A geometric law of iterated logarithm for Brownian motions (Q1909420)

Let \(B(t)\), \(t \geq 0\), be a \(d\)-dimensional Brownian motion, and denote by \(C_t\) the closed convex hull of the set \(\{B_s: 0 \leq s \leq t\}\). A law of the iterated logarithm is proved for this family of random sets. First one has \[ \limsup_{t\to 0} (2t \log \log 1/t)^{-1/2} C_t = U^d \quad \text{a.s.}, \] where \(U^d\) is the Euclidean unit ball in dimension \(d\). Putting \(h^+ (A,B) = \text{sup} \{d(x,B); x\in A\}\) for two subsets \(A\), \(B\) of the Euclidean space of dimension \(d\), one has \[ \lim \inf_{t \to 0} h^+ ((2t \log \log 1/t)^{-1/2} C_t, U^d) = 0 \quad \text{a.s.} \] which gives a lower bound.