Conditional marginal expected shortfall

Abstract

In the context of bivariate random variables $\left (Y^{(1)},Y^{(2)}\right )$, the marginal expected shortfall, defined as $\mathbb {E}\left (Y^{(1)}|Y^{(2)} \ge Q_{2}(1-p)\right )$ for p small, where Q₂ denotes the quantile function of Y⁽²⁾, is an important risk measure, which finds applications in areas like, e.g., finance and environmental science. Our paper pioneers the statistical modeling of this risk measure when the random variables of main interest $\left (Y^{(1)},Y^{(2)}\right )$ are observed together with a random covariate X, leading to the concept of the conditional marginal expected shortfall. The asymptotic behavior of an estimator for this conditional marginal expected shortfall is studied for a wide class of conditional bivariate distributions, with heavy-tailed marginal conditional distributions, and where p tends to zero at an intermediate rate. The finite sample performance is evaluated on a small simulation experiment. The practical applicability of the proposed estimator is illustrated on flood claim data.

Appendix A

Lemma 1

Assume $(\mathcal {D})$ and $({\mathscr{H}})$ and x₀ ∈Int(S_X). Let (t_n)_n≥ 1 and (h_n)_n≥ 1 be arbitrary sequences satisfying $t_{n} \to \infty$ and h_n → 0 such that ${h}_{n}^{\eta _{\gamma _{j}} \wedge \eta _{\varepsilon _{j}}} \ln t_{n} \to 0$, as $n \to \infty$, and 0 ≤ η < 1. Then

$$ \begin{array}{@{}rcl@{}} \left | \frac{t_{n} \overline{F}_{j}(U_{j}(t_{n}/y|x_{0})|x)}{y^{\eta}} -y^{1-\eta} \right | \to 0, \text{ as } n \to \infty, \end{array} $$

uniformly in y ∈ (0,T] and x ∈ B(x₀,h_n).

Lemma 2

Assume $(\mathcal {D})$, $({\mathscr{H}})$, $(\mathcal {K})$ and $(\mathcal {R})$ with x → R(y₁,y₂|x) being a continuous function, and x₀ ∈ Int(S_X) such that f_X(x₀) > 0. Consider sequences $k \to \infty$ and h → 0 as $n \to \infty$ in such a way that k/n → 0 and $h^{\eta _{\gamma _{1}}\wedge \eta _{\gamma _{2}} \wedge \eta _{\varepsilon _{1}}\wedge \eta _{\varepsilon _{2}} }\ln n/k \to 0$. Then, as $n \to \infty$

$$ \begin{array}{@{}rcl@{}} \mathbb{E} (T_{n}(y_{1},y_{2}|x_{0})) & \to & f_{X}(x_{0}) R(y_{1},y_{2}|x_{0}), \\ kh^{d} \mathbb{V}ar(T_{n}(y_{1},y_{2}|x_{0})) & \to & \|K\|_{2}^{2} f_{X}(x_{0}) R(y_{1},y_{2}|x_{0}). \end{array} $$

The proof of these lemmas and all the subsequent ones are given in Appendix A.5.

1.1 A.1 Proof of Theorem 1

To prove the result we will make use of empirical process theory with changing function classes, see for instance van der Vaart and Wellner (1996). To this aim we start by introducing some notation. Let P be the distribution measure of $\left (Y^{(1)},Y^{(2)},X\right )$, and denote the expected value under P, the empirical version and empirical process as follows

$$ \begin{array}{@{}rcl@{}} Pf:=\int fdP,\mathbb{P}_{n}f:=\frac{1}{n}\sum\limits_{i=1}^{n}f\left( {Y}_{i}^{(1)}, {Y}_{i}^{(2)}, X_{i}\right),\mathbb{G}_{n}f:=\sqrt{n}(\mathbb{P}_{n}-P)f, \end{array} $$

for any real-valued measurable function $f:\mathbb {R}^{2}\times \mathbb {R}^{d}\to \mathbb {R}$. For a function class $\mathcal {F}$, let $N_{[]}(\varepsilon , \mathcal {F}, L_{2}(P))$, denote the minimal number of ε −brackets needed to cover $\mathcal {F}$. The bracketing integral is then defined as

$$ \begin{array}{@{}rcl@{}} J_{[]}(\delta,\mathcal{F},L_{2}(P)) ={\int}_{0}^{\delta} \sqrt{\ln N_{[]}(\varepsilon, \mathcal{F}, L_{2}(P))} d\varepsilon. \end{array} $$

We introduce our sequence of classes $\mathcal {F}_{n}$ on $\mathbb {R}^{2}\times \mathbb {R}^{d}$ as

$$ \begin{array}{@{}rcl@{}} \mathcal{F}_{n}&:=&\left\{(u,z)\rightarrow f_{n,y}(u,z), y\in(0,T]^{2}\right\} \end{array} $$

where

Denote also by F_n an envelope function of the class $\mathcal {F}_{n}$. Now, according to Theorem 19.28 in van der Vaart (1998) the weak convergence of the stochastic process Eq. 3 follows from the following four conditions. Let $\rho _{x_{0}}$ be a semimetric, possibly depending on x₀, making (0,T]² totally bounded. We have to prove that

along with the pointwise convergence of the covariance function.

