4
G. Pistone
Domination relation between Young functions imply continuous injection properties for the corresponding Orlicz spaces. We say that Φ 2 eventually dominates Φ 1 ,
written Φ 1 ≺ Φ 2 , if there is a constant κ such that Φ 1 (x) ≤ Φ 2 (κ x) for all x larger
than some ¯
x. As, in our case, μ is a probability measure, the continuous embedding
L Φ 2 (μ) → L Φ 1 (μ) holds if, and only if, Φ 1 ≺ Φ 2 . See a proof in [1, Theorem 8.2].
If Φ 1 ≺ Φ 2 , then (Φ 2 ) ∗ ≺ (Φ 1 ) ∗ . With reference to our examples (1.4) and (1.5), we
see that exp 2 and (cosh −1) are equivalent. They both are eventually dominated by
gauss 2 (1.6) and eventually dominate all powers (1.3).
A special case occurs when there exists a function C such that Φ(ax) ≤ C(a)Φ(x)
for all a ≥ 0. This is true, for example, for a power function and in the case of the
functions (exp 2 ) ∗ and (cosh −1) ∗ . In such a case, the conjugate space and the dual
space are equal and bounded functions are a dense set.
The spaces corresponding to case (1.3) are ordinary Lebesgue spaces. The cases
(1.4) and (1.5) provide isomorphic B-spaces L (cosh −1) (μ) ↔ L exp 2 (μ) which are
of special interest for us as they provide the model spaces for our non-parametric
version of IG, see Sect. 1.2.3.
A function f belongs to L cosh −1 (μ) if, and only if, it is sub-exponential, that is,
there exist constants C 1 , C 2 > 0 such that
P μ (| f | ≥ t) ≤ C 1 exp (−C 2 t) , t ≥ 0 .
Sub-exponential random variable are of special interest in applications because they
admit an explicit exponential bounds in the Law of Large Numbers. Random variables
whose square is sub-exponential are called sub-gaussian. There is a large literature
on this subject, see, for example, [4, 30–32].
We will be led to use a further notation. For each Young function Φ, the function
Φ(x) = Φ(x
2
) is again a Young function such that f L Φ (μ) ≤ λ if, and only if,
| f |
2
L Φ (μ)
≤ λ
2 . We denote the resulting space by L
2
Φ (μ). For example, gauss 2
and cosh −1 are ≺-equivalent , hence the isomorphisn L gauss 2 (μ) ↔ L
2
(cosh −1) (μ).
As an application of this notation, consider that for each increasing convex Φ
it holds Φ( f g) ≤ Φ(( f
2
+ g
2
)/2) ≤ (Φ( f
2
) + Φ(g
2
))/2. It follows that when the
L
2
Φ (μ)-norm of f and of g is bounded by one, the L Φ (μ)-norm of f , g, and f g,
are all bounded by one. The need to control the product of two random variables in
L (cosh −1) (μ) appears, for example, in the study of the covariant derivatives of the
statistical bundle, see [6, 7, 14, 28].
1.2.2 Calculus of the Gaussian Space
From now on, our base probability space is the Gaussian probability space (R
n
, γ ),
γ (z) = (2π)
n/2 exp
− |z|
2
/2
. We will use a few simple facts about the analysis of
the Gaussian space, see [15, Chap. V].
G. Pistone
Domination relation between Young functions imply continuous injection properties for the corresponding Orlicz spaces. We say that Φ 2 eventually dominates Φ 1 ,
written Φ 1 ≺ Φ 2 , if there is a constant κ such that Φ 1 (x) ≤ Φ 2 (κ x) for all x larger
than some ¯
x. As, in our case, μ is a probability measure, the continuous embedding
L Φ 2 (μ) → L Φ 1 (μ) holds if, and only if, Φ 1 ≺ Φ 2 . See a proof in [1, Theorem 8.2].
If Φ 1 ≺ Φ 2 , then (Φ 2 ) ∗ ≺ (Φ 1 ) ∗ . With reference to our examples (1.4) and (1.5), we
see that exp 2 and (cosh −1) are equivalent. They both are eventually dominated by
gauss 2 (1.6) and eventually dominate all powers (1.3).
A special case occurs when there exists a function C such that Φ(ax) ≤ C(a)Φ(x)
for all a ≥ 0. This is true, for example, for a power function and in the case of the
functions (exp 2 ) ∗ and (cosh −1) ∗ . In such a case, the conjugate space and the dual
space are equal and bounded functions are a dense set.
The spaces corresponding to case (1.3) are ordinary Lebesgue spaces. The cases
(1.4) and (1.5) provide isomorphic B-spaces L (cosh −1) (μ) ↔ L exp 2 (μ) which are
of special interest for us as they provide the model spaces for our non-parametric
version of IG, see Sect. 1.2.3.
A function f belongs to L cosh −1 (μ) if, and only if, it is sub-exponential, that is,
there exist constants C 1 , C 2 > 0 such that
P μ (| f | ≥ t) ≤ C 1 exp (−C 2 t) , t ≥ 0 .
Sub-exponential random variable are of special interest in applications because they
admit an explicit exponential bounds in the Law of Large Numbers. Random variables
whose square is sub-exponential are called sub-gaussian. There is a large literature
on this subject, see, for example, [4, 30–32].
We will be led to use a further notation. For each Young function Φ, the function
Φ(x) = Φ(x
2
) is again a Young function such that f L Φ (μ) ≤ λ if, and only if,
| f |
2
L Φ (μ)
≤ λ
2 . We denote the resulting space by L
2
Φ (μ). For example, gauss 2
and cosh −1 are ≺-equivalent , hence the isomorphisn L gauss 2 (μ) ↔ L
2
(cosh −1) (μ).
As an application of this notation, consider that for each increasing convex Φ
it holds Φ( f g) ≤ Φ(( f
2
+ g
2
)/2) ≤ (Φ( f
2
) + Φ(g
2
))/2. It follows that when the
L
2
Φ (μ)-norm of f and of g is bounded by one, the L Φ (μ)-norm of f , g, and f g,
are all bounded by one. The need to control the product of two random variables in
L (cosh −1) (μ) appears, for example, in the study of the covariant derivatives of the
statistical bundle, see [6, 7, 14, 28].
1.2.2 Calculus of the Gaussian Space
From now on, our base probability space is the Gaussian probability space (R
n
, γ ),
γ (z) = (2π)
n/2 exp
− |z|
2
/2
. We will use a few simple facts about the analysis of
the Gaussian space, see [15, Chap. V].