Conditional persistence of Gaussian random walks

Let {X-n}(n >= 1) be a sequence of i.i.d. standard Gaussian random variables, let S-n =Sigma(n)(i=1) X-i be the Gaussian random walk, and let T-n = Sigma(n)(i=1) S-i 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) T-k <= 0 | T-n = 0, S-n = 0}less than or similar to n(-1/2), P{max(1 < k < 2n) T-k <= 0 | T-2n = 0, S-2n = 0}greater than or similar to n(-1/2)/logn, for n --> infinity, which partially proves a conjecture by Caravenna and Deuschel [3].