Tempered stable laws as random walk limits

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by  Arijit Chakrabarty

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TEMPERED STABLE LAWS AS RANDOM WALK LIMITS
ARIJIT CHAKRABARTY AND MARK M. MEERSCHAERT
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Abstract. Stable laws can be tempered by modifying the L´evy measure to cool
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the probability of large jumps. Tempered stable laws retain their signature power
0
2 law behavior at infinity, and infinite divisibility. This paper develops random walk
models that converge to a tempered stable law under a triangular array scheme.
n
Since tempered stable laws and processes are useful in statistical physics, these
a
J randomwalkmodelscanprovideabasicphysicalmodelforthe underlyingphysical

5 phenomena.
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]
R
P 1. Introduction
.
h
t Tempered stable laws were introduced in physics as a model for turbulent velocity
a
m fluctuations (Koponen (1995); Novikov (1994)). They have also been used in finance
[
(Carr et al. (2002, 2003)) and hydrology (Meerschaert et al. (2008)) as a model of

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transient anomalous diffusion (Baeumer and Meerschaert (2010)). The general class
v
4
oftemperedstabledistributionsforrandomvectorswasdeveloped byRosin´ski (2007).
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4 In short, the L´evy measure of a stable law is modified in the tail to reduce the
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. probability of large jumps. Often this is done in such a way that all moments exist,
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0 but tempering by a power law of higher order is also useful (Sokolov et al. (2004)).
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1 This paper develops random walk models that converge to a tempered stable law.
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v Starting with a random walk in the domain of attraction of a stable law, the basic
i
X
ideaistomodifythetailsofthejumpstomimicthetemperingfunctionofthelimit. A
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a triangular array scheme is essential, since the limit is no longer stable. The results of
this paper areintended to forma useful randomwalk model for natural processes that
are well described by a tempered stable. The main result of this paper is Theorem
4.3, which shows that the weak limit of the row sum of that triangular array is a
tempered stable distribution. In Theorem 4.8, we show that the random walk process
12010 Mathematics Subject Classification. Primary: 60F05; 60E07.
Key words and phrases. Random walk, tempered stable law, triangular array, infinitely divisible
law.
AC was partially supported by the Centenary Postdoctoral Fellowship at the Indian Institute of
Science. MMMwaspartiallysupportedbyNSFgrantsDMS-0125486,DMS-0803360,EAR-0823965
and NIH grant R01-EB012079-01.
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2 ARIJITCHAKRABARTYANDMARK M. MEERSCHAERT
converges to the L´evy process generated by the tempered stable distribution in the
sense of finite-dimensional distributions.
Section 2 gives a brief background of stable distributions and their domains of
attraction. In Section 3, we define tempered stable distributions and the triangular
array model. In Section 4, we state and prove the results regarding the convergence
of the random walk to the tempered stable distribution.
2. Stable limits for random walks
Recall that a random vector X on Rd is infinitely divisible if and only if its char-
acteristic function E[eihλ,Xi] = eψ(λ), where
(2.1) ψ(λ) = iha,λi− 1hλ,Qλi+ eihλ,xi −1−ihλ,xi1(kxk ≤ 1) M(dx),
2 Zx6=0(cid:16) (cid:17)
where a ∈ Rd, Q is a nonnegative definite symmetric d×d matrix with entries in R,
and M is a σ-finite Borel measure on Rd \{0} such that min{1,kxk2}M(dx) <
x6=0
∞. The triple [a,Q,M] is called the L´evy represenRtation, and it is unique
(Meerschaert and Scheffler, 2001, Theorem 3.1.11). The measure M is known as
the L´evy measure of X.
A Rd valued random vector X is said to be stable if and only if for all n ≥ 1, there
exist b > 0 and a ∈ Rd so that X + ... + X =d b X + a , where X ,X ,... are
n n 1 n n n 1 2
i.i.d. copies of X. Clearly, a stable random vector is infinitely divisible. It is well
known that given a stable random vector, either it is Gaussian, or the Gaussian part
is completely absent, i.e., in the L´evy representation (2.1), Q = 0. In this paper,
“stable random vector” will refer to the latter case, i.e., non-Gaussian stable random
vectors. For such a random vector X, P(kXk > ·) regularly varies with index −α
for some 0 < α < 2. Sometimes, X is also referred to as an α-stable random vector.