Proof of Eq. 9.

Let $\rho _{x_{0}} (y,\bar y) := |y_{1}-\bar y_{1}|+|y_{2}-\bar y_{2}|$. Denote $A_{n,y}:= \left \{ \overline {F}_{1}\left (Y^{(1)}|x_{0}\right ) \le (k/n) y_{1}, \overline {F}_{2}\left (Y^{(2)}|x_{0}\right ) \le (k/n) y_{2} \right \}$. We have then

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We consider now three cases. □

Case 1

$y_{1} \wedge \bar y_{1} \le \delta _{n}$. Assume without loss of generality that $y_{1} \le \bar y_{1}$. By expanding the square in the above conditional expectation and using the fact that, e.g., $A_{n,y} \subset \{ \overline {F}_{1}(Y^{(1)}|x_{0}) \le (k/n) y_{1} \}$, we obtain the following inequality

which, after substituting in Eq. 13 leads to

$$ \begin{array}{@{}rcl@{}} &&P(f_{n,y}-f_{n,\bar y})^{2}\\ & \le & 3 \frac{n}{k} {\int}_{S_{K}} K^{2}(v) \frac{P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right) \le (k/n) y_{1}|X=x_{0}-hv\right)}{y_{1}^{2\eta}}f_{X}(x_{0}-hv) dv \\ & & + \frac{n}{k} {\int}_{S_{K}} K^{2}(v) \frac{P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right) \le (k/n) \bar y_{1}|X=x_{0}-hv\right)}{\bar {y}_{1}^{2\eta}}f_{X}(x_{0}-hv) dv. \end{array} $$

Now note that

$$ \begin{array}{@{}rcl@{}} P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right) \le (k/n) y_{1}|X=x_{0}-hv\right) = \overline{F}_{1}\left (U_{1}(n/(ky_{1})|x_{0})|x_{0}-h v \right ), \end{array} $$

which, together with the result of Lemma 1, motivates the following decomposition

$$ \begin{array}{@{}rcl@{}} &&P(f_{n,y}-f_{n,\bar y})^{2}\\ & \le & 3y_{1}^{1-2\eta} {\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv \\ & & + 3 {\int}_{S_{K}} K^{2}(v) \left [ \frac{1}{{y}_{1}^{2\eta}} \frac{n}{k} \overline{F}_{1}\left (U_{1}(n/(ky_{1})|x_{0})|x_{0}-h v \right )-{y}_{1}^{1-2\eta} \right ] f_{X}(x_{0}-hv) dv \\ & & + \bar {y}_{1}^{1-2\eta} {\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv \\ & & + {\int}_{S_{K}} K^{2}(v) \left [ \frac{1}{\bar {y}_{1}^{2\eta}} \frac{n}{k} \overline{F}_{1}\left (U_{1}(n/(k \bar y_{1})|x_{0})|x_{0}-h v \right )-\bar {y}_{1}^{1-2\eta} \right ] f_{X}(x_{0}-hv) dv. \end{array} $$

Using Lemma 1 and the fact that $\rho _{x_{0}}(y, \bar y)\leq \delta _{n}$ which implies $\bar y_{1} \le 2 \delta _{n}$, we get

$$ \begin{array}{@{}rcl@{}} P(f_{n,y}-f_{n,\bar y})^{2} & \le & 5 \delta_{n}^{1-2\eta} {\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv +o(1), \end{array} $$

where the o(1) term does not depend on y₁ and $\bar y_{1}$.

