I have run into the following question: Let $X_0=1$, and $X_i\sim$Uni$(0,X_{i-1})$ for $i>0$, where Uni$(a,b)$ is the uniform distribution on $(a,b)$, and calculate $\lim\limits_{n\to\infty}(\ln X_n)/n$ (a.s.) - this can be handled by writing $X_n=Y_1Y_2\ldots Y_n$ with $Y_i$ iid Uni$(0,1)$ - so the question is played back to the law of large numbers.
My question would be the following: is there any article or book out there discussing the following situation - a generalisation of the previously mentioned exercise:
Let $\Theta\subseteq\mathbb{R}$ and $\mathcal{X}(\Theta)=\{X(\vartheta)|\forall\vartheta\in\Theta\}$ a collection of random variables, such that $F_{\vartheta_1}(x)=F_{\vartheta_2} (x)$ Lebesgue-as $\Rightarrow \vartheta_1=\vartheta_2$ and $\forall \vartheta\in\Theta:$ Ran$X(\vartheta)\subseteq\Theta$. Eg. $\mathcal{X}(\mathbb{R})=\{\mathcal{N}(\mu,1)|\mu\in\mathbb{R}\}$, the family of normal distributions with variance $1$.
Let $A_0$ be given, and define $A_n\sim X(A_{n-1})$ (so we take a value from the distribution belonging to the parameter given by the previous element of the series.
I would be interested in finding functions $f(n)$ and $g(n)$ such that $$ \lim_{n\to\infty}\frac{f(A_n)}{g(n)}\to h \, \text{a.s. or in }L_2 $$ Where $h$ is a constant - so something like the law of large numbers.