In probability theory, a subgaussian distribution, the distribution of a subgaussian random variable, is a probability distribution with strong tail decay. More specifically, the tails of a subgaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives subgaussian distributions their name. Often in analysis, we divide an object (such as a random variable) into two parts, a central bulk and a distant tail, then analyze each separately. In probability, this division usually goes like "Everything interesting happens near the center. The tail event is so rare, we may safely ignore that." Subgaussian distributions are worthy of study, because the gaussian distribution is well-understood, and so we can give sharp bounds on the rarity of the tail event. An analogous class, sometimes called subexponential distributions, is also useful; note however that the more established meaning of subexponential is almost the opposite -- that decay is slower than exponential rather than that the tail is lighter than exponential -- and so care must be taken with that term. Formally, the probability distribution of a random variable X {\displaystyle X} is called subgaussian if there is a positive constant C such that for every t ≥ 0 {\displaystyle t\geq 0} ,
P ( | X | ≥ t ) ≤ 2 exp ( − t 2 / C 2 ) {\textstyle \mathbb {P} (|X|\geq t)\leq 2\exp {(-t^{2}/C^{2})}} . There are many equivalent definitions. For example, a random variable X {\displaystyle X} is sub-Gaussian iff its distribution function is bounded from above (up to a constant) by the distribution function of a Gaussian:
P ( | X | ≥ t ) ≤ c P ( | Z | ≥ t ) ∀ t > 0 {\displaystyle \mathbb {P} (|X|\geq t)\leq c\,\mathbb {P} (|Z|\geq t)\quad \forall t>0}
where c ≥ 0 {\displaystyle c\geq 0} is constant and Z {\displaystyle Z} is a mean zero Gaussian random variable.
Definitions
Subgaussian norm The subgaussian norm of X {\displaystyle X} , denoted as ‖ X ‖ ψ 2 {\displaystyle \Vert X\Vert _{\psi _{2}}} , is ‖ X ‖ ψ 2 = inf { c > 0 : E [ exp ( X 2 c 2 ) ] ≤ 2 } . {\displaystyle \Vert X\Vert _{\psi _{2}}=\inf \left\{c>0:\mathbb {E} \left[\exp {\left({\frac {X^{2}}{c^{2}}}\right)}\right]\leq 2\right\}.} In other words, it is the Orlicz norm of X {\displaystyle X} generated by the Orlicz function Φ ( u ) = e u 2 − 1. {\displaystyle \Phi (u)=e^{u^{2}}-1.} By condition ( 2 ) {\displaystyle (2)} below, subgaussian random variables can be characterized as those random variables with finite subgaussian norm.
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