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 property, or at least 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.)
