2 On Normalization Functions and ϕ-Families of Probability Distributions
21
2.2 Revisiting Deformed Exponentials
In this section we revisit some fundamental concepts about the deformed exponential
functions and probability distributions so we can proceed with the study about the
characterization of them, in terms of the normalization function.
2.2.1 Deformed Exponential Functions
Definition 2.1 A deformed exponential is a convex function ϕ : R → [0, ∞) such
that lim u→−∞ ϕ(u) = 0 and lim u→∞ ϕ(u) = ∞.
It is easy to verify that the ordinary exponential and Tsallis q-exponential are
deformed exponential functions [10]. The notion of deformed exponential exp φ was
firstly introduced in [13] and studied after in [9, 14, 15].
Fixing an increasing function φ that is strictly positive in (0, ∞) and
1
φ
is integrable
we have the φ-logarithm defined as as:
ln φ (u) =
u
1
1
φ(x)
dx, u > 0.
The function ln φ is concave, negative in (0, 1), positive on (1, +∞) and the inverse
is the function exp φ [13]. Every deformed exponential exp φ (·) satisfies the Definition
2.1 and the deformed exponential ϕ(·) proposed in [27] is a deformed exponential
exp φ when ϕ(0) = 1.
Vigelis and Cavalcante in [27] used a σ -finite, non-atomic measure space
(T, ,, μ), and a deformed exponential ϕ to introduce a non-parametric generalization of the exponential statistical manifold proposed by [3, 18]. This generalization
was subsequently investigated in [2, 20, 26] stressing the geometric aspects of the
new family of distributions.
A very important aspect the authors realized in [27] is the need of an additional
condition over the function to find the domain of the ϕ-families of probability distributions F
ϕ
c , that is a generalization of the exponential families of probabilites
distributions E( p) [3, 18, 19].
This condition is given as [27]:
Condition 2.1 There exists a function u 0 : T → (0, ∞) such that
T
ϕ(c + λu 0 )dμ < ∞,
for all λ > 0,
(2.1)
for every function c : T → R, for which
T ϕ(c)dμ = 1.