Universal Persistence for Local Time of One-dimensional Random Walk
نویسنده
چکیده
We prove the power law decay p(t, x) ∼ t−φ(x,b)/2 in which p(t, x) is the probability that the fraction of time up to t in which a random walk S of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds x throughout s ∈ [1, t]. Here φ(x, b) = P(Lévy(1/2, κ(x, b)) < 0) for κ(x, b) = √ 1−xb− √ 1+x √ 1−xb+ √ 1+x and b = bS > 0 measuring the asymptotic asymmetry between positive and negative excursions of the walk (with bs = 1 for symmetric increments).
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