I'm confused about the definition of pseudo-Poisson processes, as given in Kallenberg's Foundations of Modern Probability, Chapter 10, pg. 191. He writes...
A pseudo-Poisson process is a process of the form $X = Y \circ N$ a.s., where $Y$ is a discrete-time Markov process in $S$ and $S$ is an independent, homogenous Poisson process.
Previously, he defined homogenous Poisson processes on the real line $\mathbb{R}_+$ as Poisson process $\xi$ s.t. $E\xi = c\lambda$, presumably for Lebesgue measure $\lambda$, where $c > 0$ some constant, noting that $N_t = \xi[0,t]$ is a time- and space-homogenous Markov process.
However, I am not sure at all how one composes a Markov process with a Poisson process (a random subset, or a random measure). How does one interpret the definition of a pseudo-Poisson process intuitively and rigorously?
I think indices would be helpful here. You can view $\{Y[k]\}_{k=0}^{\infty}$ as the DTMC (taking values in some state space $S$). Let $N(t)$ for $t \geq 0$ be the process that counts the number of arrivals during $[0,t]$. It is assumed that $N(t)$ is a nondecreasing staircase function that is integer valued, increases by 1 at arrival times, and has initial condition $N(0)=0$.
Then we can define $X(t) = Y[N(t)]$ for $t \geq 0$.
For example, suppose $S=\{red, green, blue, brown\}$ and suppose:
$Y[0]=red$, $Y[1]=green$, $Y[2]=blue$, $Y[3]=green$.
The first four arrivals are at times $1.2$, $3.6$, $5.8$, $5.9$.
Then: $$X(t)=\left\{\begin{array}{cc} red & \quad \mbox{if } t \in [0, 1.2)\\ green & \quad \mbox{if } t\in [1.2, 3.6)\\ blue & \quad \mbox{if } t\in [3.6, 5.8)\\ green & \quad \mbox{if } t \in [5.8, 5.9) \end{array}\right.$$ The value of $X(t)$ for $t\geq 5.9$ depends on the values of $\{Y[4], Y[5], Y[6], \ldots\}$ and on the arrival times after time $5.9$.
Intuitively, you can just view the values of the discrete time sequence $\{Y[0], Y[1], Y[2], Y[3], ...\}$ as being laid out as blocks on the timeline $t\geq 0$, where the state of $X(t)$ is constant over these blocks of time. The first block starts at time $0$ and each new block starts at the (real-valued) times $\{T_1, T_2, T_3, ...\}$ that are the random arrival times of the $N(t)$ process.