  Sub-Gaussian and subexponential distributions are introduced and applied to
study the fluctuation-response relation out of equilibrium. A bound on the
difference in expected values of an arbitrary sub-Gaussian or subexponential
physical quantity is established in terms of its sub-Gaussian or subexponential
norm. Based on that, we find that the entropy difference between two states is
bounded by the energy fluctuation in these states. Moreover, we obtain
generalized versions of the thermodynamic uncertainty relation in different
regimes. Operational issues concerning the application of our results in an
experimental setting are also addressed, and nonasymptotic bounds on the errors
incurred by using the sample mean instead of the expected value in our
fluctuation-response inequalities are derived.
