1 IG of Poincaré inequalities …
13
It follows that for all f, g ∈ C
2
poly (R
n
) we derive from Eq. (1.22)
Cov γ ( f, g) =
∞
0
e
−t
P t ∇ f (x) · ∇g(x) γ (x) dx dt .
(1.23)
We use here a result of [27, Proposition 5]. Let |·| 1 and |·| 2 be two norms on R
n ,
such that |x · y| ≤ |x| 1 |y| 2 . For a Young function Φ, consider the norm of L Φ (γ )
and the conjugate space endowed with the dual norm,
f L Ψ,∗( γ ) = sup
f g γ
Φ(g) γ ≤ 1
.
The following inequality that includes the standard Poincaré case when Φ(u) =
u
2
/2.
Proposition 4 Given a couple of conjugate Young function Φ, Ψ , and norms |·| 1 ,
|·| 2 on R
n such that x · y ≤ |x| 1 |y| 2 , x, y ∈ R
n , for all f, g ∈ C
1
poly (R
n
), it holds
Cov γ ( f, g)
≤
|∇ f | 1
L Φ (γ )
|∇g| 2
L Ψ,∗( γ )
.
The case of our interest here is Φ = cosh −1, Ψ = (cosh −1) ∗ . As (cos −1) ∗ ≺
(cosh −1), it follows, in particular, that Cov γ ( f, f ) is bouded by a constant times
|∇ f ||
2
L cosh −1( γ ) .
1.4 Discussion and Conclusions
We have collected here a list of possible applications of the information geometry
of the Gaussian space that has been introduced in [13, 27] and further developed in
the present paper.
1.4.1 Sub-exponential Random Variables
Let f ∈ C
2
poly (R
n
) be a random variable of the Gaussian space. Assume moreover
that f is globally Lipschitz, that is,
|∇ f (x)| ≤ f Lip(R n ) |x|
where f Lip(R n ) is the Lipschitz semi-norm, that is, the best constant. It follows
from Eq. (1.17) that f ∈ L (cosh −1) (γ ) and the norm admits a computable bound.
If p is any probability density of the maximal exponential model of γ , that is,
it is connected to 1 by an open exponential arc, then Proposition 1 implies that