  We report on a fundamental role of a non-normalized formal steady state,
i.e., an infinite invariant density, in a semi-Markov process where the state
is determined by the inter-event time of successive renewals. The state
describes certain observables found in models of anomalous diffusion, e.g., the
velocity in the generalized L\'evy walk model and the energy of a particle in
the trap model. In our model, the inter-event-time distribution follows a
fat-tailed distribution, which makes the state value more likely to be zero
because long inter-event times imply small state values. We find two scaling
laws describing the density for the state value, which accumulates in the
vicinity of zero in the long-time limit. These laws provide universal behaviors
in the accumulation process and give the exact expression of the infinite
invariant density. Moreover, we provide two distributional limit theorems for
time-averaged observables in these non-stationary processes. We show that the
infinite invariant density plays an important role in determining the
distribution of time averages.
