Appendix A. A.1 Proof of Theorem 1

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Appendix A

Throughout the proofs, let K > 0 be a constant that may change from line to line. All o_{(P )}- and O_{(P )}-symbols are to be understood with respect to n → ∞. Recall the definition of Fn as the σ- field generated by {εu, xu, u ≤ n}. We write Ei[ · ] = E[ · | Fi]. Without further mention, we assume conditions (C1)–(C11) to hold throughout, even though some of the following results may hold under a subset of these conditions. The proof of Theorem 1 provided in Appendix A.1 requires three propositions, which are proved in Appendix A.2.

A.1 Proof of Theorem 1

The quantity of interest in Theorem 1 is zb_α_n_,n zαn,n

= bh^1/δ_n+1^◦( bθ_n) σn+1

· bzαn

zαn

, z ∈ {DRM, ξ}. (A.1)

For the first term on the right-hand side of (A.1) we require Proposition 1. We have

bh^1/δ_n+1^◦( bθn)

σ_n+1 − 1 = O_P(n^−1/2).

Since log(1 + x) = x + o(x) as x → 0, Proposition 1 implies log

bh^1/δ_n+1^◦( bθn)/σn+1

= OP(n^−1/2).

With this and (A.1), we get

log bz_α_n_,n zαn,n

!

= log





bh^1/δ_n+1^◦( bθ_n) σn+1



+ log

zbαn

zαn

= OP(n^−1/2) + log

bzαn

zαn

. Hence, since√

k/ log{k/(n[1 − αn])} = o(n^1/2), Theorem 1 follows if we show that 1

bγ

√k

log{k/(n[1 − αn])}log

bz_α_n zαn

d

−→

(n→∞)N (0, 1), z ∈ {DRM, ξ}. (A.2)

To prove (A.2), we require two results. First, that high—but within-sample—quantiles can be esti- mated consistently based on the standardized residuals:

Proposition 2. We have

√ k log





Ub_(k+1) U (n/k)



= OP(1).

Second, the Hill (1975) estimatorbγ based on the standardized residuals must be√

k-consistent for the tail shape γ of the innovation distribution (see condition (C11)):

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Proposition 3. We have

√

k (bγ − γ) −→^d

(n→∞)N (0, γ²).

Now, for both z = DRM and z = ξ, (A.2) follows using Propositions 2 and 3 as in the proof of Theorem 1 in Hoga (2019+). This proves Theorem 1.

A.2 Proofs of Propositions 1–3

The proofs of Propositions 1–3 draw on work of Kim and Lee (2016), who show Proposition 3 for APARCH models without covariates (i.e., with π^◦ = 0). In the following, we require the QMLE to be√

n-consistent, i.e.,

√n

θb_n− θ^◦

= O_P(1). (A.3)

This, however, follows directly from Francq and Thieu (2019, Theorem 2) under conditions (C1)–

(C9).

We require some additional notation. Define the q × q-matrix

B = B(θ) =





β1 . . . βq−1 βq

I_q−1 0_q−1



,

where I_q−1 denotes the (q − 1) × (q − 1)-identity matrix and 0_q−1 a (q − 1) × 1-vector of zeros. Denote the (a, b)-th element of B by B(a, b) and put B₀ = B(θ^◦) for short. Write







hi(θ) ... h_i−q+1(θ)







=





β₁ . . . β_q−1 β_q Iq−1 0q−1











hi−1(θ) ... h_i−q(θ)





 +





ω +Pp

j=1fj(εi−j) + π⁰xi−1

0_q−1



, (A.4)

where fj(x) = α+,j(x)^δ₊^◦+ α−,j(x)^δ₋^◦. For brevity, set hi = hi(θ^◦) = σ_i^δ^◦. Finally, define the neighbor- hoods of θ^◦

N_n(η) =θ : |θ − θ^◦| ≤ η/√ n

and N_n⁻(η) = N_n(η) \ {θ^◦}.

To prove Propositions 1–3, we require a series of lemmas. Some lemmas (e.g., Lemmas 1 and 4) are directly from the literature and are reproduced here for easier reference. Other lemmas (e.g., Lemmas 2 and 3) are extensions of the results in Kim and Lee (2016) to APARCH–X models.

Lemma 1 (Francq and Thieu, 2019, Lemma 2). There exists s > 0, such that E |ε₁|^s< ∞.

Lemma 2 (cf. Kim and Lee, 2016, Lemma 9). Let η > 0. Then, there exist r ∈ (0, 1) and K > 0,

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such that for large n

sup

θ∈Nn(η)

B^j(1, b) ≤ Kr^j for j ∈ N, b = 1, . . . , q. (A.5) Furthermore, there exists a sequence of positive Fi−1-measurable random variables {Wi}, such that E W_i^ν < ∞ for some ν > 0, and

sup

θ∈Nn(η)

h_i(θ) ≤ W_i. (A.6)

