\overline {S})$ are of this form.

Lemma 11 (Continuous affine functions)

A function $\phi \in \mathcal {C}(\mathcal {P}(\overline {S}))$ is affine if and only if ϕ = l_f for some $f\in \mathcal {C}(\overline {S})$.

Proof

Let $\phi :\mathcal {P}(\overline {S})\to \mathbb {R}$ be affine and continuous. Since ϕ is continuous, setting $f(x):=\phi (\delta _{x}) (x\in \overline {S})$ defines a continuous function $f:\overline {S}\to \mathbb {R}$. Since ϕ is affine, ϕ(μ) = l_f(μ) whenever μ is a finite convex combination of delta measures. Since such measures are dense in $\mathcal {P}(\overline {S})$ and ϕ is continuous, we conclude that ϕ = l_f. □

Lemma 12 (Lower semi-continuous convex functions)

Let $C\subset \mathcal {C}(\overline {S})$ be convex, closed, and nonempty. Then

$$ \phi:=\sup_{f\in C} l_f $$

(A.1)

defines a lower semi-continuous convex function $\phi :\mathcal {P}(\overline {S})\to (-\infty ,\infty ]$. Conversely, each such ϕ is of the form (A.1).

Proof

It is straightforward to check that (A.1) defines a lower semi-continuous convex function $\phi :\mathcal {P}(\overline {S})\to (-\infty ,\infty ]$. To prove that every such function is of the form (A.1), let $\mathcal {C}(\overline {S})^{\prime }$ denote the dual of the Banach space $\mathcal {C}(\overline {S})$, i.e., $\mathcal {C}(\overline {S})^{\prime }$ is the space of all continuous linear forms $l\!:\!\mathcal {C}(\overline {S})\!\to \!\mathbb {R}$. We equip $\mathcal {C}(\overline {S})^{\prime }$ with the weak-∗ topology, i.e., the weakest topology that makes the maps l ↦ l(f) continuous for all $f\in \mathcal {C}(\overline {S})$. Then $\mathcal {C}(\overline {S})^{\prime }$ is a locally convex topological vector space and by the Riesz-Markov-Kakutani representation theorem, we can view $\mathcal {P}(\overline {S})$ as a convex compact metrizable subset of $\mathcal {C}(\overline {S})^{\prime }$. Now any lower semi-continuous convex function $\phi :\mathcal {P}(\overline {S})\to (-\infty ,\infty ]$ can be extended to $\mathcal {C}(\overline {S})^{\prime }$ by putting ϕ := ∞ on the complement of $\mathcal {P}(\overline {S})$. Applying [6, Theorem I.3] we obtain that ϕ is the supremum of all continuous affine functions that lie below it. By Lemma 11, we can restrict ourselves to continuous affine functions of the form l_f with $f\in \mathcal {C}(\overline {S})$. It is easy to see that $\{f\in \mathcal {C}(\overline {S}):l_{f}\leqslant \phi \}$ is closed and convex, proving that every lower semi-continuous convex function $\phi :\mathcal {P}(\overline {S})\to (-\infty ,\infty ]$ is of the form (A.1). □

We define

$$ \mathcal{C}_{\text{cv}}\left( \mathcal{P}(\overline{S})\right) :=\left\{\phi\in\mathcal{C}(\mathcal{P}(\overline{S})):\phi\text{ is convex}\right\} $$

(A.2)

If two probability measures $\rho _{1},\rho _{2}\in \mathcal {P}(\mathcal {P}(S))$ satisfy the equivalent conditions of the following theorem, then we say that they are ordered in the convex order, and we denote this as ρ₁ ⩽_cvρ₂. The fact that ⩽_μ defines a partial order will be proved in Lemma 15 below. The convex order can be defined more generally for $\rho _{1},\rho _{2}\in \mathcal {P}(C)$ where C is a convex space, but in the present paper we will only need the case $C=\mathcal {P}(\overline {S})$.

Theorem 13 (The convex order for laws of random probability measures)

Let S be a Polish space and let $\overline {S}$ be a metrizable compactification of S. Then, for $\rho _{1},\rho _{2}\in \mathcal {P}(\mathcal {P}(S))$, the following statements are equivalent.

