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Is there a way to compare the hazard function that comes from a stochastic process with the hazard function which comes from a subset of that process?

Simulations of a stochastic process generate self-similar structures such that the survival in a short simulation should really just be a subset of a much longer simulation.

Is there a way to show that the survival functions come from the same process?

I have tried to look for convergence as I let the simulation run longer and longer. But since this scales the event times by the simulation time I have been unable to prove convergence.

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