Proof: Kim and Lee (2016, Lemma 9) show (A.5), so it remains to prove (A.6). Under assumption (C5),

h_i(θ) = ω

∞

X

j=0

B^j(1, 1) +

∞

X

j=0

B^j(1, 1)

p

X

l=1

f_l(ε_i−j−l) +

∞

X

j=0

B^j(1, 1)π⁰x_i−j−1 (A.7)

is the strictly stationary solution to the stochastic recurrence equation in (A.4). Clearly, f_l(ε_i−j−l) ≤ max{α_+,l, α−,l}[(ε_i−j−l)^δ₊^◦+ (ε_i−j−l)^δ₋^◦]

≤ K|ε_i−j−l|^δ^◦

for some K > 0 independent of θ, by compactness of Θ stipulated in (C3). Thus, also exploiting (A.5), we obtain

hi(θ) ≤ K

∞

X

j=0

r^j+ K

∞

X

j=0 p

X

l=1

r^j|ε_i−j−l|^δ^◦+ K

∞

X

j=0 M

X

m=1

r^jxm,i−j−1.

For a suitable choice of s > 0, the s-th moment of the right-hand side r.v. is finite by Lemma 1 and (C1). Since the r.v. on the right-hand side does not depend on θ, (A.6) follows. Lemma 3 (cf. Kim and Lee, 2016, Lemma 10). There exists r ∈ (0, 1) and a F₀-measurable r.v. V ≥ 0 with E V^ν < ∞ for some ν > 0 and for large n

sup

θ∈Nn(η)

bhi(θ) − hi(θ)

≤ rⁱV for all i = 1, 2, . . . .

The above lemma can be proven as Lemma 10 in Kim and Lee (2016), since the difference bhi(θ) − h_i(θ) does not depend on the covariates. The next lemma is required to derive some inequalities.

Lemma 4 (Kim and Lee, 2016, Lemma 11). Let ε > 0. Then, for each j ∈ N, B^j((1 + ε)θ^◦)(1, 1) ≤ (1 + ε)^jB₀^j(1, 1)

and there exists n₀ ∈ N, such that for all n ≥ n0 and j ∈ N B₀^j(1, 1) ≤ (1 + ε)^j inf

θ∈Nn(η)B^j(1, 1). (A.8)

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Lemma 5 (cf. Kim and Lee, 2016, Lemma 12). Let η > 0. Then, for every w > 0

lim sup

n→∞

E

sup

θ∈Nn⁻(η)

|h₁(θ^◦) − h₁(θ)|

|θ^◦− θ|h₁(θ)

w

< ∞.

Proof: Let w > 0. Using (A.7), we write h1(θ^◦) − h1(θ)

h1(θ) = ω^◦P∞

j=0B₀^j(1, 1) − ωP∞

j=0B^j(1, 1) h1(θ)

+

∞

X

j=0 p

X

l=1

{B₀^j(1, 1) − B^j(1, 1)}fl(ε1−j−l) h₁(θ)

+

∞

X

j=0

B₀^j(1, 1){(π^◦)⁰x1−j−1− π⁰x1−j−1}/h₁(θ)

+

∞

X

j=0

{B₀^j(1, 1) − B^j(1, 1)}π⁰x1−j−1/h1(θ)

=: (I) + (II) + (III) + (IV ).

Thanks to Lemma 1, the terms |(I)|/|θ^◦− θ| and |(II)|/|θ^◦− θ| can be bounded by r.v.s with finite w-th moment, exactly as in the proof of Lemma 12 in Kim and Lee (2016).

Consider (III). From (A.7) and the non-negativity of the coefficients in θ,

h1(θ) ≥ ω + B^j(1, 1)πmxm,1−j−1 (A.9) for any j ≥ 0 and m = 1, . . . , M . Using the non-negativity of B₀^j(1, 1), we bound

|(III)| =

∞

X

j=0 M

X

m=1

B₀^j(1, 1){π^◦_m− π_m}x_m,1−j−1/h1(θ)

≤ |θ^◦− θ|

∞

X

j=0 M

X

m=1

B₀^j(1, 1)x_m,1−j−1 ω + B^j(1, 1)πmxm,1−j−1

≤ |θ^◦− θ|

∞

X

j=0 M

X

m=1

B₀^j(1, 1) B^j(1, 1)

1 πm

B^j(1, 1)^π_ω^mx_m,1−j−1 1 + B^j(1, 1)^π_ω^mxm,1−j−1

≤ K|θ^◦− θ|

∞

X

j=0 M

X

m=1

{(1 + ε)r^s}^j|x_m,1−j−1|^s.

where we used (A.9) and |π_m^◦ − π_m| ≤ |θ^◦ − θ| in the second step, and (A.8) and the inequality x/(1 + x) ≤ x^sfor all x ≥ 0 and s ∈ (0, 1] in the final step. If s is chosen such that Ekx₁k^ws < ∞ and (e.g.) ε = (1 − r^s)/(2r^s), then the w-th moment of |(III)|/|θ^◦− θ| is finite.

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Finally, consider

|(IV )| ≤

∞

X

j=0 M

X

m=1

|B₀^j(1, 1) − B^j(1, 1)|πmxm,1−j−1/h1(θ).