• (i) $\displaystyle \int \phi \mathrm {d}\rho _{1}\leqslant \int \phi \mathrm {d}\rho _{2}$ forall $\phi \in \mathcal {C}_{\mu }\left (\mathcal {P}(\overline {S})\right )$.

• (ii) There exists an S-valued random variable X defined on some probability space $({\Omega },\mathcal {F},\P )$ and sub-Ω-fields $\mathcal {F}_{1}\subset \mathcal {F}_{2}\subset \mathcal {F}$ such that $\displaystyle \rho _{i}=\P \left [\P [X\in \cdot |\mathcal {F}_{i}]\in \cdot \right ] (i = 1,2)$.

Proof

For any probability kernel P on $\mathcal {P}(\overline {S})$, measure $\rho \in \mathcal {P}(\overline {S})$, and function $\phi \in \mathcal {C}(\mathcal {P}(\overline {S}))$, we define $\rho P\in \mathcal {P}(\mathcal {P}(\overline {S}))$ and $P\phi \in B(\mathcal {P}(\overline {S}))$ by

$$ \rho P:=\int\rho(\mathrm{d}\mu)P(\mu, \cdot ) \quad\text{and}\quad P\phi:=\int P(\cdot ,\mathrm{d}\mu)\phi(\mu). $$

(A.3)

By definition, a dilation is a probability kernel P such that Pl_f = l_f for all $f\!\in \!\mathcal {C}(\overline {S})$.

As in the proof of Lemma 12, we can view $\mathcal {P}(\overline {S})$ as a convex compact metrizable subset of the locally convex topological vector space $\mathcal {C}(\overline {S})^{\prime }$. Then [11, Theorem 2] tells us that (i) is equivalent to:

• (iii) There exists a dilation P on $\mathcal {P}(\overline {S})$ such that ρ₂ = ρ₁P.

To see that this implies (ii), let ξ₁,ξ₂ be $\mathcal {P}(\overline {S})$-valued random variables such that ξ₁ has law ρ₁ and the conditional law of ξ₂ given ξ₁ is given by P. Let $\mathcal {F}_{1}$ be the Ω-field generated by ξ₁, let $\mathcal {F}_{2}$ be the Ω-field generated by (ξ₁,ξ₂), and let X be an $\overline {S}$-valued random variable whose conditional law given $\mathcal {F}_{2}$ is given by ξ₂. Then

$$ \P\left[\P[X\in \cdot |\mathcal{F}_2]\in \cdot \right] =\P[\xi_2\in \cdot ]=\rho_1P=\rho_2. $$

(A.4)

Since P is a dilation

$$\begin{array}{@{}rcl@{}} \mathbb{E}[f(X) | \mathcal{F}_1]&=&\mathbb{E}\left[\mathbb{E}[f(X) | \mathcal{F}_2] \left| \mathcal{F}_1\right.\right] =\mathbb{E}\left[l_f(\xi_2) \left| \mathcal{F}_1\right.\right]\\ &=&\int P(\xi_1,\mathrm{d}\mu)l_f(\mu)=l_f(\xi_1) \end{array} $$

(A.5)

for all $f\in \mathcal {C}(\overline {S})$, and hence

$$ \P\left[\P[X\in \cdot |\mathcal{F}_1]\in \cdot \right] =\P\left[\xi_1\in \cdot \right]=\rho_1. $$

(A.6)

We note that since $\rho _{1},\rho _{2}\in \mathcal {P}(\mathcal {P}(S))$, we have $\xi _{1},\xi _{2}\in \mathcal {P}(S)$ a.s. and hence X ∈ S a.s. This proves the implication (iii)⇒(ii).

To complete the proof, it suffices to show that (ii)⇒(i). By Lemma 12, each $\phi \in \mathcal {C}_{\mu }(\mathcal {P}(\overline {S}))$ is of the form ϕ = sup f∈Cl_f for some $C\subset \mathcal {C}(\overline {S})$. Then (ii) implies

$$\begin{array}{@{}rcl@{}} \displaystyle\int\phi \mathrm{d}\rho_1 &=&\mathbb{E}\left[\sup\limits_{f\in C}\mathbb{E}[f(X) | \mathcal{F}_1]\right] =\mathbb{E}\left[\sup\limits_{f\in C}\mathbb{E}\left[\mathbb{E}[f(X) | \mathcal{F}_2] \left| \mathcal{F}_1\right.\right]\right]\\ \displaystyle\quad&\leqslant& \mathbb{E}\left[\mathbb{E}\left[\sup\limits_{f\in C}\mathbb{E}[f(X) | \mathcal{F}_2] \left|\vphantom{\sup\limits_{f\in C}} \mathcal{F}_1\right.\right]\right] =\mathbb{E}\left[\sup\limits_{f\in C}\mathbb{E}[f(X) | \mathcal{F}_2]\right]\\ &=&\int\phi \mathrm{d}\rho_2. \end{array} $$

