Two LIL-type results for the Brownian bridge
X\({}_ t\) is a Brownian bridge. Suppose H(t) is a nonnegative function defined for t close to zero such that H(t) is nondecreasing for t close to zero, and g(t) defined as \(\sqrt{t(1-t)}H(t)\) is nondecreasing for t close to zero. H is called upper class at zero if for almost all sample paths \(\omega\) there is a positive quantity \(\epsilon\) (\(\omega)\) such that for all \(t<\epsilon\), \(X_ t<g(t)\). Define \(T_ g\) as inf\(\{\) \(t: X_ t\geq g(t)\}\). The following two theorems are proved. Theorem 1. H is upper class at zero if and only if \(\int_{0+}\exp (- H^ 2(t))H(t)t^{-1}dt\) is finite. Theorem 2. If H is upper class at zero then as \(v\to 0\), \(.0907(1+o(1))I_ g(v)\leq p_ v\leq e(1+o(1))I_ g(v)\), where for \(v>0\), \(p_ v=P(T_ g\leq v)\) and \(I_ g(v)=(2\pi)^{- 1/2}\int^{v}_{0+}\exp (-H^ 2(t))H(t)t^{-1}dt\).
scientific article; zbMATH DE number 3965096