Case 2

$y_{1} \wedge \bar y_{1} > \delta _{n}$ and $y_{2} \wedge \bar y_{2} \le \delta _{n}$. Assume without loss of generality that $y_{2} \le \bar y_{2}$. Similarly to the approach followed in Case 1, we obtain

and thus

$$ \begin{array}{@{}rcl@{}} &&{\kern0pt}P(f_{n,y}-f_{n,\bar{y}})^{2}\\ & \le & \frac{3 y_{2}}{\left( y_{1} \wedge \bar y_{1}\right)^{2 \eta}} {\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv \\ & & + \frac{3{y}_{2}^{2\eta}}{(y_{1} \wedge \bar y_{1})^{2 \eta}}{\int}_{S_{K}} K^{2}(v) \left [ \frac{1}{{y}_{2}^{2\eta}} \frac{n}{k} \overline{F}_{2}\left (U_{2}(n/(ky_{2})|x_{0})|x_{0}-h v \right )-{y}_{2}^{1-2\eta} \right ] f_{X}(x_{0}-hv) dv \\ & & + \frac{\bar y_{2}}{(y_{1} \wedge \bar y_{1})^{2 \eta}} {\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv \\ & & + \frac{\bar{y}_{2}^{2\eta}}{(y_{1} \wedge \bar y_{1})^{2 \eta}}{\int}_{S_{K}} K^{2}(v) \left [ \frac{1}{\bar{y}_{2}^{2\eta}} \frac{n}{k} \overline{F}_{2}\left (U_{2}(n/(k\bar y_{2})|x_{0})|x_{0}-h v \right )-\bar{y}_{2}^{1-2\eta} \right ] f_{X}(x_{0}-hv) dv. \end{array} $$

Again by Lemma 1 and using that $\bar y_{2} \le 2 \delta _{n}$ we have that

$$ \begin{array}{@{}rcl@{}} P(f_{n,y}-f_{n,\bar y})^{2} & \le & 5 \delta_{n}^{1-2\eta} {\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv +o(1), \end{array} $$

where the o(1) term does not depend on y₂ and $\bar y_{2}$.

Case 3

$y_{1} \wedge \bar y_{1} > \delta _{n}$ and $y_{2} \wedge \bar y_{2} > \delta _{n}$. Let $y \vee \bar y$ denote the vector with the component-wise maxima of y and $\bar y$, and similarly $y \wedge \bar y$ is the vector with the component-wise mimima of y and $\bar y$. Then

Note that

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which leads to

$$ \begin{array}{@{}rcl@{}} &&{}\left.{P(f_{n,y}-f_{n,\bar{y}})^{2}}\right. \\ &&\le\left. \frac{\left( y_{1}^{\eta}-\bar y_{1}^{\eta}\right)^{2}}{(y_{1} \bar y_{1})^{2\eta}} \frac{n}{k} {\int}_{S_{K}}K^{2}(v) P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right)\le (k/n) y_{1} \wedge \bar y_{1},\overline{F}_{2}\left( Y^{(2)}|x_{0}\right)\le (k/n) y_{2} \wedge \bar y_{2}\right|X=x_{0}-h v\right) \\ & & \times f_{X}(x_{0}-hv) dv \\ &&\left.\left. +\frac{1}{(y_{1} \wedge \bar y_{1})^{2\eta}} \frac{n}{k} {\int}_{S_{K}}K^{2}(v) \left[P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right)\le (k/n) y_{1} \vee \bar y_{1},\overline{F}_{2}\left( Y^{(2)}|x_{0}\right)\le (k/n) y_{2} \vee \bar y_{2}\right|X=x_{0}-h v\right) \right.\right. \\ && \left.\left.- P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right)\le (k/n) y_{1} \wedge \bar y_{1},\overline{F}_{2}\left( Y^{(2)}|x_{0}\right)\le (k/n) y_{2} \wedge \bar y_{2}\right|X=x_{0}-h v\right)\right] f_{X}(x_{0}-hv) dv \\ & &=: Q_{1,n}+Q_{2,n}. \end{array} $$

As for Q_1,n, we easily obtain

$$ \begin{array}{@{}rcl@{}} Q_{1,n} \le \frac{\left( y_{1}^{\eta}-\bar y_{1}^{\eta}\right)^{2}}{(y_{1} \bar y_{1})^{2\eta}} {\int}_{S_{K}}K^{2}(v) \frac{n}{k} \overline{F}_{1}\left (U_{1}(n/(k y_{1}\!\wedge\! \bar y_{1})|x_{0})|x_{0} - h v \right )f_{X}(x_{0}-hv) dv. \end{array} $$

