Abstract
We consider a hybrid method to simulate the return time to the initial state in a critical-case birth--death process. The expected value of this return time is infinite, but its distribution asymptotically follows a power-law. Hence, the simulation approach is to directly simulate the process, unless the simulated time exceeds some threshold and if it does, draw the return time from the tail of the power law.
Highlights
We consider a hybrid method to simulate the return time to the initial state in a criticalcase birthdeath process
In [4] a very similar model that can generate both patterns were proposed. They model the amount of types alive at a given time instance as follows
Each type has at birth, independently of the other types, a tness value attached to it
Summary
The poweRlaw::rplcon(1,Tmin , α) draws a single value from a power law supported on (Tmin, ∞) with density and cumulative distribution functions (Eq (1), (4), [2]) equalling α − 1 t −α t −α+1 p(t) =. As we do not know from what value the tail asymptotics are a good approximation we cannot a priori be sure whether the threshold will be exceeded with probability p, nor if after p−1 the sampling of H from the limit will be accurate.
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