138
H. Matsuzoe and A. Takatsu
A q-Gaussian density corresponds to a normal (Gaussian) distribution for q = 1,
and a Student t-distribution for 1 < q < 3. In the both cases, the support of each
q-Gaussian measure is the whole of R and
p q (x; ξ) = p q (x; μ, σ ) =
1
Z q σ
exp q
−
1
3 − q
x − μ
σ
2
.
6.3.2 Sufficient Conditions for (6.4)
In order to give a rigorous treatment of an escort expectation associated with the
a-refined q-logarithmic function, we only deal with the case 1 ≤ q < 3. Set
q :=
σ > 0
1
Z q σ
< 1
,
S q := { p q (·; ξ) | ξ ∈ R × q }.
It holds for ξ = (μ, σ ) ∈ R × q and x ∈ R that
ln q
1
Z q σ
< ln q (1) = 0,
ln q
p q (x; ξ)
∈ (−∞, 0).
Definition 5 For 1 ≤ q < 3 and ξ ∈ R × q , define q (·; ξ) : R → (−∞, 0) by
q (x; ξ) := ln q
p q (x; ξ)
,
which is called the q-likelihood function of p q (·; ξ).
For 1 ≤ q < 3, a ∈ R \ {0} and ξ ∈ R × q , we define a measure ν q,a;ξ on R as
the absolutely continuous measure with respect to the one-dimensional Lebesgue
measure with Radon–Nikodym derivative
dν q,a;ξ
dx
(x) =
1
ln
q,a
p q (x; ξ)
.
Since the inverse function of ln q,a is exp q,a , Lemma 1 in the case n = 1 leads to
dν q,a;ξ
dx
(x) = exp
q,a (ln q,a ( p q (x; ξ ))) =
− q (x; ξ)
1−a χ q ( p q (x; ξ )).
A direct computation leads to the relation that