The L´evy representation of the random vector X is [a,0,M ] for some a ∈ Rd where
0
M (dr,ds) = r−α−1drσ(ds), and σ is a finite non-zero Borel measure on the unit
0
sphere Sd−1 = {x ∈ Rd : kxk = 1}, see for example (Meerschaert and Scheffler, 2001,
Theorem 7.3.16). The measure σ is known as the spectral measure of X. For more
details on stable distributions, the reader is referred to Samorodnitsky and Taqqu
(1994).
TEMPERED STABLE LAWS AS RANDOM WALK LIMITS 3
The domain of attraction of an α-stable random vector X consists of Rd valued
random vectors H such that there exist a sequence of positive numbers (b ) and a
n
sequence (a ) in Rd satisfying
n
(2.2) b−1(H +···+H )−a ⇒ X
n 1 n n
as n → ∞, where ⇒ denotes weak convergence and H ,H ,... are i.i.d. copies of
1 2
H. A necessary and sufficient condition for (2.2) is that V(r) = P(kHk > r) varies
regularly with index −α, and
P(kHk > r, H ∈ D) σ(D)
(2.3) P H ∈ D kHk > r = kHk →
kHk V(r) σ(Sd−1)
(cid:0) (cid:12) (cid:1)
(cid:12)
as r → ∞ for all Borel subsets D of Sd−1 such that σ(∂D) = 0, see for example
(Meerschaert and Scheffler, 2001, Theorem 7.3.16). When α > 1, m = E(H) exists,
and we can center to zero expectation in (2.2) by setting a = nb−1m. Then the limit
n n
X also has zero mean, and its log-characteristic function
(2.4) ψ(λ) = eihλ,xi −1−ihλ,xi M(dx).
Zx6=0(cid:16) (cid:17)
When 0 < α < 1, no centering is required: Set a = 0 in (2.2); then X is a centered
n
stable law with log-characteristic function
(2.5) ψ(λ) = eihλ,xi −1 M(dx).
Zx6=0(cid:16) (cid:17)
See for example (Meerschaert and Scheffler, 2001, Theorem 8.2.16).
Suppose that X is a stable random vector. Let {X(t)} denote the L´evy process
associated with X, so that X(0) = 0 almost surely, X(t) has stationary, independent
increments, and X(1) = X in distribution. Suppose that (2.2) holds with a = 0, let
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b(c) = b , and S(c) = ⌈c⌉ H for c ≥ 0. Then, as c → ∞, we also have process
⌈c⌉ j=1 j
convergence {b(c)−1S(ct)P} ⇒ {X(t)} in the sense of finite dimensional distri-
t≥0 t≥0
butions (Meerschaert and Scheffler, 2001, Example 11.2.18) as well as convergence in
the Skorokhod space D([0,∞),Rd) of right continuous functions with left hand lim-
its, in the J topology (Meerschaert and Scheffler, 2004, Theorem 4.1). The random
1
vectors X(t) have smooth density functions P(x,t) that solve a fractional differential
equation ∂ P(x,t) = D∇αP(x,t) for anomalous diffusion (Meerschaert et al. (1999)).
∂t σ
The fractional derivative operator ∇α f(x) is defined, for suitable functions f(x)
M
with Fourier transform fˆ(λ) = eihλ,xif(x)dx, as the inverse Fourier transform of
R
4 ARIJITCHAKRABARTYANDMARK M. MEERSCHAERT
(−ihλ,si)ασ(ds)fˆ(λ), and D > 0 is a positive constant that depends on choice
ksk=1
Rof the norming sequence b(c). The random walk S provides a physical model for par-
n
ticle jumps, whose ensemble behavior is approximated by the stable density functions
P(x,t). For example, the random walk can be simulated to solve the fractional diffu-
sion equation, a numerical method known as particle tracking (Zhang et al. (2006)).
The purpose of this paper is to develop analogous random walk models for tempered
stables.
