A generalization of functional limit theorems on the Riemann zeta process

$\zeta(\cdot)$ being the Riemann zeta function, $\zeta_{\sigma}(t) := \frac{\zeta(\sigma + i t)}{\zeta(\sigma)}$ is, for $\sigma > 1$, a characteristic function of some infinitely divisible distribution $\mu_{\sigma}$. A process with time parameter $\sigma$ having $\mu_{\sigma}$ as its marginal at time $\sigma$ is called a Riemann zeta process. Ehm [2] has found a functional limit theorem on this process being a backwards Levy process. In this paper, we replace $\zeta(\cdot)$ with a Dirichlet series $\eta(\cdot;a)$ generated by a nonnegative, completely multiplicative arithmetical function $a(\cdot)$ satisfying (3), (4) and (5) below, and derive the same type of functional limit theorem as Ehm on the process corresponding to $\eta(\cdot;a)$ and being a backwards Levy process.