Conditional persistence of Gaussian random walks
نویسندگان
چکیده
Let {Xn}n≥1 be a sequence of i.i.d. standard Gaussian random variables, let Sn = ∑n i=1 Xi be the Gaussian random walk, and let Tn = ∑n i=1 Si be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence: P { max 1≤k≤n Tk ≤ 0 ∣∣∣ Tn = 0, Sn = 0} . n, P { max 1≤k≤2n Tk ≤ 0 ∣∣∣ T2n = 0, S2n = 0} & n−1/2 logn , for n→∞, which partially proves a conjecture by Caravenna and Deuschel [3].
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