Now, by the mean value theorem, applied to $\left (y_{1}^{\eta }-\bar y_{1}^{\eta }\right )^{2}$, and a decomposition motivated by Lemma 1,

$$ \begin{array}{@{}rcl@{}} &&{}Q_{1,n} \\ &\le & (y_{1} \wedge \bar y_{1})^{-1-2 \eta} (y_{1}-\bar y_{1})^{2} {\int}_{S_{K}}K^{2}(v)f_{X}(x_{0}-hv) dv \\ &&\left.+ (y_{1} \wedge \bar y_{1})^{-2} (y_{1}-\bar y_{1})^{2} {\int}_{S_{K}}K^{2}(v) \left[ \frac{1}{(y_{1} \wedge \bar{y}_{1})^{2\eta}} \frac{n}{k} \overline{F}_{1}\left( \vphantom{{\int}_{S_{K}}} U_{1}(n/(k y_{1} \wedge \bar y_{1})|x_{0})\right|x_{0}-h v \right)\right. \\ &-&\left. (y_{1} \wedge \bar y_{1})^{1-2\eta} \right ] \times f_{X}(x_{0}-hv) dv. \end{array} $$

This then gives

$$ \begin{array}{@{}rcl@{}} Q_{1,n} & \le & {\delta}_{n}^{1-2\eta} {\int}_{S_{K}}K^{2}(v)f_{X}(x_{0}-hv) dv +o(1), \end{array} $$

where the o(1) term does not depend on y₁ and $\bar y_{1}$.

Concerning Q_2,n, we have the following inequality

$$ \begin{array}{@{}rcl@{}} &&{}Q_{2,n} \\ &\le&\left. \frac{1}{(y_{1} \wedge \bar y_{1})^{2\eta}} \frac{n}{k} {\int}_{S_{K}}K^{2}(v) P\left( (k/n) y_{1} \wedge \bar y_{1} \le \overline{F}_{1}\left( Y^{(1)}|x_{0}\right)\le (k/n) y_{1} \vee \bar y_{1}\right|X=x_{0}-h v\right) f_{X}(x_{0}-hv) dv\\ & &\left. + \frac{1}{(y_{1} \wedge \bar y_{1})^{2\eta}} \frac{n}{k} {\int}_{S_{K}}K^{2}(v) P\left( (k/n) y_{2} \wedge \bar{y}_{2} \le \overline{F}_{2}\left( Y^{(2)}|x_{0}\right)\le (k/n) y_{2} \vee \bar y_{2}\right|X=x_{0}-h v\right) f_{X}(x_{0}-hv) dv \\ & =:& Q_{2,1,n}+Q_{2,2,n}. \end{array} $$

We only give details about Q_2,1,n, the term Q_2,2,n can be handled analogously. Direct computations give

$$ \begin{array}{@{}rcl@{}} Q_{2,1,n} = \frac{1}{(y_{1} \wedge \bar y_{1})^{2\eta}} \frac{n}{k} {\int}_{S_{K}}K^{2}(v) {\int}_{U_{1}(n/(k (y_{1} \vee \bar y_{1}))|x_{0})}^{U_{1}(n/(k (y_{1} \wedge \bar y_{1}))|x_{0})}f_{1}(y|x_{0}-hv)dy f_{X}(x_{0}-hv)dv, \end{array} $$

and, after substituting $u=(n/k) \overline {F}_{1}(y|x_{0})$, we have

$$ \begin{array}{@{}rcl@{}} Q_{2,1,n} =\frac{1}{(y_{1} \wedge \bar y_{1})^{2\eta}} {\int}_{S_{K}}K^{2}(v) {\int}_{y_{1} \wedge \bar y_{1}}^{y_{1} \vee \bar y_{1}} \frac{f_{1}(U_{1}(n/(ku)|x_{0})|x_{0}-hv)}{f_{1}(U_{1}(n/(ku)|x_{0})|x_{0})}du f_{X}(x_{0}-hv)dv. \end{array} $$

Using Eq. 1 and arguments similar to those used in the proof of Lemma 1 one obtains for n large and some small κ > 0,

$$ \begin{array}{@{}rcl@{}} \frac{f_{1}(U_{1}(n/(ku) |x_{0})|x_{0}-h v)}{f_{1}(U_{1}(n/(ku)|x_{0})|x_{0})} \le C u^{-\kappa} , \end{array} $$

where C does not depend on u. Then, for n large enough,

$$ \begin{array}{@{}rcl@{}} Q_{2,1,n} & \le& \frac{C}{(y_{1} \wedge \bar y_{1})^{2\eta}} {\int}_{y_{1} \wedge \bar y_{1}}^{y_{1} \vee \bar y_{1}}u^{-\kappa} du {\int}_{S_{K}} K^{2}(v) f_{X}(x_{0}-hv) dv \\ & \le & \frac{C}{(y_{1} \wedge \bar y_{1})^{2\eta}} (y_{1}\wedge \bar y_{1})^{-\kappa}\left( y_{1} \vee \bar y_{1}-y_{1} \wedge \bar y_{1}\right) \\ & \le & C \delta_{n}^{1-2\eta-\kappa} \\ & =& o(1), \end{array} $$

for a small κ ∈ (0, 1 − 2η).

