On almost sure behavior of stable subordinators over rapidly increasing sequences

Let $(X(t),\ t\geq 0)$ with $X(0)=0$ be a stable subordinator with index $0<\alpha<1$ and let $(t_k)$ be an increasing sequence such that $t_{k+1}/t_k\to\infty$ as $k\to\infty$. Let $(a_t)$ be a positive nondecreasing function of $t$ such that $a(t)/t\leq 1$. Define $Y(t)=X(t+a(t))-X(t)$ and $Z(t)=X(t)-X(t-a(t))$, $t>0$. We obtain law-of-the-iterated-logarithm results for $(X(t_k)),(Y(t_k))$ and $Z(t_k)$, properly normalized.