From the equation at the top of p. 7 in the supplementary material of Kim and Lee (2016), there exists n0 ∈ N, such that for all n ≥ n0

sup

θ∈Nn(η)

|B^j₀(1, 1) − B^j(1, 1)| ≤ K|θ^◦− θ|j{B^j((1 + ε)θ^◦)(1, 1)}. (A.10) Further, by (A.5) and Lemma 4 there exists n₁∈ N, such that for all n ≥ n1 and θ ∈ N_n(η)

B^j((1 + ε)θ^◦)

{B^j(1, 1)}^1−s ≤ (1 + ε)^2j{B^j(1, 1)}^s≤ K[(1 + ε)²r^s]^j. (A.11) Then, for all n ≥ max{n0, n1} and θ ∈ N_n(η),

|(IV )|

(A.9)

≤

∞

X

j=0 M

X

m=1

|B₀^j(1, 1) − B^j(1, 1)|π_mx_m,1−j−1 ω + B^j(1, 1)πmxm,1−j−1 (A.10)

≤ K|θ^◦− θ|

∞

X

j=0 M

X

m=1

jB^j((1 + ε)θ^◦)(1, 1) B^j(1, 1)

B^j(1, 1)^π_ω^mx_m,1−j−1 1 + B^j(1, 1)^π_ω^mxm,1−j−1

≤ K|θ^◦− θ|

∞

X

j=0 M

X

m=1

jB^j((1 + ε)θ^◦)(1, 1)

{B^j(1, 1)}^1−s x^s_m,1−j−1

(A.11)

≤ K|θ^◦− θ|

∞

X

j=0 M

X

m=1

j[(1 + ε)²r^s]^j|x_m,1−j−1|^s,

where the third inequality follows from x/(1 + x) ≤ x^s for all x ≥ 0 and s ∈ (0, 1]. Again, if s and ε are chosen such that Ekx_ik^ws < ∞ and (1 + ε)²r^s < 1, then the w-th moment of |(IV )|/|θ^◦ − θ| is finite. Combining the results for (I)–(IV ), the conclusion follows.

We can now prove the first proposition:

Proof of Proposition 1: We show that ^b^hⁿ⁺¹^{( b}^θⁿ⁾

σ_n+1^δ◦ − 1 = O_P(n^−1/2). From this, the conclusion follows easily using the mean value theorem. Recall that hn+1= hn+1(θ^◦) = σ^δ_n+1^◦ . Write

sup

θ∈Nn(η)

bh_n+1(θ) h_n+1 − 1

≤ sup

θ∈Nn(η)

bh_n+1(θ) − h_n+1(θ) h_n+1

+ sup

θ∈Nn(η)

h_n+1(θ) − h_n+1 h_n+1

=: (I) + (II). (A.12)

Consider (I). From Lemma 3, for r ∈ (0, 1) sup

θ∈Nn(η)

|bh_i(θ) − h_i(θ)| ≤ rⁱV, i = 1, 2, . . .

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for n large and V ≥ 0 with E |V |^ν < ∞ for some ν > 0. Hence, by Markov’s inequality, P

(√ n sup

θ∈Nn(η)

|bhn+1(θ) − hn+1(θ)| > ε )

≤ Pn

rⁿ⁺¹V > ε/√ n

o

≤ (ε/√

n)^−νE[rⁿ⁺¹V ]^ν

≤ Kn^ν/2(r^ν)ⁿ⁺¹E |V |^ν −→

(n→∞)0.

Since h_n+1≥ ω^◦ > 0 almost surely, we conclude that (I) = o_P(n^−1/2).

For (II), consider

h_n+1(θ) − h_n+1

h_n+1 = h_n+1(θ) − h_n+1

h_n+1(θ) ·h_n+1(θ) h_n+1 . By Markov’s inequality and Lemma 5,

P (√

n sup

θ∈Nn(η)

|h_n+1(θ) − hn+1| h_n+1(θ) > K

)

≤ K⁻¹E



 sup

θ∈Nn⁻(η)

|h_n+1(θ) − hn+1|

|θ^◦− θ|h_n+1(θ)





√n|θ^◦− θ|

= O(K⁻¹) −→

(K→∞)0.

Thus,

sup

θ∈Nn⁻(η)

|h_n+1(θ) − hn+1|

h_n+1(θ) = OP(n^−1/2).

Furthermore, we obtain sup

θ∈Nn(η)

hn+1(θ) h_n+1

≤ 1 ω^◦ sup

θ∈Nn(η)

hn+1(θ)

≤ W_n+1/ω^◦= OP(1),

since Wn+1 = OP(1) by Lemma 2. Thus, (II) = OP(n^−1/2). Combining this with (I) = oP(n^−1/2), we get from (A.12) that

sup

θ∈Nn(η)

bh_n+1(θ) hn+1

− 1

= OP(n^−1/2).

By (A.3),

Pn

θb_n∈ N/ _n(η)o

≤ Pn√

n| bθ_n− θ^◦| ≥ ηo

≤ ε/2 (A.13)

for η > 0 sufficiently large. The desired result follows. The proofs of Propositions 2 and 3 require additional notation. Recalling that h_i= h_i(θ^◦), set

Ubi(θ) = Ui

"

1 +hi(θ) − bhi(θ) bhi(θ)

#1/δ^◦

1 +hi− h_i(θ) hi(θ)

1/δ^◦

, A_i(ξ, θ) = Iⁿ

Ubi(θ)>e^ξ/

√

kU (n/k)o,

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Ai(ξ) = Iⁿ

Ui>e^ξ/

√ kU (n/k)

o.

