  We consider renewal stochastic processes generated by non-independent events
from the perspective that their basic distribution and associated generating
functions obey the statistical-mechanical structure of systems with interacting
degrees of freedom. Based on this fact we look briefly into the less known case
of processes that display phase transitions along time. When the density
distribution \psi_{n}(t) for the occurrence of the n-th event at time t is
considered to be a partition function, of a 'microcanonical' type for n
'degrees of freedom' at fixed 'energy' t, one obtains a set of four partition
functions of which that for the generating function variable z and Laplace
transform variable \epsilon, conjugate to n and t, respectively, plays a
central role. These partition functions relate to each other in the customary
way and in accordance to the precepts of large deviations theory, while the
entropy, or Massieu potential, derived from \psi_{n}(t) satisfies an Euler
relation. We illustrate this scheme first for an ordinary renewal process of
events generated by a simple exponential waiting time distribution \psi (t).
Then we examine a process modelled after the so-called Hamiltonian Mean Field
(HMF) model that is representative of agents that perform a repeated task with
an associated outcome, such as an opinion poll. When a sequence of (many)
events takes place in a sufficiently short time the process exhibits clustering
of the outcome, but for larger times the process resembles that of independent
events. The two regimes are separated by a sharp transition, technically of the
second order. Finally we point out the existence of a similar scheme for random
walk processes.