(A.7)

□

The n-th moment measure ρ⁽ⁿ⁾ associated with a probability law $\rho \in \mathcal {P}(\mathcal {P}(\overline {S}))$ has been defined in (4.1). The following lemma links the first and second moment measures to the convex order.

Lemma 14 (First and second moment measures)

Let S be a Polish space. Assume that $\rho _{1},\rho _{2}\in \mathcal {P}(\mathcal {P}(S))$ satisfy ρ₁ ⩽_μ ρ₂. Then $\rho ^{(1)}_{1}=\rho ^{(1)}_{2}$ and

$$ \int\rho^{(2)}_1(\mathrm{d} x,\mathrm{d} y)f(x)f(y)\leqslant\int\rho^{(2)}_2(\mathrm{d} x,\mathrm{d} y)f(x)f(y) \qquad\left( f\in B(S)\right). $$

(A.8)

If ρ₁ ⩽_μ ρ₂ and (A.8) holds with equality for all bounded continuous $f:S\to \mathbb {R}$, then ρ₁ = ρ₂.

Proof

By Theorem 13, there exists an $\overline {S}$-valued random variable X defined on some probability space $({\Omega },\mathcal {F},\P )$ and sub-Ω-fields $\mathcal {F}_{1}\subset \mathcal {F}_{2}\subset \mathcal {F}$ such that $\displaystyle \rho _{i}=\P \left [\P [X\in \cdot |\mathcal {F}_{i}]\in \cdot \right ] (i = 1,2)$. Since for each f ∈ B(S)

$$\begin{array}{@{}rcl@{}} \int\rho^{(1)}_1(\mathrm{d} x)f(x)&=&\mathbb{E}\left[\mathbb{E}[f(X) |\mathcal{F}_1]\right]=\mathbb{E}[f(X)] =\mathbb{E}\left[\mathbb{E}[f(X) |\mathcal{F}_2]\right]\\ &=&\int\rho^{(1)}_2(\mathrm{d} x)f(x), \end{array} $$

(A.9)

we see that $\rho ^{(1)}_{1}=\rho ^{(1)}_{2}$. Fix f ∈ B(S) and set $M_{i}:=\mathbb {E}[f(X) |\mathcal {F}_{i}] (i = 1,2)$. Then

$$\begin{array}{@{}rcl@{}} \displaystyle\int\rho^{(2)}_2(\mathrm{d} x,\mathrm{d} y)f(x)f(y) &=&\mathbb{E}\left[\mathbb{E}[f(X) |\mathcal{F}_2]^2\right] =\mathbb{E}[M_2^2]\\ &=&\mathbb{E}[M_1^2]+\mathbb{E}\left[(M_2-M_1)^2\right] \geqslant\mathbb{E}[M_1^2]\\ &=&\int\rho^{(2)}_1(\mathrm{d} x,\mathrm{d} y)f(x)f(y), \end{array} $$

(A.10)

proving (A.8). Let $\overline {S}$ be a metrizable compactification of S. If ρ₁ ≤_μρ₂ and (A.8) holds with equality for all bounded continuous $f:S\to \mathbb {R}$, then (A.10) tells us that M₁ = M₂ for each $f\in \mathcal {C}(\overline {S})$, i.e.,

$$ \mathbb{E}[f(X) |\mathcal{F}_1]=\mathbb{E}[f(X) |\mathcal{F}_2]\text{ a.s.\ for each }f\in\mathcal{C}(\overline{S}). $$

(A.11)

