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Sub-Gaussian and subexponential distributions are introduced and applied tostudy the fluctuation-response relation out of equilibrium. A bound on thedifference in expected values of an arbitrary sub-Gaussian or subexponentialphysical quantity is established in terms of its sub-Gaussian or subexponentialnorm. Based on that, we find that the entropy difference between two states isbounded by the energy fluctuation in these states. Moreover, we obtaingeneralized versions of the thermodynamic uncertainty relation in differentregimes. Operational issues concerning the application of our results in anexperimental setting are also addressed, and nonasymptotic bounds on the errorsincurred by using the sample mean instead of the expected value in ourfluctuation-response inequalities are derived.

Sub-Gaussian and subexponential fluctuation-response inequalities