3. Tempered random walks
This section develops random walk models that converge to a tempered stable,
using a triangular array scheme. As in Rosin´ski (2007), we define a d-dimensional
proper tempered α-stable random vector to be an infinitely divisible random vector
with L´evy representation [a,0,M] with
(3.1) M(dr,ds) = r−α−1q(r,s)drσ(ds)
for r > 0 and s ∈ Sd−1. Here α ∈ (0,2), σ is a finite Borel measure on the unit
sphere Sd−1 = {x ∈ Rd : kxk = 1}, and q : (0,∞) × Sd−1 is a Borel measur-
able function such that for all s ∈ Sd−1, q(·,s) is non-increasing, q(0+,s) = α and
q(∞,s) = 0. We also assume that q is continuous in the second variable, i.e., q(r,·)
is continuous for all r > 0. In Rosin´ski (2007), the assumption is that q(0+,s) = 1.
However, a simple reparametrization yields q(0+,s) = α. It is also assumed in
Rosin´ski (2007) that q(·,s) is completely monotone, but we do not need that as-
sumption in this paper. Note that tempered stable random vectors are full dimen-
sional, since the L´evy measure is not concentrated on any lower dimensional subspace
(Meerschaert and Scheffler, 2001, Proposition 3.1.20). Let H be a random vector in
the domain of attraction of X such that (without loss of generality) P(H = 0) = 0.
We will define a random walk that approximates the tempered stable using a condi-
tional tempering of the jumps. Define a function π : (0,∞)×Sd−1 → R by
∞
(3.2) π(u,s) = uα r−α−1q(r,s)dr.
Z
u
TEMPERED STABLE LAWS AS RANDOM WALK LIMITS 5
From the fact that the function q is bounded above by α, it is immediate that the
integral on the right hand side is finite. Clearly,
∂π(u,s) ∞
= αuα−1 r−α−1q(r,s)dr−u−1q(u,s)
∂u Z
u
∞
= αuα−1 r−α−1{q(r,s)−q(u,s)}dr ≤ 0,
Z
u
the inequality following from the fact that q(·,s) is non-increasing. Thus π(·,s) is also
non-increasing. A simple application of the L’Hoˆpital’s rule yields that π(0+,s) = 1
and π(∞,s) = 0.
Define a family of probability measures on (0,∞) by Π(du,s) = − ∂ π(u,s)du.
∂u
Since q is a measurable function, for every Borel set A ⊂ (0,∞), Π(A,·) is a mea-
surable function from Sd−1 to R. Hence there exists a random variable T > 0 whose
conditional distribution given H is Π(·,H/kHk).
Now we construct the tempered random walk. For t > 0, define
H
(3.3) Ht = (kHk∧tT).
kHk
Let {(H : T ) : j ≥ 1} be i.i.d. copies of (H,T). Suppose v → ∞ is a sequence of
j j n
positive numbers. Define Y := (kH k∧v T )H /kH k for n,j ≥ 1, and let
nj j n j j j
(3.4) S (k) := Y +···+Y .
n n1 nk
The next section shows that, for suitably chosen truncationthresholds v , therandom
n
walk (3.4) is asymptotically tempered stable.
4. Limits of tempered random walks
The results in this section show that for suitable truncation thresholds v and
n
centering, the random walk (3.4) converges to a tempered stable. The form of the
centering is then related to thecase with no tempering. We beginwith a few technical
results. Recall that for sigma-finite Borel measures µ ,µ on Γ = Rd \ {0}, µ →v µ
n n
(vagueconvergence) meansthatµ (D) → µ(D)forBorelsetsD ⊂ Γthatarebounded
n
away from zero, for which µ(∂D) = 0.
Lemma 4.1. Suppose that Ht is defined by (3.3). Then
P(t−1Ht ∈ ·) 1
v
→ M(·)
P(kHk > t) σ(Sd−1)
6 ARIJITCHAKRABARTYANDMARK M. MEERSCHAERT
on Γ as t → ∞, where M is as in (3.1).
For theproof, we shall need the following result fromweak convergence. This result
is similar to Theorem 1.3.4 in van der Vaart and Wellner (1996), which states the
corresponding result for probability measures. Although the result is well known, we
include the proof here for completeness, since we could not locate a suitable reference.
Lemma 4.2. Suppose (µ ) is a sequence of measures on some metric space S
n
equipped with the Borel sigma-field, converging weakly to some finite measure µ.
