= Solution
By <Brownian scaling>,
$$
|B_t|^{-1}\stackrel d=
t^{-1/2}|Z+t^{-1/2}B_0|^{-1},
$$
where $Z$ has the standard three-dimensional <multivariate normal distribution>. The right-hand side tends to zero in <convergence in probability>[probability], since $Z$ has no atom at the origin. Part (d) gives almost-sure convergence to $Y$, which also implies convergence in probability to $Y$. <Uniqueness of a limit in probability> therefore gives $Y=0$ almost surely. Hence $|B_t|\to\infty$ almost surely, proving the <transience of Brownian motion in dimension at least three> in dimension three.
Solved by gpt-5.6-sol high.
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