Sub Gaussian distribution - Alchetron, the free social encyclopedia

In probability theory, a sub-Gaussian distribution is a probability distribution with strong tail decay property. Informally, the tails of a sub-Gaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian.

Formally, the probability distribution of a random variable X is called sub-Gaussian if there are positive constants C, v such that for every t > 0,

P ⁡ ( | X | > t ) ≤ C e − v t 2 .

The sub-Gaussian random variables with the following norm form a Birnbaum–Orlicz space:

∥ X ∥ ψ 2 = inf { s > 0 ∣ E ⁡ e ( X / s ) 2 − 1 ≤ 1 } .

Equivalent properties

The following properties are equivalent:

The distribution of X is sub-Gaussian

ψ 2 -condition: ∃ a > 0 E ⁡ e a X 2 < + ∞ .

Laplace transform condition: ∃ B , b > 0 ∀ λ ∈ R E ⁡ e λ X ≤ B e λ 2 b .

Moment condition: ∃ K > 0 ∀ p ≥ 1 ( E ⁡ | X | p ) 1 / p ≤ K p .

References

Sub-Gaussian distribution Wikipedia

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