Then, for all bounded non-negative upper semicontinuous functions f, we have
limsup fdµ ≤ fdµ.
n→∞ n
R R
Proof. Since f is bounded and non-negative, we can assume without loss of generality
that 0 ≤ f < 1. For k ≥ 1, denote f := 1 k 1 f−1 i−1,1 . It is easy to see
k k i=1 k
that f ≤ f ≤ f + 1. Thus, for fixed k ≥ 1, P (cid:0) (cid:2) (cid:1)(cid:1)
k k
k
1 i−1
limsup fdµ ≤ limsup f dµ ≤ limsupµ f−1 ,1
n k n n
Z Z k (cid:18) (cid:20) k (cid:19)(cid:19)
n→∞ n→∞ Xi=1 n→∞
k
1 i−1 1
≤ µ f−1 ,1 ≤ fdµ+ µ(S),
k (cid:18) (cid:20) k (cid:19)(cid:19) Z k
Xi=1
by the Portmanteau Theorem (Theorem 2.1, Billingsley (1968)) and the observation
that f−1 i−1,1 is a closed set because f is upper semicontinuous. Since µ is a finite
k
measure,(cid:2)this co(cid:1)mpletes the proof. (cid:3)
Proof of Lemma 4.1. Note that if q(r,·) is continuous for all r > 0, then the same
is true for π(u,·) for all u > 0. To see this, suppose that the former holds. Fix
u > 0 and s ,s ∈ Sd−1 such that s → s. By the assumption on q and the dominated
n n
convergence theorem, it follows that lim ∞r−α−1q(r,s )dr = ∞r−α−1q(r,s)dr.
n→∞ u n u
Then π(u,s ) → π(u,s), so π(u,·) is continuRous for every u > 0. ARsimilar argument
n
shows that ∂π(u,·) is continuous for every u > 0.
∂u
For the proof, we shall use the fact that as t → ∞,
H α
(4.1) P(kHk > t)−1P t−1kHk ∈ dr, ∈ ds →v r−α−1drσ(ds)
(cid:20) kHk (cid:21) σ(Sd−1)
TEMPERED STABLE LAWS AS RANDOM WALK LIMITS 7
which is a restatement of Theorem 8.2.18 in Meerschaert and Scheffler (2001). It
suffices to show that for every closed set A ⊂ Rd \{0},
(4.2) limsupP(t−1Ht ∈ A)P(kHk > t)−1 ≤ σ(Sd−1)−1M(A),
t→∞
and that for every ε > 0,
(4.3) lim P(t−1Ht ∈ Bc)P(kHk > t)−1 = σ(Sd−1)−1M(Bc),
ε ε
t→∞
where B is the closed ball of radius r centered at origin: B := {x ∈ Rd : kxk ≤ r}.
r r
Fix a closed set A ⊂ Rd \{0} and note that
∞ ∞ H
P(t−1Ht ∈ A) = 1 ((r ∧u)s)Π(du,s) P t−1kHk ∈ dr, ∈ ds
A
Z0 ZSd−1Z0 (cid:0) kHk (cid:1)
Fix sequences r and s so that r > 0, r → r > 0, s ∈ Sd−1 and s → s. Then
n n n n n n
∞
limsup 1 ((r ∧u)s )Π(du,s )
A n n n
Z
n→∞ 0
∞ ∂π(u,s )
≤ 1− liminf1Ac((rn ∧u)sn) − n du
Z n→∞ (cid:18) ∂u (cid:19)
0
∞ ∂π(u,s) ∞
= 1− − liminf1Ac((rn ∧u)sn)du ≤ 1A((r ∧u)s)Π(du,s)
Z (cid:18) ∂u (cid:19) n→∞ Z
0 0
byFatou’sLemma, continuity of ∂ π(u,·),andthefactthatAc isopen. Then (r,s) 7→
∂u
∞1 ((r ∧u)s)Π(du,s) is upper semicontinuous. From (4.1), it follows that for all
0 A
εR > 0, the restriction of P(kHk > t)−1P [t−1kHk ∈ dr,H/kHk ∈ ds] to Bc converges
ε
weakly to that of αr−α−1drσ(ds)/σ(Sd−1). Thus, by Lemma 4.2 and the fact that A
is bounded away from zero, it follows that
P(t−1Ht ∈ A) ∞ ∞ α
limsup ≤ 1 ((r∧u)s)Π(du,s) r−α−1drσ(ds)
t→∞ P(kHk > t) Z0 ZSd−1Z0 A σ(Sd−1)
∞ r α
= 1 (us)Π(du,s) r−α−1drσ(ds)
A
Z0 ZSd−1(cid:26)Z0 (cid:27) σ(Sd−1)
∞ ∞ α
+ 1 (rs)Π(du,s) r−α−1drσ(ds) =: I +I .
