Almost-sure law of large numbers for Bessel and Dunkl processes with drift

Prove that a Dunkl process or Bessel process with regular drift vector \(\lambda\) and arbitrary admissible starting point \(x\) satisfies \(X_t^\lambda/t\to\lambda\) almost surely as \(t\to\infty\), under the conditions of the paper’s probability-convergence law of large numbers.

Background

The paper derives convergence in probability, $X_t^\lambda/t\to\lambda$, from its central limit theorems for Dunkl and Bessel processes with regular drift. It compares this result with an earlier strong law involving a transformation $m_1(X_t^\lambda)$, but explains that the two results cannot currently be combined to obtain almost-sure convergence of the untransformed process. The authors explicitly conjecture that the stronger almost-sure statement nevertheless holds.