I have been working with a continuous-time birth and death process in my research, of the linear type discussed in this paper of Karlin: https://www.jstor.org/stable/24900526. I am struggling to correctly understand a question of conditional probabilities.
The setup is as follows. The birth and death process concerns two stochastic variables: a state $j(t)$ and a cumulative population $M(t)$. As the state undergoes nearest-neighbor transitions $j \to j \pm 1$, the cumulative population shares only the positive transitions: $M \to M + 1$ iff $j\to j + 1$. There are of course other parameters which we suppress for simplicity.
If $j(T) = 0$, the process terminates and is "absorbed at zero". We know the distribution of these absorption times $T \sim d\mu(T)$ (for any initial state $j(0)$) and therefore the probability that a given process was absorbed before time $T$, which is just the CDF of the distribution $d\mu(T)$.
One can also compute the probabilities of eqn (26) in Karlin: $$ R^i_n = \text{probability of being absorbed on the $n$-th transition given }j(0) = i $$ If one knows the $R^i_n$, they can also calculate the probability distribution of $M$ at absorption. This is because for an process which absorbs, the total number of transitions $n_T$ is related to the value of $M$ at absorption via $$ n_T = 2M(T) - i $$ assuming that $j(0) = M(0) = i$.
We therefore know the following, given $j(0) = F(0) = i$.
- The probability that a process is absorbed by time $T$
- The distribution of absorption times $d\mu(T)$
- The distribution of $M(T)$ at the time of absorption, for processes which absorb. Equivalently the asymptotic distribution $dM(\infty)$, assuming absorption occurs.
The question is: Is this enough information to describe the distribution of values of $M(t)$ at an arbitrary time $t$?
I am not understanding whether the probabilities $R_n^i$ are conditioned on absorption happening, or if they are the joint probability of absorption and $n$ transitions. I want to understand what time-dependent statements about the distribution of values of $M(t)$ can be made knowing the information bulleted above, if any.