In particular, bU_i( bθ_n) = bU_i is the standardized residual. Note in the definition of the bU_i(θ)’s that the quantities in square brackets are positive almost surely and, hence, the power is well defined for any δ^◦ > 0. We also set

C_i(ξ, θ) =



log (

Ub_i(θ) U (n/k)

) + ξ

√ k





+

,

C_i(ξ) =

"

log

U_i U (n/k)

+ ξ

√ k

#

+

.

Lemma 6 (cf. Kim and Lee, 2016, Lemma 13). Let η > 0 and mn → ∞ with m_n < n, as n → ∞.

Then, there exist r0 ∈ (0, 1), ν₀ > 0, a Fi−1-measurable r.v. Vi ≥ 0 with sup_i∈NE |Vi|^ν⁰ < ∞, and F_i−1-measurable ∆n,i= ∆n,i(η) ≥ 0, such that

1 n

n

X

i=1

E ∆_n,i= O(1) and max

i=1,...,n

∆n,i

√n = o_P(1). (A.14)

Moreover, there exists η0> 0, such that with probability approaching 1 we have, as n → ∞,

Ai(ξ, η, −η0) ≤ Ai(ξ, θ) ≤ Ai(ξ, η, η0), (A.15) C_i(ξ, η, −η₀) ≤ C_i(ξ, θ) ≤ C_i(ξ, η, η₀) (A.16) for all θ ∈ Nn(η), i = mn, . . . , n and ξ ∈ R, where

A_i(ξ, η, η₀) = I⁽

Ui[^1+rⁱ0η0Vi]

1+^η0∆n,i^√_n

>e^ξ/

√ kU (n/k)

),

C_i(ξ, η, η₀) =





log







U_i1 + r₀ⁱη₀V_ih

1 +^η⁰^√^∆_n^n,ii U (n/k)





 + ξ

√ k







+

.

Proof: Recall (A.7) and set

∆_n,i= sup

θ∈Nn⁻(θ)

hi− h_i(θ)

|θ − θ^◦|h_i(θ) ∈ F_i−1. (A.17) Due to bh_i(θ) ≥ inf_θ∈N_n_(η)ω > 0 (recall (C3)) and Lemma 3, we obtain for large n and all i = 1, 2, . . .

sup

θ∈Nn(η)

h_i(θ) − bh_i(θ) bhi(θ)

≤ rⁱ₀V₀, (A.18)

where V₀ ≥ 0 is F₀-measurable with E[V₀]^ν⁰ < ∞ for some ν₀ > 0, and r₀ is chosen in the obvious way. Setting V_i = V₀, the claims (A.14) and (A.15) follow as in the proof of Lemma 13 in Kim and Lee (2016), using Lemma 5. The proof of (A.15) relies on the fact that there exists η₀ > 0, such that

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with probability approaching 1 (w.p.a. 1), (Ui)+

h

1 + rⁱ₀(−η0)Vi

i

1 +(−η₀)∆_n,i

√n

≤ ( bUi(θ))+≤ (U_i)+

h

1 + r₀ⁱη0Vi

i

1 +η₀∆_n,i

√n

for all θ ∈ N_n(η) and i = m_n, . . . , n. From this, (A.16) easily follows as well. Lemma 7 (cf. Kim and Lee, 2016, Lemma 14). Let η0 > 0, δ0 ∈ (0, 1), and r₁ ∈ (0, 1). If m_n < n and m_n→ ∞ as n → ∞ and η₀∈ R, we have that for i = mn, . . . , n

n kw_i

E_i−1[A_i(ξ, η, η₀) − A_i(ξ)]

≤ K max

r₁ⁱ, |η₀|∆_n,i

√n

, (A.19)

n kw_i

E_i−1[C_i(ξ, η, η₀) − C_i(ξ)]^`

≤ K max

r₁ⁱ, |η₀|∆_n,i

√n

, ` ∈ {1, 2}, (A.20) where, with r₀ from Lemma 6,

wi:= I

|η0|^∆n,i^√

n ≤δ0, rⁱ₀|η0|V_i<rⁱ₁

. The proof of Lemma 7 requires two additional lemmas.

Lemma 8. For all C0 > 0 and ξ ∈ R, the survivor function F (x) = P {U1> x} satisfies F

Ce^ξ/

√

kU (n/k)

= C^−1/γk n

"

1 −ξ/γ

√ k + o

1

√ k

#

uniformly in C ≥ C0.

Proof: From p. 46 of Einmahl et al. (2016), it follows that for any x₀ > 0 sup

x≥x0

n

kF (xU (n/k)) − x^−1/γ A(n/k)x^−1/γ

= O(1) and, hence,

n

kF (xU (n/k)) − x^−1/γ = x^−1/γO(A(n/k)) uniformly in x ≥ x0. With x = Ce^ξ/

√

k and √

kA(n/k) = o(1) from (C11), F (Ce^ξ/

√

kU (n/k)) = C^−1/γe^−ξ/(γ

√

k)+ o(1/√ k).

