Y

σ^{2}

when

lo g E e^{λY} \leq \frac{σ ^{2} λ ^{2}}{2} (λ \in R) .

The Chernoff bound, applied to

Y

and

- Y

, yields

P (∣ Y ∣ \geq t) \leq 2 e^{- t^{2} / (2 σ^{2})} .

u = t^{2} / (2 σ^{2})

now give

E ∣ Y ∣^{q} = q \int_{0}^{\infty} t^{q - 1} P (∣ Y ∣ \geq t) d t \leq 2 q \int_{0}^{\infty} t^{q - 1} e^{- t^{2} / (2 σ^{2})} d t = 2Γ (\frac{q}{2} + 1) (2 σ^{2})^{q /2} .

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.