The orbit of a dynamical system issued from an initial point is said to have historic behaviour if for some continuous function on the state space, the time average does not exist. In this talk, the following 2 problems about the (non-)negligibleness of the set of initial points with random historic behaviour will be discussed:

(1) Is the set of initial points with historic behaviour a residual subset of the state space for random expanding maps on the circle?
(2) Does there exist a random (non-i.i.d.) diffeomorphism for which the set of initial points with historic behaviour is a positive Lebesugue measure set?

As an application of a tool in the proof of our main results, random zeta functions shall be briefly addressed.

This is partly a joint work with S. Kiriki and T. Soma.