The conclusion now follows from the Taylor expansion e^−ξ/(γ

√ k)=

1 −^ξ/γ^√

k + o(^√¹

k)

.

Lemma 9. For any C > 0 and ξ ∈ R, E

"

log

U₁C U (n/k)

+ ξ

√ k

#

+

= C^1/γk n

"

γ + ξ

√ k + o

1

√ k

# .

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Proof: By Davidson (1994, Theorem 9.14), E

"

log

U1C U (n/k)

+ ξ

√ k

#

+

= Z ∞

0

F (C⁻¹e^xe^−ξ/

√

kU (n/k)) dx.

Hence, using Lemma 8 and the equivalent formulation of (C11) in terms of F (·) from de Haan and Ferreira (2006, Theorem 2.3.9), the conclusion follows as in Lemma 4 in Kim and Lee (2011). Proof of Lemma 7: We first prove (A.19). For w_i = 0 the inequality is trivial. So it remains to consider w_i = 1. Set

Ui

h

1 + r₀ⁱη0Vi

i

1 +η0∆n,i

√n

=: UiC, where C is Fi−1-measurable due to Lemma 6. From Lemma 8,

n k

E_i−1[A_i(ξ, η, η₀) − A_i(ξ)]

= n k

Pn

U_i > C⁻¹e^ξ/

√

kU (n/k) | F_i−1o

− Pn

U_i > e^ξ/

√

kU (n/k) | F_i−1o

≤ K(C^1/γ− 1),

where we have used the Fi−1-measurability of C. The conclusion follows from the definition of C and the mean value theorem.

It remains to show (A.20). Again, assume wi = 1. First, consider the case ` = 1. From Lemma 9 and the F_i−1-measurability of C,

n k

E_i−1C_i(ξ, η, η₀) − C_i(ξ) ≤ K

h

1 + rⁱ₀η₀V_ii1/γ

1 +η0∆n,i

√n

1/γ

− 1

≤ K max

rⁱ₁, |η₀|∆_n,i

√n

, where the last inequality follows from the mean value theorem.

Now, let ` = 2. Use the inequality |x+− y₊|^r≤ |x − y|^rI{max{x,y}>0} to obtain that if wⁱ = 1,

E_i−1C_i(ξ, η, η₀) − C_i(ξ)2

= E_i−1







"

log

U_iC U (n/k)

+ ξ

√ k

#

+

−

"

log

U_i U (n/k)

+ ξ

√ k

#

+







2

≤ E_i−1







log²(C) I⁽

max

log

n UiC U (n/k)

o +^√^ξ

k, log n Ui

U (n/k)

o +^√^ξ

k

>0 )







= log²(C) Ei−1Iⁿ

log{Uimax{C,1}/U (n/k)}+^√^ξ

k>0

o. (A.21)

Write

E_i−1Iⁿ

log{Uimax{C,1}/U (n/k)}+^√^ξ

k>0o

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= Ei−1I^



 log

(

Ui[^1+r0ⁱmax{η0,0}Vi]

1+max{η0,0}∆n,i√ n

/U (n/k)

) +^√^ξ

k>0







≤ E_i−1I⁽

Ui>e^−ξ/

√

k[^1+rⁱ0|η0|V_i]⁻¹

1+^|η0|∆n,i^√_n

−1

U (n/k) )

= k n h

1 + rⁱ₀|η₀|V_ii1/γ

1 +|η₀|∆_n,i

√n

1/γ"

1 +ξ/γ

√ k + o

1

√ k

#

≤ Kk n

n

1 + (2^1/γ − 1)rⁱ₀|η₀|V_io

1 + (2^1/γ − 1)|η₀|∆_n,i

√n

≤ Kk

n, (A.22)

where we have used Lemma 8 in the third step, the inequality (1 + x)^r≤ 1 + (2^r− 1)x for x ∈ [0, 1]

and r ≥ 1 in the fourth step, and wi= 1 in the fourth and final step.

It remains to bound log²(C) = log²

1 + rⁱ₀η0Vi

n

1 +^η⁰^√^∆_n^n,i o

. For η0 > 0, we use log(1+x) ≤ x for x ≥ 0 to get

log²(C) ≤ log²





"

1 + max

r₀ⁱη0Vi,η0√∆n,i

n

#2



≤ K

"

max

r₁ⁱ, |η₀|∆_n,i

√n

#2

. For η0 ≤ 0, we use log(1 + x) > x/(1 + x) for x > −1 to arrive at

log²(C) ≤ log²





"

1 − max

r₀ⁱ|η₀|V_i,|η₀|∆_n,i

√n

#2



≤ 4







− maxn

r₁ⁱ, ^|η⁰^√^|∆_n^n,io 1 − maxn

rⁱ₁, ^|η⁰^√^|∆_n^n,io







2

≤ K

"

max

r₁ⁱ, |η₀|∆_n,i

√n

#2

Hence, observing that w_i= 1, we get for general η₀∈ R, log²(C) ≤ K max

rⁱ₁, |η₀|∆_n,i

√n

. (A.23)

Combining (A.21) with (A.22) and (A.23), the claim for ` = 2 follows. Lemma 10 (cf. Kim and Lee, 2016, Lemma 15). Let η > 0, η₀ ∈ R, and

Bi(ξ, η, η0) = Ai(ξ, η, η0) − Ai(ξ), D_i(ξ, η, η₀) = C_i(ξ, η, η₀) − C_i(ξ).

