6 Electronic Communications in Probability
Proposition 1. If − α = ℓ ∈ {1, 2, . . .}, then for any θ ∈ [0, ξ] and any µ with θ
µ
∈ Θ
ℓ
, we have π
µ
r w
θ
∼ C
µ
r
ℓ
sin ℓθ + δ as r → 0, where C
µ
is a finite nonzero constant independent of r and θ .
Our next result is that π
µ
defined by 7 does not change sign on S. Note that this resolves Conjecture 1 in Dai and Harrison [6] for the special class of SRBMs studied in this paper.
Proposition 2. The function
π
µ
does not change sign on S.
4 Properties of
π
µ In this section, we prove the properties of
π
µ
claimed in Section 3. The proof of the main result, Theorem 1, is deferred to Sections 5 and 6.
4.1 Proof of Lemma 1
We first divide 10 by sin θ
µ
− δ − kξ sinδ + k − 1ξ, which is nonzero as a consequence of the assumption on
µ in conjunction with the identity δ + ε = π − ℓξ. Again using this identity, we find after some elementary trigonometry that 10 is equivalent to, with
ω
k
= θ
µ
− 2δ − kξ, c
k
sink ξ sinω
ℓ+k
= −c
k−1
sin ℓ + 1 − kξ sinω
k−1
. To show that this holds for the c
k
defined in 9, we observe that 〈µ, Ref
k
− Rot
k
v
1
〉 sinω
2k−1
= 〈µ, Ref
k−1
− Rot
k−1
v
1
〉 sinω
2k
and that 〈µ, Rot
i
− Rot
j
e
1
〉 = −2 sin j − iξ sinω
i+ j
. After some algebra we also find that
sink ξ sinω
ℓ+k
sinω
2k−1
Y
0≤i j≤ℓ; i, j6=k
〈µ, Rot
i
− Rot
j
e
1
〉 =
sin ℓ − k + 1ξ sinω
k−1
sinω
2k
Y
0≤i j≤ℓ; i, j6=k−1
〈µ, Rot
i
− Rot
j
e
1
〉, and the claim follows.
4.2 Proof of Proposition 1
For simplicity we suppose that k µk = 1. We first investigate the behaviour near zero of π
µ
x, for which we rewrite e
〈µ,x〉
π
µ
x using the determinantal representation 7. A key ingredient is the identity e
r cos η
= I r + 2
P
∞ n=1
cosn ηI
n
r, where I
n
is the modified Bessel function of the first kind. Using this identity, after absorbing e
〈µ,x〉
into the first row, we rewrite the elements on this
Reflected Brownian motion in a wedge 7
row as e
〈µ,x〉
π
µ j
x =
e
kxk〈µ,Rot
j
w
θ
〉
− 〈µ, I − Ref
j
v
1
〉 e
kxk〈µ,Rot
j
w
θ
〉
− e
kxk〈µ,Ref
j
w
θ
〉
〈µ, Rot
j
− Ref
j
v
1
〉 11
= I
kxk + 2
∞
X
n=1
T
n
cosω
2 j
− θ I
n
kxk − 2
〈µ, I − Ref
j
v
1
〉 〈µ, Rot
j
− Ref
j
v
1
〉
∞
X
n=1
T
n
cosω
2 j
− θ − T
n
cosω
2 j
+ θ
I
n
kxk =
I kxk + 2
∞
X
n=1
T
n
cosω
2 j
− θ I
n
kxk − 2〈µ, I − Ref
j
v
1
〉
∞
X
n=1
sinn θ U
n−1
cosω
2 j
I
n
kxk, where we again set
ω
k
= θ
µ
− 2δ − kξ, and T
n
and U
n
are the Chebyshev polynomials of the first and second kind, respectively.
In conjunction with some trigonometry, the above reasoning shows that
e
〈µ,x〉
π
µ j
x = I kxk +
2 sin
δ
∞
X
n=1
h
j,n
θ I
n
kxk, where h
j,n
θ is defined as 1
2 sinn
θ + δU
n
cosω
2 j
− 〈µ, v
1
kv
1
k〉 sinnθ U
n−1
cosω
2 j
+ 1
2 sinn
θ − δU
n−2
cosω
2 j
. We use the convention U
−1
x = 0. Therefore, e
〈µ,x〉
π
µ
x can be expanded in terms of modified Bessel functions of the first kind, and for n ≥ 1 the coefficient in front of I
n
kxk is proportional to ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
h
0,n
θ h
1,n
θ h
2,n
θ · · ·
h
ℓ,n
θ U
ℓ−1
cosω U
ℓ−1
cosω
2
U
ℓ−1
cosω
4
· · · U
ℓ−1
cosω
2 ℓ
.. .
.. .
.. .
U cosω
U cosω
2
U cosω
4
· · · U
cosω
2 ℓ
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ .
12
To see how this follows from 7, note that we may apply elementary determinantal operations to replace a row with elements cos
ω
2 j m
by U
m
cosω
2 j
. The term I kxk is not present in the
expansion in view of the last row in 12 with ones. The condition
θ
µ
∈ Θ
ℓ
guarantees that none of the cos ω
2 j
are equal, and we conclude that the coefficient of I
n
kxk vanishes for n ℓ and that it is proportional to sinℓθ + δ for n = ℓ. Since I
ℓ
r ∼ C r
ℓ
for some constant C 6= 0 as r → 0, this yields π
µ
r w
θ
∼ C
µ
r
ℓ
sin ℓθ + δ.
4.3 Proof of Proposition 2
