  The notion of typicality in statistical mechanics is essential to
characterize a macroscopic system. An overwhelming majority of the pure state
looks almost identical if we neglect macroscopic non-local correlations,
suggesting that thermal equilibrium is the collection of the typical
properties. Quantum entanglement, which characterizes a non-local correlation,
also has a typical behavior in equilibrium systems. However, it remains elusive
whether there is a typical behavior of entanglement in dynamical
non-equilibrium systems. To investigate the typicality, we consider a situation
where a system in a pure state starts to share entanglement with its
environment system due to the interaction between them. Assuming the initial
state is randomly chosen from an ensemble of pure states, a criteria for the
typicality of the R\'enyi entropies is presented. In addition, it is
analytically proven that the second R\'enyi entropy has a typical behavior in
two cases. The first one is an energy dissipation process in a multiple-qubit
system which is initially in a random pure state in an energy shell. Since the
typical behavior is qualitatively the same as the prediction of the Page curve
conjecture, it gives the first proof of the Page curve conjecture in a
dynamical process. In the second case, the typicality is proven for any
dynamics described by a multiple-product of a single-qudit channel when the
system is initially in a pure state randomly chosen from the whole Hilbert
space. This result shows that entanglement typicality is not a specific feature
of energy dissipating processes.