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Then, for any K > 0 and integer sequence mn< n tending to infinity with mn= o(n),

sup

ξ∈[−K,K]

√1 k

n

X

i=mn

Bi(ξ, η, η0)

= oP(1), (A.24)

sup

ξ∈[−K,K]

√1 k

n

X

i=mn

Di(ξ, η, η0)

= oP(1). (A.25)

Proof: The proof of (A.24) is similar to that of (A.25), only simpler. Hence, we only show (A.25).

Let ρ > 0 and define ξ_u = uρ ∈ [−K, K] for u ∈ Z. Then, for ξ ∈ (ξu, ξ_u+1],

√1 k

n

X

i=mn

Di(ξ, η, η0) ≤ 1

√ k

n

X

i=mn

C_i(ξu+1, η, η0) − Ci(ξu)

= 1

√ k

n

X

i=mn

D_i(ξu+1, η, η0) − Ei−1Di(ξu+1, η, η0)

+ 1

√ k

n

X

i=mn

Ei−1Di(ξu+1, η, η0)

+ 1

√ k

n

X

i=mn

n

Ci(ξu+1) − Ci(ξu) − EC_i(ξu+1) − Ci(ξu)o

+ 1

√ k

n

X

i=mn

EC_i(ξu+1) − Ci(ξu) , and

√1 k

n

X

i=mn

Di(ξ, η, η0) ≥ 1

√ k

n

X

i=mn

C_i(ξu, η, η0) − Ci(ξu+1)

= 1

√ k

n

X

i=mn

D_i(ξu, η, η0) − Ei−1Di(ξu, η, η0)

+ 1

√ k

n

X

i=mn

Ei−1Di(ξu, η, η0)

+ 1

√ k

n

X

i=mn

n

Ci(ξu) − Ci(ξu+1) − EC_i(ξu) − Ci(ξu+1)o

+ 1

√ k

n

X

i=mn

EC_i(ξu) − Ci(ξu+1) . Hence, to prove (A.25) it is enough to show that for any δ > 0,

n→∞limP







max

u∈Z : uρ∈[−K,K]

√1 k

n

X

i=mn

D_i(ξ_u, η, η₀) − E_i−1D_i(ξ_u, η, η₀)

> δ







= 0, (A.26)

n→∞limP







max

√1 k

n

X

i=mn

> δ







= 0, (A.27)

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n→∞limP







max

√1 k

n

X

i=mn

n

Ci(ξu+1) − Ci(ξu) − EC_i(ξu+1) − Ci(ξu)o

> δ







= 0, (A.28)

n→∞limP







max

√1 k

n

X

i=mn

EC_i(ξu+1) − Ci(ξu)

> δ







= 0. (A.29)

Let δ > 0. To show (A.29), use Lemma 9 to write

√1 k

n

X

i=mn

EC_i(ξu+1) − Ci(ξu) = 1

√ k

n

X

i=1



 k n

ξu− ξ_u+1

√

k + o

1

√ k

!



= |ξ_u− ξ_u+1| + o(1).

It follows that (A.29) holds for ρ = δ/2. Hence, we fix ρ = δ/2 in the remainder of the proof.

To prove (A.27), note that Pn

i=mn{∆_n,i/√

n} = O_P(√

n). This follows since for any ε > 0, we get from Markov’s inequality and (A.14) that

P







n

X

i=mn

∆√n,i

n ≥ K√ n







≤ 1 K

1 nE





n

X

i=mn

∆n,i



< ε for K sufficiently large. Due to this and (A.20), we obtain that

√1 k

n

X

i=mn

wiEi−1Di(ξu, η, η0)

≤ K

√ k

k n

n

X

i=mn

max

rⁱ₁, |η₀|∆_n,i

√n

≤ K

√k n







n

X

i=mn

r₁ⁱ +

n

X

i=mn

∆_n,i

√n







= OP

pk/n

= oP(1).

Since w_m_n = . . . = w_n= 1 w.p.a. 1, we also have

√1 k

n

X

i=mn

E_i−1D_i(ξ_u, η, η₀)

= o_P(1).

This and subadditivity imply

P







max

√1 k

n

X

i=mn

> δ







≤ X

P







√1 k

n

X

i=mn

> δ

|{u ∈ Z : uρ ∈ [−K, K]}|







= o(1), (A.30)

where | · | applied to a set denotes cardinality.

