144
H. Matsuzoe and A. Takatsu
Proof Since we have that
n <
1
2
+
1
q − 1
+ n − 1
for1 < q < 3,
we apply Lemme 5 together with the change of variables to have that
p q (x; ξ)
(n−1)(q−1)+q
· x
2γ
∈ L
1
(dx)
for 0 ≤ γ ≤ n.
Moreover, the fact that
− q (x; ξ) ≥ − ln q
1
Z q σ
> 0
for x ∈ R completes the proof of the corollary.
Combining the computation that
∂
∂μ
q (x; μ, σ ) =
2
(3 − q)
·
1
(Z q σ ) 1−q σ
x − μ
σ
,
∂
∂σ
q (x; μ, σ ) = −
1
(Z q σ ) 1−q σ
1 −
x − μ
σ
2
,
(6.7)
with Corollary 6 in the case n = 2, we conclude that
x →
∂
∂s 1
∂
∂s 2
ρ
(q,a)
(x; ξ 1 , ξ 2 )
(ξ,ξ )
is integrable on R for ξ ∈ R × q,a . Since the function x → ρ
(q,a)
(x; ξ 1 , ξ 2 ) is integrable on R for (ξ 1 , ξ 2 ) ∈ (R × q,a )
2 , the dominated convergence theorem implies
that
∂
∂s 1
∂
∂s 2
D (q,a) ( p q (·; ξ 1 ), p q (·; ξ 2 )
(ξ,ξ )
= −
R
∂
∂s 2
ln q,a
p q (x; ξ 2 )
·
∂
∂s 1
ln q,a
p q (x; ξ 1
· exp
q,a
ln q,a
p q (x; ξ 1
(ξ,ξ )
dx
= −
1
j=0
b 2
j
R
∂
∂s 2
q (x; ξ 2 ) ·
∂
∂s 1
q (x; ξ 1 )
(ξ,ξ )
·
− q (x, ξ)
− j p q (x; ξ) 2q−1 dx
for s i ∈ {μ i , σ i }. This quantity provides a Riemannian metric on S q,a .
Definition 7 For s, t ∈ {μ, σ }, define a function g
(q,a)
st
: R × q,a → R by
g
(q,a)
st
(ξ ) :=
R
∂
∂s
ln q,a
p q (x; ξ)
·
∂
∂t
ln q,a
p q (x; ξ)
· exp
q,a
ln q,a
p q (x; ξ)
dx.