Combining all the above we have verified Eq. 9.

Proof of Eq. 10. A natural envelope function of the class ${\mathcal {F}}_{n}$ is

This yields

Concerning Q_3,n(T) we obtain by direct integration and a slight adjustment of Lemma 1, for large n

$$ \begin{array}{@{}rcl@{}} Q_{3,n}(T) & = & \frac{1}{1-2\eta} \left (\frac{n}{k} \right )^{1-2\eta}{\int}_{S_{K}}K^{2}(v) \left[\overline{F}_{1}(U_{1}(n/(kT) |x_{0})|x_{0}-hv)\right]^{1-2\eta} f_{X}(x_{0}-hv)dv\\ & = & \frac{T^{1-2\eta}}{1-2\eta} {\int}_{S_{K}}K^{2}(v) f_{X}(x_{0}-hv)dv\\ & & + \frac{1}{1-2\eta} {\int}_{S_{K}}K^{2}(v) \left [ \left (\frac{n}{k} \overline{F}_{1}(U_{1}(n/(kT) |x_{0})|x_{0}-hv)\right )^{1-2\eta} -T^{1-2\eta} \right ] f_{X}(x_{0}-hv)dv\\ &\le& C T^{1-2\eta-\kappa}, \end{array} $$

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for κ < 1 − 2η.

Concerning Q_4,n(T), combining $(\mathcal {D})$ with $({\mathscr{H}})$ gives the following bound, for n large and y ≥ U₁(n/(kT)|x₀),

$$ \begin{array}{@{}rcl@{}} \left | \left (\frac{ \overline{F}_{1}(y|x_{0}-hv)}{\overline{F}_{1}(y|x_{0})} \right )^{2\eta}-1 \right | & \le & C_{1} \left (h^{\eta_{A_{1}}}+y^{C_{2} h^{\eta_{\gamma_{1}}}}h^{\eta_{\gamma_{1}}} \ln y + |\delta_{1}(y|x_{0})| h^{\eta_{B_{1}}} \right . \\ & & \left .+ |\delta_{1}(y|x_{0})|y^{C_{3} h^{\eta_{\varepsilon_{1}}}}h^{\eta_{\varepsilon_{1}}} \ln y \right ). \end{array} $$

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Each of the terms in the right-hand side of the above inequality needs now to be used in Q_4,n(T), leading to the terms Q_4,j,n(T), j = 1,…, 4, studied below. First

$$ \begin{array}{@{}rcl@{}} &&{\kern15pt}Q_{4,1,n}(T):=\\ &&h^{\eta_{A_{1}}} \left (\frac{n}{k} \right )^{1-2\eta} {\int}_{S_{K}}K^{2}(v) {\int}_{U_{1}(n/(kT)|x_{0})}^{\infty} \frac{1}{(\overline{F}_{1}(y|x_{0}-hv))^{2\eta}}dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv. \end{array} $$

This term is clearly of smaller order than Q_3,n(T) studied above and hence Q_4,1,n(T) = O(1).

For the second term in the right-hand side of Eq. 16 we need to study

$$ \begin{array}{@{}rcl@{}} &&Q_{4,2,n}(T):=\\ &&h^{\eta_{\gamma_{1}}} \left (\frac{n}{k} \right )^{1-2\eta} {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(T)}^{\infty} y^{\xi_{1,n}}\ln y \frac{1}{(\overline{F}_{1}(y|x_{0}-hv))^{2\eta}}dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv \end{array} $$

where t_n(T) := U₁(n/(kT)|x₀) and $\xi _{1,n}:=C_{2} h^{\eta _{\gamma _{1}}}$. Let $p_{n}(y):=\xi _{1,n} y^{\xi _{1,n}-1}\ln y+y^{\xi _{1,n}-1}$. Applying integration by parts on the inner integral gives, for n large enough,

$$ \begin{array}{@{}rcl@{}} &&Q_{4,2,n}(T)\\ & = & \left (\frac{n}{k} \right )^{1-2\eta} \frac{ h^{\eta_{\gamma_{1}}} \ln ({t_{n}(T)}) [t_{n}(T)]^{\xi_{1,n}}}{1-2\eta} {\int}_{S_{K}}K^{2}(v) \left[\overline{F}_{1}(t_{n}(T)|x_{0}-hv)\right]^{1-2\eta}f_{X}(x_{0}-hv)dv \\ & & + \left (\frac{n}{k} \right )^{1-2\eta} \frac{h^{\eta_{\gamma_{1}}}}{1-2\eta} {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(T)}^{\infty} p_{n}(y) \left[\overline{F}_{1}(y|x_{0}-hv)\right]^{1-2\eta}dyf_{X}(x_{0}-hv)dv \\ & =: & Q_{4,2,1,n}(T)+Q_{4,2,2,n}(T). \end{array} $$