Next, we show (A.26). Since {wiD_i(ξu, η, η0) − Ei−1Di(ξu, η, η0) } is a zero-mean sequence with

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zero autocorrelation, we get using Chebyshev’s inequality

P







√1 k

n

X

i=mn

wiD_i(ξu, η, η0) − Ei−1Di(ξu, η, η0)

> ε







≤ 1 ε²kVar





n

X

i=mn

wiD_i(ξu, η, η0) − Ei−1Di(ξu, η, η0)





= 1 ε²k

n

X

i=mn

Var

w_iD_i(ξ_u, η, η₀) − E_i−1D_i(ξ_u, η, η₀)

= 1 ε²k

n

X

i=mn

Eh

w_iD_i(ξ_u, η, η₀) − E_i−1D_i(ξ_u, η, η₀) i2

= 1 ε²k

n

X

i=mn

Eh

w_iE_i−1D_i(ξ_u, η, η₀) − E_i−1D_i(ξ_u, η, η₀) 2i

≤ 1 ε²k

n

X

i=mn

Eh

w_iE_i−1D_i(ξ_u, η, η₀) 2i

≤ K n

n

X

i=mn

E

"

max

r₁ⁱ,|η₀|∆_n,i

√n

#

≤ K n

n

X

i=mn

rⁱ₁+ K

√n 1 n

n

X

i=mn

E ∆n,i= o(1)

where we used the law of the iterated expectation in the fourth line and (A.20) in the sixth line. Since wmn = . . . = wn= 1 w.p.a. 1, this again implies

√1 k

n

X

i=mn

D_i(ξu, η, η0) − Ei−1Di(ξu, η, η0)

= oP(1).

From this, (A.26) follows as in (A.30).

Finally, (A.28) is a consequence of Lemma 5 in Kim and Lee (2009). Note that their condition (A2) is implied by our assumption (C11)—as can be easily seen from de Haan and Ferreira (2006, Theorem 2.3.9)—and their condition (A3) holds for χ = 0 and ω = 0 for the i.i.d. sequence U_i. Lemma 11 (cf. Kim and Lee, 2016, Proposition 4). Let K > 0. Then,

sup

ξ∈[−K,K]

√1 k

n

X

i=1

Iⁿ

Ui>e^ξ/

√ kU (n/k)

o− Iⁿ

Ubi>e^ξ/

√ kU (n/k)

o

= oP(1), (A.31)

sup

ξ∈[−K,K]

√1 k

n

X

i=1

(

log{U_i/U (n/k)} + ξ

√ k

+

−

log{ bU_i/U (n/k)} + ξ

√ k

+

)

= o_P(1). (A.32)

Proof: We only show (A.32), the proof of (A.31) being similar and simpler. Let m_n < n tend to infinity with mn = o(√

k) as n → ∞. Then, we obtain from (A.16) that for any η > 0 there exists

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η0 > 0, such that w.p.a. 1,

Ci(ξ, η, −η0) − Ci(ξ) ≤ Ci(ξ, θ) − Ci(ξ) ≤ Ci(ξ, η, η0) − Ci(ξ) ∀ i = m_n, . . . , n, θ ∈ Nn(η).

Hence, from (A.25),

sup

θ∈Nn(η)

sup

ξ∈[−K,K]

√1 k

n

X

i=mn

C_i(ξ, θ) − C_i(ξ)

= o_P(1).

Thus, for any ε > 0 and δ > 0 there exists n₀ ∈ N, such that for all n ≥ n0

P





 sup

ξ∈[−K,K]

√1 k

n

X

i=mn

n

C_i(ξ, bθ_n) − C_i(ξ)o

≥ δ







≤ P





 sup

θ∈Nn(η)

sup

ξ∈[−K,K]

√1 k

n

X

i=mn

C_i(ξ, θ) − C_i(ξ)

≥ δ, bθ_n∈ N_n(η)





 + Pn

θb_n∈ N/ _n(η)o

≤ ε/2 + ε/2 = ε, where we used (A.13).

It remains to show that sup

ξ∈[−K,K]

√1 k

mn

X

i=1

n

C_i(ξ, bθ_n) − C_i(ξ)o

= o_P(1). (A.33)

Use the inequality |x₊− y₊|^r ≤ |x − y|^rI{max{x,y}>0} to bound

√ k

+

−

√ k

+

≤

log( bU_i) − log(U_i) . Consider

| log( bU_i(θ)) − log(U_i)| =

1 δ^◦log

(

1 +h_i(θ) − bh_i(θ) bhi(θ)

) + 1

δ^◦log

1 +h_i− h_i(θ) hi(θ)

.

Using log(1 + x) ≤ x for x > −1, we obtain the upper bounds log

(

1 +h_i(θ) − bh_i(θ) bhi(θ)

)

≤ sup

θ∈Nn(η)

|h_i(θ) − bh_i(θ)|

bhi(θ)

(A.18)

≤ V₀,

log

≤ sup

θ∈Nn(η)

|h_i− h_i(θ)|

hi(θ)

(A.17)

≤ η max

i=1,...,n

∆√n,i

n. Using log(1 + x) ≥ x/(1 + x) for x > −1, we obtain the lower bounds

log (

1 +hi(θ) − bhi(θ) bh_i(θ)

)

≥ − sup

θ∈Nn(η)

|h_i(θ) − bhi(θ)|

h_i(θ)

(A.18)

≥ −KV₀,

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log

≥ − sup_θ∈N_n_(η)^|hⁱ_h^−hⁱ^(θ)|

i(θ)

h

1 − sup_θ∈N_n_(η)^|hⁱ_h^−hⁱ^(θ)|

i(θ)

i

+ (A.17)

