  General upper bounds on fluctuations of trajectory observables were recently
obtained. It turned out that the size of fluctuations of dynamical observable
is limited from below and from above. For the moment generating function of
general upper bounds on the size of fluctuations, the moments (average value
and variance) of the size of fluctuations are obtained. A more complex and
interesting task is to obtain the first-passage time for process of upper
bounds of reaching fluctuations of observables. Characteristic functions,
average values and variances of the first-passage time of reaching fluctuations
of observables of the trajectory of the Markov chain of positive and negative
levels are also obtained. Some general issues of the relationship between the
theory of random processes (using the example of the risk theory used) and
thermodynamics of trajectories are also considered.
