Some asymptotic laws for multidimensional BM

In this chapter, we first build upon the knowledge gained in Chapter 5 about the asymptotic windings of planar BM around one point, together with the Kallianpur-Robbins ergodic theorem for planar BM, to extend Spitzer’s theorem:

$${2 \theta_t \over {log \;t}} \; {{\longrightarrow}\!\!\!\!\!\!\!\!\!\!}_{t \rightarrow \infty}^{(law)}C_1$$

into a multidimensional result for the winding numbers (θ ¹ _t , θ ²₁ ,…, θ ⁿ _t ) of planar BM around n points (all notations which may be alluded to, but not defined in this chapter are found in Pitman-Yor [75]).