Z0 ZSd−1(cid:26)Zr A (cid:27)σ(Sd−1) 1 2
Note that Lemma 4.2 applies since Π(du,s) is a probability measure for each s.
A change of the order of integration yields that
1 ∞ ∂π(u,s)
I = 1 (us)u−α − σ(ds)du.
1 σ(Sd−1) Z0 ZSd−1 A (cid:18) ∂u (cid:19)
8 ARIJITCHAKRABARTYANDMARK M. MEERSCHAERT
It is immediate that I = σ(Sd−1)−1 ∞ 1 (rs)απ(r,s)r−α−1drσ(ds). Thus,
2 0 Sd−1 A
R R
P(t−1Ht ∈ A)
limsup
P(kHk > t)
t→∞
(4.4)
1 ∞ ∂π(r,s)
≤ 1 (rs) απ(r,s)−r r−α−1drσ(ds).
σ(Sd−1) Z0 ZSd−1 A (cid:26) ∂r (cid:27)
From (3), it follows that απ(r,s)−r ∂ π(r,s) = q(r,s). Plugging this in (4.4) yields
∂r
that
P(t−1Ht ∈ A) 1 ∞ M(A)
limsup ≤ 1 (rs)q(r,s)r−α−1drσ(ds) = ,
A
t→∞ P(kHk > t) σ(Sd−1) Z0 ZSd−1 σ(Sd−1)
thus showing (4.2). For (4.3), note that as t → ∞,
∞ H
P(t−1Ht ∈ Bc) = π(ε,s)P t−1kHk ∈ dr, ∈ ds
ε Zε ZSd−1 (cid:18) kHk (cid:19)
∞ α
∼ P(kHk > t) π(ε,s) r−α−1drσ(ds)
Zε ZSd−1 σ(Sd−1)
1 P(kHk > t)
= P(kHk > t) ε−απ(ε,s)σ(ds) = M(Bc),
σ(Sd−1) ZSd−1 σ(Sd−1) ε
by (4.1), the fact that π(ε,·) is continuous, and the definition of π. (cid:3)
The following theorem is the main result of this paper. It shows that the tempered
random walk (3.4) converges weakly to a tempered stable, for suitable tempering
constants v and suitable centering vectors a .
n n
Theorem 4.3. For n ≥ 1 let b := inf{x : P(kHk > x) ≤ n−1} . If the sequence (v )
n n
satisfies
(4.5) lim v−1b = σ(Sd−1)1/α,
n n
n→∞
then,
(4.6) v−1S (n)−a ⇒ ρ
n n n
where ρ is an infinitely divisible probability measure on Rd with L´evy representation
[0,0,M], and (a ) is defined by
n
(4.7) a := n xP(v−1Y ∈ dx).
n Z n n1
{kxk<1}
TEMPERED STABLE LAWS AS RANDOM WALK LIMITS 9
Proof. Note that nP(kHk > b ) → 1 (Resnick, 2007, p. 24). Since P(kHk > ·) varies
n
regularly with index −α, (4.5) implies P(kHk > v )/P(kHk > b ) → σ(Sd−1). Then
n n
(4.8) lim nP(kHk > v ) = σ(Sd−1).
n
n→∞
An appeal to Lemma 4.1 shows that
(4.9) nP(v−1Y ∈ ·) →v M(·),
n n1
and then it suffices to check (Meerschaert and Scheffler, 2001, Theorem 3.2.2):
(4.10) limlimsupnv−2EkY 1(kY k ≤ v δ)k2 = 0.
n n1 n1 n
δ↓0 n→∞
For this, note that kY k21(kY k ≤ v δ) ≤ kH k21(kH k ≤ v δ) + v2T21(v T ≤
n1 n1 n 1 1 n n 1 n 1
v δ,kH k > v δ) ≤ kH k21(kH k ≤ v δ)+v2δ21(kH k > v δ). Since P(kHk > ·) is
n 1 n 1 1 n n 1 n
regularly varying with index −α and α < 2, by Karamata’s theorem (Resnick, 2007,
Theorem 2.1) it follows that E[kHk21(kHk ≤ v δ)] ∼ (v δ)2P(kHk > v δ)α/(2 −
n n n
α) ∼ δ2−αv2P(kHk > v )α/(2 − α) as n → ∞. Using the regular variation of
n n
P(kHk > ·) once again, it is immediate that E[v2δ21(kHk > v δ)] = v2δ2P(kHk >
n n n
v δ) ∼ δ2−αv2P(kHk > v ). To complete the proof, use (4.8) to obtain C < ∞ such
n n n
that for all δ > 0,
(4.11) limsupnv−2EkY 1(kY k ≤ v δ)k2 ≤ Cδ2−α.
