Let $Z_0(t)$ be a Markov chain and $X(t)$ a subordinator. Set $Z(t) = Z_0(X(t))$ and let $\alpha$ be a nonstable state of $Z_0$. It is shown, via an example, that it is possible for $\alpha$ to be a stable state of $Z(t)$ even when the total mass of the Levy measure of $X$ is unbounded.
"Can a Nonstable State Become Stable by Subordination?." Ann. Probab. 5 (3) 463 - 466, June, 1977. https://doi.org/10.1214/aop/1176995806