Let $ (M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $ (\Omega,\mathcal{F},\mathbb{P})$ . Define the probability measure $ \mu$ on $ \mathbb{R}^\mathbb{N}$ to be the law of $ (M_n)_{n \geq 1}$ , and for each $ n$ define the probability measure $ \mu_n$ on $ \mathbb{R}^n$ to be the law of $ (M_1,\ldots,M_n)$ .

For each $ n \in \mathbb{N}$ , let $ (\mu_\mathbf{x})_{\mathbf{x} \in \mathbb{R}^n}$ be an $ \mathbb{R}^n$ -indexed family of probability measures $ \mu_\mathbf{x}$ on $ \mathbb{R}^\mathbb{N}$ such that for all $ A \in \mathcal{B}(\mathbb{R})^{\otimes \mathbb{N}}$ , the map $ \mathbf{x} \mapsto \mu_\mathbf{x}(A)$ is measurable and $ $ \mu(A) \ = \ \int_{\mathbb{R}^n} \int_{\mathbb{R}^\mathbb{N}} \mathbf{1}_A(x_1,x_2,\ldots) \, \mu_{(x_1,\ldots,x_n)}(d(x_{n+1},x_{n+2},\ldots)) \, \mu_n(d(x_1,\ldots,x_n)). $ $ It is not hard to show that for $ \mu_n$ -almost all $ \mathbf{x}$ , the sequence $ (\pi_i)_{i \geq 1}$ of projections $ \pi_i \colon (x_r)_{r \geq 1} \mapsto x_i$ is a martingale over $ \mu_\mathbf{x}$ . For each $ \mathbf{x} \in \mathbb{R}^n$ , let $ [a_\mathbf{x}^M,b_\mathbf{x}^M]$ be the smallest closed interval that is assigned full measure by $ \pi_{i\ast}\mu_\mathbf{x}$ for every $ i \geq 1$ .

Definition: We say that $ (M_n)_{n \geq 1}$ has the extreme convergence property if for $ \mathbb{P}$ -almost all $ \omega \in \Omega$ , either $ $ M_{n+1}(\omega) – a_{(M_1(\omega),\ldots,M_n(\omega))}^M \to 0 \ \textrm{ as } \ n \to \infty $ $ or $ $ b_{(M_1(\omega),\ldots,M_n(\omega))}^M – M_{n+1}(\omega) \to 0 \ \textrm{ as } \ n \to \infty. $ $ Definition: We say that $ (M_n)_{n \geq 1}$ has the extreme-convergence decomposability property if there exists a probability space $ (Y,\mathcal{Y},\xi)$ , a filtration $ (\mathcal{F}_n)_{n \geq 1}$ of sub-$ \sigma$ -algebras of $ \mathcal{F}$ , and a uniformly bounded sequence $ (M_n’)_{n \geq 1}$ of functions $ M_n’ \colon Y \times \Omega \to \mathbb{R}$ , with $ M_n’$ being $ (\mathcal{Y} \otimes \mathcal{F}_n)$ -measurable, such that

• for each $ y \in Y$ , $ (M_n'(y,\cdot))_{n \geq 1}$ is a martingale with respect to the filtration $ (\mathcal{F}_n)_{n \geq 1}$ , and has the extreme convergence property;
• for each $ n$ , for $ \mathbb{P}$ -almost all $ \omega$ , $ $ M_n(\omega) \ = \ \int_Y M_n'(y,\omega) \, \xi(dy). $ $

My vauge intuition is that most uniformly bounded martingales arising in practice will have the extreme-convergence decomposability property.

Have there been any studies on the extreme convergence property, as defined above, or on similar concepts? In particular, are there any reasonably verifiable conditions guaranteeing that a uniformly bounded martingale has the extreme convergence property or extreme-convergence decomposability property (or similar properties)?

Motivation: Suppose we have a random homeomorphism $ (f_\alpha)_{\alpha \in I}$ of a compact metric space $ X$ , defined over a probability space $ (I,\mathcal{I},\nu)$ , and a probability measure $ \rho$ on $ X$ such that $ $ \rho(A) \ = \ \int_I \rho(f_\alpha(A)) \, \nu(d\alpha) $ $ for all $ A \in \mathcal{B}(X)$ . Then over $ \nu^{\otimes \mathbb{N}}$ , the stochastic process $ $ \rho(f_{\alpha_n} \circ \ldots \circ f_{\alpha_1}(A)) $ $ is a martingale (for any $ A$ ), and therefore converges $ \nu^{\otimes \mathbb{N}}$ -almost surely as $ n \to \infty$ . In the case that $ X$ is a circle, we very often have that the limit of this martingale is supported on $ \{0,1\}$ (at least if $ A$ is connected, and I think for general $ A$ ), so that the stochastic flow is almost surely contractive outside a random repelling singleton; and even when the stochastic flow does not have this behaviour, I believe that typically the above martingale can be expressed as an equal-weight convex combination of martingales $ \rho(f_{\alpha_n} \circ \ldots \circ f_{\alpha_1}(A_i))$ each with limit supported on $ \{0,1\}$ , where the sets $ A_i$ partition $ A$ . (I am basing these claims on known results for random circle homeomorphisms, see e.g. here and references therein.) I am interested in trying to extend some aspects of these results to more general $ X$ . (In particular, my ultimate ideal goal is to show that “generically”, in some sense, a minimal random homeomorphism that is locally contractive under i.i.d. iterates is almost-globally contractive under i.i.d. iterates.)