We obtain, for n large enough

$$ \begin{array}{@{}rcl@{}} Q_{4,2,1,n}(T) &\le& C h^{\eta_{\gamma_{1}}} \ln (t_{n}(T)) [t_{n}(T)]^{\xi_{1,n}} T^{1-2\eta-\kappa}\\ &=&O(1), \end{array} $$

since for distributions satisfying $(\mathcal {D})$ one has that

$$ \begin{array}{@{}rcl@{}} U_{1}(y|x_{0}) = (A_{1}(x_{0}))^{\gamma_{1}(x_{0})}y^{\gamma_{1}(x_{0})}(1+a_{1}(y|x_{0})) \end{array} $$

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where |a₁(.|x₀)| is regularly varying with index equal to − γ₁(x₀)β₁(x₀), and by using the fact that $h^{\eta _{\gamma _{1}}} \ln (n/k)\to 0$ as $n\to \infty$.

Now consider Q_4,2,2,n(T). We have

$$ \begin{array}{@{}rcl@{}} Q_{4,2,2,n}(T) &=& \frac{h^{\eta_{\gamma_{1}}} T^{1-2\eta}}{1-2\eta} {\int}_{S_{K}} K^{2}(v) {\int}_{t_{n}(T)}^{\infty} p_{n}(y) \left (\frac{\overline{F}_{1}(y|x_{0}-hv)}{\overline{F}_{1}(y|x_{0})} \right )^{1-2\eta}\left (\frac{\overline{F}_{1}(y|x_{0})}{\overline{F}_{1}(t_{n}(T)|x_{0})} \right )^{1-2\eta}dy\\ && \times f_{X}(x_{0}-hv) dv. \end{array} $$

For n large and y ≥ t_n(T), with $\xi _{2,n}=Ch^{\eta _{\varepsilon _{1}}}$,

$$ \begin{array}{@{}rcl@{}} \left (\frac{\overline{F}_{1}(y|x_{0}-hv)}{\overline{F}_{1}(y|x_{0})} \right )^{1-2\eta} \le C y^{\xi_{1,n}}\left (1+y^{\xi_{2,n}}h^{\eta_{\varepsilon_{1}}} \ln y \right ). \end{array} $$

Substituting u = y/t_n(T) we get

$$ \begin{array}{@{}rcl@{}} &&Q_{4,2,2,n}(T) \leq C h^{\eta_{\gamma_{1}}} T^{1-2\eta} [t_{n}(T)]^{1+\xi_{1,n}}\\ && \times {\int}_{S_{K}} K^{2}(v) {\int}_{1}^{\infty} p_{n}(t_{n}(T)u) u^{\xi_{1,n}} \left (1+(t_{n}(T)u)^{\xi_{2,n}}h^{\eta_{\varepsilon_{1}}} \ln (t_{n}(T)u) \right )\left (\frac{\overline{F}_{1}(t_{n}(T)u|x_{0})}{\overline{F}_{1}(t_{n}(T)|x_{0})} \right )^{1-2\eta}du \\ &&\times f_{X}(x_{0}-hv) dv. \end{array} $$

Since $\overline {F}_{1}(.|x_{0})$ is regularly varying, we can apply the Potter bound (see, e.g., de Haan and Ferreira 2006, Proposition B.1.9), and obtain, for n large enough and 0 < δ < 1/γ₁(x₀)

$$ \begin{array}{@{}rcl@{}} Q_{4,2,2,n}(T) & \le & C h^{\eta_{\gamma_{1}}} T^{1-2\eta} [t_{n}(T)]^{2\xi_{1,n}}{\int}_{S_{K}} K^{2}(v)f_{X}(x_{0}-hv) dv \\ & & \times {\int}_{1}^{\infty} \left (\xi_{1,n}u^{\xi_{1,n}-1}\ln (t_{n}(T))+ \xi_{1,n}u^{\xi_{1,n}-1}\ln u +u^{\xi_{1,n}-1} \right ) u^{\xi_{1,n}- (1/\gamma_{1}(x_{0})-\delta)(1-2\eta)} \\ & & \times \left (1+(t_{n}(T)u)^{\xi_{2,n}}h^{\eta_{\varepsilon_{1}}} \ln (t_{n}(T)u) \right ) du. \end{array} $$

