Strong law and central limit theorem

A lot is known about the behaviour of the volume of $W^a(t)$ as $t \to \infty$. For instance,

$E\vert W^a(t)\vert \sim \left\{\begin{array}{ll} \sqrt{8t/\pi} &(d=1)\\ 2\pi t/\log t &(d=2)\\ \kappa_a t &(d \geqslant 3) \end{array}\right.$ (2)

with $\kappa_a = a^{d-2}2\pi^{d/2}/\Gamma(\frac{d-2}{2})$ the Newtonian capacity of $B_a(0)$ (associated with the Green's function of $(-\Delta/2)^{-1}$) and

$Var \vert W^a(t)\vert \asymp \left\{\begin{array}{ll} t &(d=1)\\ t^2/\log^4 t &(d=2)\\ t \log t &(d=3)\\ t &(d \geqslant 4) \end{array}\right.$ (3)

(Spitzer [6], Le Gall [4]). Moreover, $\vert W^a(t)\vert$ satisfies the strong law and the central limit theorem for $d \geqslant 2$; the limit law is Gaussian for $d \geqslant 3$ and non-Gaussian for $d=2$ (Le Gall [5]).

Note that for $d \geqslant 2$ the Wiener sausage is a sparse object: since the Brownian motion typically travels a distance $\sqrt{t}$ in each direction, (2) shows that most of the space in the convex hull of $W^a(t)$ is not covered.