I am a bit at a loss when it comes to the mathematical framework of Yule processes. Exhausting Google I have found several terms that seem to be used interchangably, namely: Yule process, Yule-Furry process, Preferential Attachment process, Delayed Poisson Point Process, Pure birth process.
Out of the above names I have already studied Poisson point processes, both homogeneous and inhomogeneous and general counting processes. I am looking for a mathematical reference that rigorously introduces the Yule process and possibly discusses some of its properties (see excerpt below).
My first encounter with the Yule process comes from the following excerpt:
Excerpt (doi:10.1007/s10955-018-2062-5): We will require some results about Yule processes and Yule processes with mutation. Let $Y=(Y_t)_{t \in \mathbb{R}_+}$ denote the Yule process started from $Y_0=1$, such that for $t \in \mathbb{R}_+$, $Y_t$ describes the number of individuals alive at time $t$, when each individual lives forever and gives birth to children at rate $1$. The following lemma is well-known:
Lemma: The process $e^{-t}Y_t$ is a martingale. Its terminal value $W$ exists a.s. and is exp(1) distributed.
Summary: I need a mathematical reference (also for my bibliography) which properly introduces the Yule-process and my final goal is to understand the excerpt above.
'Branching Processes' by Athreya & Ney contains a few pages on the the Yule process (in section 11. of chapter III), including mentioning the convergence of $e^{-t}Y_t$ to $Exp(1)$ distribution (although it is left as an exercise - the tools are given on p109 to prove it via moment generating functions).
The Yule process is also mentioned in Ross's 'Introduction to Probability Models' with some properties given.
Admittedly I am finding the same problem of relatively sparse mentions in the literature. Hope this helps.