After tedious computations one gets

$$ \begin{array}{@{}rcl@{}} Q_{4,2,2,n}(T) & \le & C T^{1-2\eta} h^{\eta_{\gamma_{1}}} [t_{n}(T)]^{2\xi_{1,n}}\left\{1+h^{\eta_{\gamma_{1}}} \ln (t_{n}(T))+[t_{n}(T)]^{\xi_{2,n}} h^{\eta_{\varepsilon_{1}}}\ln (t_{n}(T)) \right\}\\ &=&O(1), \end{array} $$

by Eq. 17 and the fact that $h^{\eta _{\gamma _{1}}\wedge \eta _{\varepsilon _{1}}} \ln (n/k)\to 0$ as $n\to \infty$. Hence, Q_4,2,n(T) = O(1).

Finally, the two last terms Q_4,3,n(T) and Q_4,4,n(T) can be dealt with similarly as the two previous ones since

$$ \begin{array}{@{}rcl@{}} Q_{4,3,n}(T) &:=& h^{\eta_{B_{1}}} \left (\frac{n}{k} \right )^{1-2\eta} {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(T)}^{\infty} \frac{|\delta_{1}(y|x_{0})|}{(\overline{F}_{1}(y|x_{0}-hv))^{2\eta}}dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv\\ &\leq& \left( \underset{y\geq t_{n}(T)}{\sup} |\delta_{1}(y|x_{0})|\right) h^{\eta_{B_{1}}} \left (\frac{n}{k} \right )^{1-2\eta}\\ &&\times {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(T)}^{\infty} \frac{1}{(\overline{F}_{1}(y|x_{0}-hv))^{2\eta}}dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv \end{array} $$

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and

$$ \begin{array}{@{}rcl@{}} Q_{4,4,n}(T) &:=& h^{\eta_{\varepsilon_{1}}} \left (\frac{n}{k} \right )^{1-2\eta} {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(T)}^{\infty} \frac{|\delta_{1}(y|x_{0})|y^{\xi_{2,n}}\ln y}{(\overline{F}_{1}(y|x_{0}-hv))^{2\eta}}dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv\\ &\leq& \left( \underset{y\geq t_{n}(T)}{\sup} |\delta_{1}(y|x_{0})|\right) h^{\eta_{\varepsilon_{1}}} \left (\frac{n}{k} \right )^{1-2\eta}\\ && \times {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(T)}^{\infty} \frac{y^{\xi_{2,n}}\ln y}{(\overline{F}_{1}(y|x_{0}-hv))^{2\eta}}dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv. \end{array} $$

(19)

This yields Q_4,3,n(T) = O(1) and Q_4,4,n(T) = O(1). Combining all these results, we get Eq. 10.

Proof of Eq. 11. To this aim, for any α ∈ (0, 1/η − 2), we have

The terms into brackets can be studied similarly as Q_j,n(T),j = 3, 4, and thus Eq. 11 is established since $kh^{d}\to \infty$

Proof of

Eq. 12. Without loss of generality assume T = 1 and consider, for $a,\theta ,\tilde \theta <1$, the classes

$$ \begin{array}{@{}rcl@{}} \mathcal{F}_{n}^{(1)}(a) & := & \left\{ f_{n,y} \in \mathcal{F}_{n} : y_{1} \le a \right\}, \\ \mathcal{F}_{n}^{(2)}(a) & := & \left\{ f_{n,y} \in \mathcal{F}_{n} : y_{1} > a , y_{2} \le a \right\}, \\ \mathcal{F}_{n}(\ell,m) & :=& \left\{ f_{n,y} \in \mathcal{F}_{n} : \theta^{\ell+1} \le y_{1} \le \theta^{\ell}, \tilde \theta^{m+1} \le y_{2} \le \tilde \theta^{m} \right\}, \end{array} $$

where $\ell =0,\ldots , \left \lfloor \ln a/\ln \theta \right \rfloor$ and $m=0,\ldots , \left \lfloor \ln a/\ln \tilde \theta \right \rfloor$. We start by showing that $\mathcal {F}_{n}^{(1)}(a)$ is an ε −bracket, for n sufficiently large. Clearly

Then

$$ \begin{array}{@{}rcl@{}} P {u}_{1,n}^{2} & = & \left (\frac{n}{k} \right )^{1-2\eta} {\int}_{S_{K}}K^{2}(v) {\int}_{t_{n}(a)}^{\infty} \frac{1}{(\overline{F}_{1}(y|x_{0}))^{2\eta}} dF_{1}(y|x_{0}-hv) f_{X}(x_{0}-hv)dv \\ & = & Q_{3,n}(a)+Q_{4,n}(a), \end{array} $$

using the same decomposition as for $P{F_{n}^{2}}$. Thus, one can obtain the result from the above analysis of Q_3,n(T) and Q_4,n(T), taking into account that the various constants involved in these will not depend on a.