n n1 n1 n
n→∞
(cid:3)
The next two results show that the centering constants in (4.6) can be chosen in
the same way as for (2.2) when α 6= 1. We say that a tempered stable law with index
0 < α < 1 is centered if its log-characteristic function can be written in the form (2.5)
where M is given by (3.1).
Theorem 4.4. If 0 < α < 1 and v satisfies (4.5), then v−1S (n) ⇒ ρ where ρ is
n n n 1 1
centered tempered stable.
Proof. In view of Theorem 4.3 and (2.1), it suffices to show that, if a is defined by
n
(4.7), then a → xM(dx). Fix 0 < ε < 1 and note that
n {kxk<1}
R
a = n xP(v−1Y ∈ dx)+n xP(v−1Y ∈ dx) =: I +I .
n n n1 n n1 1 2
Z Z
{ε<kxk<1} {kxk≤ε}
10 ARIJITCHAKRABARTYANDMARK M. MEERSCHAERT
Clearly, by (4.9), lim I = xM(dx). Thus, it suffices to show
n→∞ 1 {ε<kxk<1}
R
(4.12) limlimsupkI k = 0.
2
ε↓0 n→∞
Note kI k ≤ nv−1E[kY k1(kY k ≤ v ε)] ≤ nv−1[E(kHk1(kHk ≤ v ε)) +
2 n n1 n1 n n n
v εP(kHk > v ε)]. Since α < 1, Karamata along with regular variation yields
n n
E(kHk1(kHk ≤ v ε)) ∼ v ε1−αP(kHk > v )α/(1− α). Then (4.12) follows, using
n n n
(cid:3)
(4.8) and regular variation.
Theorem 4.5. Suppose α > 1 and that for some β > α,
(4.13) limsup sup u1−β[α−q(u,s)] < ∞
u↓0 s∈Sd−1
where v satisfies (4.5). Then, v−1[S (n) − nE(H)] ⇒ ρ where ρ is an infinitely
n n n 2 2
divisible law with no Gaussian component, L´evy measure M and mean
∞ r
(4.14) m = −α (r −u)sΠ(du,s) r−α−1drσ(ds).
Z0 ZSd−1(cid:26)Z0 (cid:27)
Proof. Let
∞ r
(4.15) θ := α (r −u)sΠ(du,s) r−α−1drσ(ds)+ xM(dx).
Z0 ZSd−1(cid:26)Z0 (cid:27) Z{kxk≥1}
We start with showing that the integrals on the right hand side of (4.15) are well
r r
defined. Let g(r,s) := (r − u)sΠ(du,s). It is easy to see that kg(r,s)k ≤ (r −
0 0
u)Π(du,s) =: g¯(r,s). CRlearly, R
r ∂π(u,s)
(4.16) g¯(r,s) = r[1−π(r,s)]+ u du,
Z ∂u
0
and hence, ∂ g¯(r,s) = 1 − π(r,s) = rα ∞u−α−2+βu1−β[α − q(u,s)]du =
∂r r
rα 1u−α−2+βu1−β[α − q(u,s)]du + O(rα). CleRarly (4.13) holds with β replaced
r
byRβ ∧ 2. Thus, without loss of generality, we can assume that β ≤ 2. Define
K = sup{u1−β[α−q(u,s)] : s ∈ Sd−1,0 < u ≤ 1}. By hypothesis, K < ∞. Thus,
1 α−q(u,s) 1
rα u−α−2+β du ≤ Krα u−α−2+βdu ≤ K′rβ−1,
Z uβ−1 Z
r r
where K′ = K/(α +1−β) > 0 since β ≤ 2 and α > 1. Thus, as r ↓ 0, ∂ g¯(r,s) =
∂r
O(rβ−1) uniformly in s, and hence for some C < ∞,
(4.17) g¯(r,s) ≤ Crβ, r ≤ 1,s ∈ Sd−1.