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The divergence can be understood from a minimal model describing two states, both carrying a Poissonian current with different rates Γ1 Γ2. If the system switches between the states with a rate Ω the kernel of the system's master equation reads W= (Γ1 (eiχ -1) Ω Ω Γ2 (eiχ -1)). The cumulant generating function is readily found to be S (χ) = [(Γ1 + Γ2) (eiχ -1) -2Ω+ (Γ1 - Γ2) 2 (eiχ -1) +4 Ω2] /2. The normalized moments diverge as Ω→0, the second in particular reads κ2 / κ1 =1+ (Γ1 - Γ2) 2 / [2Ω (Γ1 + Γ2)].
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