@Michael - yes, if $c_k$ are i.i.d.r.v. with exponential tails.

@Nate Eldredge - not much. $\sum M_k$ just grows linearly with positive speed, and we need to obtain that $\sum f_k$ somehow resembles a sum of 0-mean r.v.'s

Sorry, I'm not a specialist in Levy processes :)

Don't know if there is a easy connection between them for Bessel processes. Frankly, I doubt one can find it.

Do you mean rather $P(\sup_{s\leq t}\|B(s)\|\geq r)$?

$\sqrt{2}$, I guess

"I don't see where you used the information about dimension in your argument." - here: "I think it's evident that the Brownian trajectory on the time interval $[0,1]$ intersects itself with positive probability."

Douglas Zare, but it's known that a BM's trajectory doesn't hit any fixed point a.s., so the complement is clearly dense in $\mathbb{R}^2$. But I agree that instead of just "intersects itself" better write something like "goes around a small ball".

By conformal invariance, this probability must be the same for all intervals (consider the time interval $[0,c]$; then the mapping $z\mapsto z/\sqrt{c}$ sends the trajectory on $[0,c]$ to trajectory on $[0,1]$). I think it's evident that the Brownian trajectory on the time interval $[0,1]$ intersects itself with positive probability.