By Lemma 10, we can choose a countable dense set $\mathcal {D}\subset \mathcal {C}(\overline {S})$. Then $\mathbb {E}[f(X){\kern .3pt}|\mathcal {F}_{1}] = \mathbb {E}[f(X){\kern .3pt}|\mathcal {F}_{2}]$ for all $f\!\in \!\mathcal {D}$ a.s. and hence $\P [X\!\in \!\cdot |\mathcal {F}_{1}] = \P [X\!\in \cdot |\mathcal {F}_{2}]$ a.s., proving that ρ₁ = ρ₂. □

The following lemma shows that the convex order is a partial order,

Lemma 15 (Convex functions are distribution determining)

If $\rho _{1},\rho _{2}\in \mathcal {P}(\mathcal {P}(\overline {S}))$ satisfy$\int \phi \mathrm {d}\rho _{1}=\int \phi \mathrm {d}\rho _{2}$ for all $\phi \in \mathcal {C}_{\mathrm cv}(\mathcal {P}(\overline {S}))$, then ρ₁ = ρ₂.

Proof

For any $f\in \mathcal {C}(\overline {S})$ and $\rho \in \mathcal {P}(\mathcal {P}(\overline {S}))$,

$$\begin{array}{@{}rcl@{}} &&\displaystyle{\int}_{\overline{S}^2}\rho^{(2)}(\mathrm{d} x,\mathrm{d} y)f(x)f(y)\\ \displaystyle&=&{\int}_{\mathcal{P}(\overline{S})}\rho(\mathrm{d}\mu){\int}_{\overline{S}^2}\mu(\mathrm{d} x)\mu(\mathrm{d} y)f(x)f(y) ={\int}_{\mathcal{P}(\overline{S})}\rho(\mathrm{d}\mu)l_f(\mu)^2. \end{array} $$

(A.12)

Therefore, since ${l_{f}^{2}}$ is a convex function, $\int \phi \mathrm {d}\rho _{1}=\int \phi \mathrm {d}\rho _{2}$ for all $\phi \in \mathcal {C}_{\mu }(\mathcal {P}(\overline {S}))$ implies equality in (A.8) and hence, by Lemma 14, ρ₁ = ρ₂. □

Appendix : B: Open Problem 12 of Aldous and Bandyopadhyay

We have seen that the use of the higer-level map from Section 4 and properties of the convex order lead to an elegant and short proof of Theorem 1, which is similar to [1, Theorem 11]. The most significant improvement over [1, Theorem 11] is that the implication (ii)⇒(i) is shown without a continuity assumption on the map T, solving Open Problem 12 of [1]. If one is only interested in solving this open problem, taking the proof of [1, Theorem 11] for granted, then it is possible to give a shorter argument that does not involve the higer-level map and the convex order.

One way to prove the implication (ii)⇒(i) in Theorem 1 is to show that nonendogeny implies the existence of a measure $\nu \in \mathcal {P}(S^{2})_{\mu }$ such that T⁽²⁾(ν) = ν and ν ≠ μ̅⁽²⁾. In [1], such a ν was constructed as the weak limit of measures ν_n which satisfied T⁽²⁾(ν_n) = ν_n+ 1; however, to conclude that T⁽²⁾(ν) = ν they then needed to assume the continuity of T⁽²⁾. Their Open Problem 12 asks if this continuity assumption can be removed.

In our proof of Theorem 1, we take $\nu =\overline {\mu }^{(2)}$, which by Theorem 5 and Lemma 14 from Appendix A satisfies ν ≠ μ̅⁽²⁾ if and only if the RTP corresponding to μ is not endogenous, and by Lemma 4.4 satisfies T⁽²⁾(ν) = ν.

Antar Bandyopadhyay told us that shortly after the publication of [1], he learned that their Open Problem 12 could be solved by adapting the proof of the implication (3)⇒(2) of [3, Théorème 9] to the setting of RTPs. To the best of our knowledge, this observation has not been published. The setting of [3, Théorème 9] are positive recurrent Markov chains with countable state space, which are a very special case of the RTPs we consider. In view of this, we sketch their argument here in our general setting and show how it relates to our argument.

