How to determine the transition matrix of the markov chain $X_n = f(X_{n-1}, \xi_n), n \geq 0$

Question

I am having trouble to find the transition matrix of the following question:

Let $X_0$ be a random variable taking values in a countable set $I \subset \mathbb{R}$. Let $(\xi)_{n \geq 0}$ be a sequence of independent, identically distributed real-valued random variables which are independent of $X_0$. Suppose that we are given a function $f:I \times \mathbb{R} \rightarrow I$. We inductively define:

$X_n = f(X_{n-1}, \xi_n),$ whereby $n \geq 0$.

We assume that $f$ is 'measurable'. This ensures that every $X_n$ is a random variable.

How do I know what my transition matrix should be of the markov chain $(X_n)_{n\geq0}$? I hope anyone can help me/give tips!

Thanks

Answer 1

$P(X_n=x_n|X_{n-1}=x_{n-1})=P(f(x_{n-1},\xi_n)=x_n)$. The evaluation of this depends on the form of $f$ and the distribution of $\xi_n$. If $\xi_n$ is a discrete random variable you can write this as $\sum_{i\in I} P(\xi_n=i)$ where $I=\{i:f(x_{n-1},i)=x_n\}$.