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I have a semi-Markov process in which the time between states is log-normally distributed, but with parameters that depend on $n$ (the mean and variance are state-dependent). In other words I have the transition time from state $n$ to $n+1$ follows $\tau_{n\rightarrow n+1}\sim \textrm{Lognormal}(\mu_n,\sigma_n^2)$. These transition times are correlated in state number, with covariance matrix $\textbf{C}$. meaning that $\tau_n$ is influenced by the particular value of $\tau_{n-1}$, while overall having a known lognormal distribution.

How can I generate a sequence of simulated transition times that respects both the state-dependent distribution of times, and their correlations?

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  • $\begingroup$ Perhaps I am missing something, but what is wrong with starting in some state $n$, deciding which state you will be jumping to, e.g. $n$, and then drawing $\tau_{n\to {n+1}}$ from the distribution as you stated it? Simply write a script to do it. $\endgroup$ Commented Oct 11 at 19:34
  • $\begingroup$ @Cryo because there are nonzero correlations between the states - while I have the distribution of $\tau_n$ for all state transitions, there is some component of the global variance that is attributable to the specific value of $\tau_{n-1}$ - which is to say, that I cannot sample independently from each distribution, I need to respect that correlation. The value of $\tau_n$ depends on the random number I generated for step $\tau_{n-1}$. Is that clear? $\endgroup$ Commented Oct 11 at 19:42
  • $\begingroup$ Can you please explain correlation in more detail? You mentioned matrix $\mathbf{C}$, but it is not clear to me how it relates to $\mu_n$ or $\sigma_n^2$. Are you saying that $\tau_{n\to n+1}$ is multivariate normal? $\endgroup$ Commented Oct 12 at 9:22
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    $\begingroup$ Is this any different to saying that $\log \tau_n$ is multivariate normal, with covariance matrix that captures your dependence on previous states? If it is not, why not generate $\log \tau_n$ and then exponentiate? $\endgroup$ Commented Oct 12 at 20:15
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    $\begingroup$ This is generally the most straightforward approach to simulating correlated random variables with arbitrary distributions by the way: start with multivariate normal, and then transform (e.g. quantile-to-quantile). A drawback is that such transform does not preserve covariance exactly, but ranks are maintained & you can always tweak the starting correlation to get the desired transformed one. $\endgroup$ Commented Oct 16 at 7:08

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