Working through a paper that uses this result about branching processes. Can't seem to figure out a way to connect the dots. Anyone have any suggestions?(adsbygoogle = window.adsbygoogle || []).push({});

Let Z_{t}be a branching process where each individual gives birth to a single offspring at rate β > 1 and dies at rate 1. The transition probabilities for Z_{t}can be computed exactly and are given by

[tex]P_{10}(t) = \frac{1 - e^{-(\beta-1)t}}{\beta - e^{-(\beta-1)t}}[/tex]

[tex]P_{1k}(t) = (1 - P_{10}(t))(1 - \eta(t))\eta(t)^{k-1}[/tex]

[tex]\eta(t) = \frac{1 - e^{-(\beta-1)t}}{1 - \frac{1}{\beta}e^{-(\beta-1)t}}[/tex]

I know that P_{1i}satisfy the following system of equations

[tex] P_i'(t) = Q P(t) [/tex]

Where Q is the transition rate matrix (i think that's what it's called) and is given by

[tex] Q = \left(

\begin{array}{ccccc}

0 & 1 & 0 & \ldots & \\

0 & -(1+\beta ) & 2 & \ddots & \vdots \\

0 & \beta & -2(1+\beta ) & 3 & \\

\vdots & \ddots & 2\beta & -3(1+\beta)& \ddots \\

& \ldots & & \ddots & \ddots \\

\end{array}

\right) [/tex]

I have verified the above solution works, but I'm not sure how one would have gotten it. Any ideas?

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# Probability transitions for branching process

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