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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, one definition of a subexponential distribution is as a
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
whose tails decay at an exponential rate, or faster: a real-valued distribution \cal D is called subexponential if, for a random variable X\sim , :(, X, \ge x)=O(e^) , for large x and some constant K>0. The subexponential norm, \, \cdot\, _, of a random variable is defined by :\, X\, _:=\inf\ \, where the infimum is taken to be +\infty if no such K exists. This is an example of a Orlicz norm. An equivalent condition for a distribution \cal D to be subexponential is then that \, X\, _<\infty. Subexponentiality can also be expressed in the following equivalent ways: # (, X, \ge x)\le 2 e^, for all x\ge 0 and some constant K>0. # (, X, ^p)^\le K p, for all p\ge 1 and some constant K>0. # For some constant K>0, (e^) \le e^ for all 0\le \lambda \le 1/K. # (X) exists and for some constant K>0, (e^)\le e^ for all -1/K\le \lambda\le 1/K. # \sqrt is sub-Gaussian.


References

* ''High-Dimensional Statistics: A Non-Asymptotic Viewpoint'', Martin J. Wainwright, Cambridge University Press, 2019, . Continuous distributions {{Probability-stub