Competing growth processes with random growth rates and random birth times

Comparing individual contributions in a strongly interacting system of stochastic growth processes can be very difficult problem. This is particularly the case when new are initiated depending on state previous ones and rates themselves random. We propose novel technique to deal with such problems show how it applied broad range examples where produces insight surprising results. The method relies two steps: In first step, which highly problem dependent, jointly embedded into continuous time so that their evolutions after initiation become approximately independent while we retain some control over times. Once an embedding achieved, second step apply Poisson limit theorem enables comparison critical window therefore allows asymptotic description extremal process. this paper prove versatile type tool obtain results for variety interesting processes. These include (a) maximal degree different types preferential attachment networks fitnesses like well-known Bianconi–Barabási tree network model Dereich, (b) most successful mutant branching evolving by selection mutation, (c) ratio between largest cycles random permutation cycle weights, also interpreted as disordered version Pitman’s Chinese restaurant