≥ −K max

i=1,...,n

∆√n,i

n,

where the latter inequality holds for all i = 1, . . . , n w.p.a. 1, due to max_i=1,...,n^∆^√^n,i_n = o_P(1). Com- bining the results for the upper and lower bounds it follows that w.p.a. 1, for all i = 1, . . . , n

| log( bU_i(θ)) − log(U_i)| ≤ K

V₀+ max

i=1,...,n

∆n,i

√n

. Consequently, since bU_i = bU_i( bθ_n) we have on the set { bθ_n∈ N_n(η)},

sup

ξ∈[−K,K]

√1 k

n

X

i=1

(

√ k

+

−

√ k

+

)

≤ Km_n

√ k

V0+ max

i=1,...,n

∆_n,i

√n

= oP(1).

This together with (A.13) implies (A.33), which concludes the proof. Define

M_n⁽¹⁾(ξ) = 1

√ k

n

X

i=1







"

log

U_i U (n/k)

+ ξ

√ k

#

+

− E

"

log

U_i U (n/k)

+ ξ

√ k

#

+





 ,

Mc_n⁽¹⁾(ξ) = 1

√k

n

X

i=1









log Ubi

U (n/k)

! + ξ

√k





+

− E

"

log

Ui

U (n/k)

+ ξ

√k

#

+





 ,

M_n⁽²⁾(ξ) = 1

√ k

n

X

i=1

( Iⁿ

Ui>e^ξ/

√

kU (n/k)o− Pn

Ui > e^ξ/

√

kU (n/k) o

) ,

Mc_n⁽²⁾(ξ) = 1

√ k

n

X

i=1

( Iⁿ

Ubi>e^ξ/

√ kU (n/k)

o− Pn

Ui > e^ξ/

√

kU (n/k) o

) .

The following two lemmas are the last ingredient for the proofs of Propositions 2 and 3.

Lemma 12 (Hsing, 1991, Theorem 3.3). We have that





Mn⁽¹⁾(0) γMn⁽²⁾(0)





−→d (n→∞)N









 0 0



, γ²



 2 1 1 1









. Lemma 13 (Kim and Lee, 2009, Lemma 3). For every K > 0 and ` ∈ {1, 2},

sup

ξ∈[−K,K]

M_n^(`)(ξ) − M_n^(`)(0)

= o_P(1).

Proof of Proposition 2: Observe that for all ξ ∈ R,

√ k log





Ub_(k+1) U (n/k)



≤ ξ ⇐⇒

n

X

i=1

Iⁿ

Ubi>e^ξ/

√

kU (n/k)o≤ k

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⇐⇒ Mc_n⁽²⁾(ξ) ≤ ξ

γ + o(1), (A.34)

where we have used Lemma 8 for the second equivalence. From (A.31) and Lemma 13, we get that Mc_n⁽²⁾(ξ) = M_n⁽²⁾(ξ) + o_P(1) = M_n⁽²⁾(0) + o_P(1)

uniformly on any ξ-compact set. Hence, (A.34) is equivalent to M_n⁽²⁾(0) + oP(1) ≤ ξ

γ + o(1).

The conclusion now follows since Lemma 12 implies Mn⁽²⁾(0) = O_P(1). Proof of Proposition 3: Define W_n=√

kh

log bU_(k+1)− log U (n/k)i

. Then,

Mc_n⁽¹⁾(−Wn) = 1

√ k

n

X

i=1



log Ubi

U (n/k)

!

− W√n

k





+

− n

√ kE

"

log

U1

U (n/k)

−W√n

k

#

+

=

√ k





 1 k

n

X

i=1

log₊( bUi/ bU_(k+1)) − γ







+ Wn+ oP(1), (A.35)

where we have used that E

"

log

U₁ U (n/k)

−W_n

√ k

#

+

= k n

γ −W_n

√

k + o_P( 1

√ k)

. This can easily be verified by using Lemma 9 and solving for the oP-term.

From Lemmas 11 and 13 both combined with W_n= O_P(1) (see Proposition 2), we get

Mc_n⁽¹⁾(−W_n) = M_n⁽¹⁾(−W_n) + o_P(1) = M_n⁽¹⁾(0) + o_P(1).

From this and (A.35),

M_n⁽¹⁾(0) − Wn=

√

k {γ − γ} + ob P(1). (A.36)

Also, from (A.34), W_n=

√ k logn

Ub_(k+1)/U (n/k)o

≤ ξ ⇐⇒ γ cM_n⁽²⁾(ξ) ≤ ξ + o(1).

Using Lemmas 11 and 13 we have that cMn⁽²⁾(ξ) = Mn⁽²⁾(ξ) + oP(1) = Mn⁽²⁾(0) + oP(1) uniformly on any ξ-compact set. Thus,

W_n≤ ξ ⇐⇒ γM_n⁽²⁾(0) ≤ ξ + o_P(1).

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Hence, with Lemma 12,





M_n⁽¹⁾(0) W_n





−→d (n→∞)N









 0 0



, γ²



 2 1 1 1









.

Combining this with (A.36) and the continuous mapping theorem, the conclusion follows.