2 On Normalization Functions and ϕ-Families of Probability Distributions
27
2.3 The Behavior of ψ Near the Boundary of B
ϕ
c
As we could see in the previous section, if the Musielak–Orlicz function c (t, u) =
ϕ(c(t) + u) − ϕ(c(t)) does not satisfy the 2 -condition, it implies that the MorseTransue space E
ϕ
c is a proper subspace of L
ϕ
c . On its turn, this implies that the
boundary of B
ϕ
c , the domain of the parametrization (2.6), is not-empty.
Actually, by that fact that B
ϕ
c = B
ϕ
c ∩ K
ϕ
c is an open set in B
ϕ
c , we conclude that
a function u ∈ B
ϕ
c belongs to the boundary of B
ϕ
c if and only if
T ϕ(c + λu) dμ <
∞ for all λ ∈ (0, 1), and
T ϕ(c + λu) dμ = ∞ for each λ > 1. It is important to
know that if the function u belongs to the boundary of B
ϕ
c , then u may or may
not belong to the set Musielak–Orlicz class ˜
L
c , that is,
T ϕ(c + u) dμ < ∞, or
T ϕ(c + u) dμ = ∞.
If the Musielak–Orlicz function c (t, u) = ϕ(c(t) + u) − ϕ(c(t)) satisfies the
2 -condition, then
T ϕ(c + u) dμ < ∞ for all u ∈ L
ϕ
c , so the set B
ϕ
c coincides
with the closed set B
ϕ
c and its boundary is empty.
In this section, we will discuss the behavior of the normalizing function ψ in two
cases. The first one supposes that the deformed exponential function ϕ(·) satisfies
the condition stated in Eq. (2.2). On the other hand, the second case assumes that
the deformed exponential function ϕ(·) does not satisfy the Condition 2.2. More
specifically, given any function u in the boundary of B
ϕ
c , denoted by ∂B
ϕ
c , we want
to know whether ψ(αu) converges to a finite value or not as α ↑ 1.
The results we are going to discuss in details in the next sections can be summarized in Table 2.2.
2.3.1 Condition 2.2 is Satisfied
Supposing that the Musielak–Orlicz function c (t, u) = ϕ(c(t) + u) − ϕ(c(t)) does
not satisfy the 2 -condition, then the boundary ∂B
ϕ
c is not-empty. We then study the
behavior of ψ assuming, in this section, that the deformed exponential ϕ satisfies the
Condition 2.2.
Table 2.2 Summary of results
Cond
Result
Condition 2 satisfied
We analyse the cases where a function u ∈ ∂B
ϕ
c belongs to the
Musielak–Orlicz class. If so, we will show that the normalizing
function converges to a finite value β near its boundary domain.
On the other hand, we also show that if u does not belong to the
Musielak–Orlicz class, the normalizing function will not
converge (tends to infinity) near its boundary domain
Condition 2 is not satisfied
We may find a function u ∈ ∂B
ϕ
c , such that it does not belong to
the Musielak–Orlicz class but the normalizing function
converges for a finite value β, near its boundary domain
27
2.3 The Behavior of ψ Near the Boundary of B
ϕ
c
As we could see in the previous section, if the Musielak–Orlicz function c (t, u) =
ϕ(c(t) + u) − ϕ(c(t)) does not satisfy the 2 -condition, it implies that the MorseTransue space E
ϕ
c is a proper subspace of L
ϕ
c . On its turn, this implies that the
boundary of B
ϕ
c , the domain of the parametrization (2.6), is not-empty.
Actually, by that fact that B
ϕ
c = B
ϕ
c ∩ K
ϕ
c is an open set in B
ϕ
c , we conclude that
a function u ∈ B
ϕ
c belongs to the boundary of B
ϕ
c if and only if
T ϕ(c + λu) dμ <
∞ for all λ ∈ (0, 1), and
T ϕ(c + λu) dμ = ∞ for each λ > 1. It is important to
know that if the function u belongs to the boundary of B
ϕ
c , then u may or may
not belong to the set Musielak–Orlicz class ˜
L
c , that is,
T ϕ(c + u) dμ < ∞, or
T ϕ(c + u) dμ = ∞.
If the Musielak–Orlicz function c (t, u) = ϕ(c(t) + u) − ϕ(c(t)) satisfies the
2 -condition, then
T ϕ(c + u) dμ < ∞ for all u ∈ L
ϕ
c , so the set B
ϕ
c coincides
with the closed set B
ϕ
c and its boundary is empty.
In this section, we will discuss the behavior of the normalizing function ψ in two
cases. The first one supposes that the deformed exponential function ϕ(·) satisfies
the condition stated in Eq. (2.2). On the other hand, the second case assumes that
the deformed exponential function ϕ(·) does not satisfy the Condition 2.2. More
specifically, given any function u in the boundary of B
ϕ
c , denoted by ∂B
ϕ
c , we want
to know whether ψ(αu) converges to a finite value or not as α ↑ 1.
The results we are going to discuss in details in the next sections can be summarized in Table 2.2.
2.3.1 Condition 2.2 is Satisfied
Supposing that the Musielak–Orlicz function c (t, u) = ϕ(c(t) + u) − ϕ(c(t)) does
not satisfy the 2 -condition, then the boundary ∂B
ϕ
c is not-empty. We then study the
behavior of ψ assuming, in this section, that the deformed exponential ϕ satisfies the
Condition 2.2.
Table 2.2 Summary of results
Cond
Result
Condition 2 satisfied
We analyse the cases where a function u ∈ ∂B
ϕ
c belongs to the
Musielak–Orlicz class. If so, we will show that the normalizing
function converges to a finite value β near its boundary domain.
On the other hand, we also show that if u does not belong to the
Musielak–Orlicz class, the normalizing function will not
converge (tends to infinity) near its boundary domain
Condition 2 is not satisfied
We may find a function u ∈ ∂B
ϕ
c , such that it does not belong to
the Musielak–Orlicz class but the normalizing function
converges for a finite value β, near its boundary domain