Let $(\omega _{\mathbf {i}},X_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ be an RTP corresponding to the map γ and a solution μ of a RDE. Construct $(Y_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ such that $(X_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ and $(Y_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ are conditionally independent and identically distributed given $(\omega _{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$. Then $X_{\varnothing }=Y_{\varnothing }$ a.s. if and only if the RTP corresponding to μ is endogenous. Let ν denote the law of $(X_{\varnothing },Y_{\varnothing })$. Then $\nu =\overline {\mu }^{(2)}$ if and only if endogeny holds. In view of this, to prove the implication (ii)⇒(i) in Theorem 1, it suffices to show that ν solves the bivariate RDE T⁽²⁾(ν) = ν. This will follow provided we show that

$$ \left( \omega_{\mathbf{i}},(X_{\mathbf{i}},Y_{\mathbf{i}})\right)_{\mathbf{i}\in\mathbb{T}} $$

(B.1)

is an RTP corresponding to the map γ⁽²⁾ and ν, i.e.,

$$\begin{array}{@{}rcl@{}} {\mathrm{(i)}}&&\text{the~} (\omega_{\mathbf{i}})_{\mathbf{i}\in\mathbb{T}} \text{~are i.i.d.,}\\ {\mathrm{(ii)}}&&\text{for each~} t\geqslant 1, \text{the~} (X_{\mathbf{i}},Y_{\mathbf{i}})_{\mathbf{i}\in{\partial\mathbb{T}_{(t)}}} \text{~are i.i.d. with common law~} \nu \\ &&\text{and independent of~} (\omega_{\mathbf{i}})_{\mathbf{i}\in\mathbb{T}_{(t)}},\\ {\mathrm{(iii)}}&&\displaystyle(X_{\mathbf{i}},Y_{\mathbf{i}}) ={\gamma}^{(2)}[\omega_{\mathbf{i}}]\left( (X_{\mathbf{i}1},Y_{\mathbf{i}1}),\ldots,(X_{\mathbf{i} k_{\mathbf{i}}},Y_{\mathbf{i} \kappa(\omega_{\mathbf{i}})})\right) \qquad(\mathbf{i}\in\mathbb{T}). \end{array} $$

(B.2)

Here (i) and (iii) are trivial. To prove property (ii), set

$$ {\Lambda}_{\mathbf{i}}:=\left( X_{\mathbf{i}},(\omega_{\mathbf{i}\mathbf{j}})_{\mathbf{j} \in \mathbb{T}}\right) \quad\text{and}\quad {\Lambda}^{(2)}_{\mathbf{i}}:=\left( X_{\mathbf{i}},Y_{\mathbf{i}},(\omega_{\mathbf{i}\mathbf{j}})_{\mathbf{j} \in \mathbb{T}}\right) \qquad(\mathbf{i}\in\mathbb{T}). $$

(B.3)

Then the $({\Lambda }_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ are identically distributed. Moreover, for each t ⩾ 1, the $({\Lambda }_{\mathbf {i}})_{\mathbf {i}\in \partial \mathbb {T}_{(t)}}$ are independent of each other and of $(\omega _{\mathbf {k}})_{\mathbf {k}\in \mathbb {T}_{(t)}}$. Recall that $(X_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ and $(Y_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ are conditionally independent and identically distributed given $(\omega _{\mathbf {k}})_{\mathbf {k}\in \mathbb {T}}$. Since the conditional law of X_i given $(\omega _{\mathbf {k}})_{\mathbf {k}\in \mathbb {T}}$ only depends on $(\omega _{\mathbf {i}\mathbf {j}})_{\mathbf {j}\in \mathbb {T}}$, the same is true for Y_i. Using this, it is not hard to see that the $({\Lambda }^{(2)}_{\mathbf {i}})_{\mathbf {i}\in \mathbb {T}}$ are identically distributed and for each t ⩾ 1, the $({\Lambda }^{(2)}_{\mathbf {i}})_{\mathbf {i}\in \partial \mathbb {T}_{(t)}}$ are independent of each other and of $(\omega _{\mathbf {k}})_{\mathbf {k}\in \mathbb {T}_{(t)}}$, and this in turn implies (ii).

In fact, since the law of $(X_{\varnothing },Y_{\varnothing })$ is the second moment measure of the random measure $\xi _{\varnothing }$ from Proposition 4, the measure ν constructed here is the same as our measure μ̲⁽²⁾. Thus, our argument and the one from [3] are both based on the same solution of the bivariate RDE.