6 Gauge Freedom of Entropies on q-Gaussian Measures
139
q (x; ξ) = ln q
1
Z q σ
−
1
(Z q σ ) 1−q (3 − q)
x − μ
σ
2
,
ln q,a ( p q (x; ξ)) = −
1
a
− q (x; ξ)
a .
(6.5)
Lemma 3 Let 1 ≤ q < 3, a ∈ R \ {0} and ξ ∈ R × q . Then for λ, γ ∈ R with
λ > 0, (λ + x
2
)
γ
∈ L
1
(ν q,a;ξ ) if and only if
either q = 1 or q > 1 with γ <
1
2
+
1
q − 1
+ a − 1.
Proof Since the decay rate of ν 1,a;ξ is o(exp(−x
ε
)) as x → ∞ for ε < 2, the lemma
holds for q = 1.
Assume q > 1. By the change of variables, it is enough to show the case
ξ = (0, 2/Z q ). Here we have that Z q σ = 2. There exist c, C, R > 0 depending on q
such that
cx
2(1−a)+
2q
1−q +2γ
<
− q (x; ξ)
1−a · χ q ( p q (x; ξ)) · (λ + x
2
)
γ
=
− ln q
1
2
+
1
2 1−q (3 − q)
Z
2
q x
2
4
1−a
·
1
2 q
1 +
q − 1
3 − q
Z
2
q x
2
4
q
1−q
· (λ + x
2
)
γ
< C x
2(1−a)+
2q
1−q +2γ
for |x| > R. This means that (λ + x
2
)
γ
∈ L
1
(ν q,a;ξ ) if and only if
2(1 − a) +
2q
1 − q
+ 2γ < −1 ⇔ γ <
1
2
+
1
q − 1
+ a − 1.
Lemma 3 in the case γ = 0 provides the condition for (q, a) such that ν q,a;ξ has
a finite mass.
Corollary 3 Let 1 ≤ q < 3, a ∈ R \ {0} and ξ ∈ R × q . Then 1 ∈ L
1
(ν q,a;ξ ) if
and only if
either q = 1 or q > 1 with
1
2
−
1
q − 1
< a.
Note that
1
2
−
1
q − 1
< 0
for1 < q < 3.
Corollary 4 Let 1 ≤ q < 3 and a ∈ R \ {0}. Then ln q,a (r ) ∈ L
1
(ν q,a;ξ ) for
ξ ∈ R × q and r ∈ S q .