Under general resetting, stochastic functionals of random walks become deterministic at long times if reset times have a finite first moment, and obey new U-shaped, W-shaped, or inverted-U limiting distributions for power-law resetting.
In the limit u → 0, f (u) ∼ u−1/2 and from (56) we see that Cβ(z) ∼ zβ+1/2 as z → 0
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Statistical properties of stochastic functionals under general resetting
Under general resetting, stochastic functionals of random walks become deterministic at long times if reset times have a finite first moment, and obey new U-shaped, W-shaped, or inverted-U limiting distributions for power-law resetting.