Concerning Q_3,n(a), according to Eq. 15, for n large

$$ \begin{array}{@{}rcl@{}} Q_{3,n}(a)&\le & C a^{1-2 \eta-\kappa}, \end{array} $$

where C does not depend on a. Now, taking a = ε^{3/(1 − 2η)}, for n large enough and ε small we have |Q_3,n(a)|≤ ε².

Concerning Q_4,n(a), we use the same decomposition as for Q_4,n(T) based on Eq. 16, which entails that, for n large enough, ε small and some small ζ > 0

$$ \begin{array}{@{}rcl@{}} Q_{4,1,n}(a)&\leq& \varepsilon^{2},\\ Q_{4,2,1,n}(a) &\leq& C h^{\eta_{\gamma_{1}}} \ln (t_{n}(a)) \left[t_{n}(a)\right]^{\xi_{1,n}} a^{1-2\eta-\kappa}\\ &\leq& C(1+|\ln a|)a^{-\zeta} a^{1-2\eta-\kappa} \\ & \leq & C a^{1-2\eta-2\kappa}, \end{array} $$

with C a constant not depending on a, since from Eq. 17 and for n large,

$$ \begin{array}{@{}rcl@{}} h^{\eta_{\gamma_{1}}} \ln t_{n}(a) \le C (1+|\ln a|). \end{array} $$

Also, for n large, and some small ζ > 0

$$ \begin{array}{@{}rcl@{}} Q_{4,2,2,n}(a) & \leq & Ca^{1-2\eta} h^{\eta_{\gamma_{1}}} [t_{n}(a)]^{2\xi_{1,n}}\left\{1+h^{\eta_{\gamma_{1}}} \ln (t_{n}(a)) + [t_{n}(a)]^{\xi_{2,n}} h^{\eta_{\varepsilon_{1}}}\ln(t_{n}(a))\right\}\\ & \leq & C a^{1-2\eta}h^{\eta_{\gamma_{1}}}a^{-\zeta}(1+|\ln a|+a^{-\zeta}(1+|\ln a|)) \\ & \leq & Ca^{1-2\eta-\kappa}, \end{array} $$

where C does not depend on a. Hence, for n large and ε small we obtain Q_4,2,2,n(a) ≤ ε². Using Eqs. 18 and 19, we have also Q_4,3,n(a) ≤ ε² and Q_4,4,n(a) ≤ ε². Combining all the terms we get $Pu_{1,n}^{2} \leq \varepsilon ^{2}$ for n large.

Next consider $\mathcal {F}_{n}^{(2)}(a)$. Then

and

$$ \begin{array}{@{}rcl@{}} Pu_{2,n}^{2} & = & \frac{1}{a^{2\eta}} \frac{n}{k} {\int}_{S_{K}} K^{2}(v) \overline{F}_{2}\left( U_{2}\left( \frac{n}{ka}\left|\vphantom{\frac{n}{ka}}\right.x_{0}\right)\left|\vphantom{\frac{n}{ka}}\right.x_{0}-hv\right)f_{X}(x_{0}-hv)dv \\ & \le & \varepsilon^{2}, \end{array} $$

when n is large enough and for ε small.

Finally, we consider $\mathcal {F}_{n}(\ell ,m)$. We obtain the following bounds

Then

The difference of the indicator functions can be decomposed as in Eq. 14, and subsequent calculations follow arguments similar to those used in the verification of Eq. 9, Case 3. Taking 𝜃 = 1 − ε³ and $\tilde \theta =1-a$, gives for n large enough and ε small that $P(\overline {u}_{n}-\underline {u}_{n})^{2}\le \varepsilon ^{2}$.

Combining the above, for n large and ε small one obtains that the cover number by bracketing is of the order ε^{− 4 − 3/(1 − 2η)}, and hence Eq. 12 is satisfied.

To conclude the proof, we comment on the pointwise convergence of the covariance function, which is given by $P f_{n,y}f_{n,\bar y}-P f_{n,y} Pf_{n,\bar y}$. We have

as $n \to \infty$, by the arguments used in the proof of Lemma 2. Also

as $n \to \infty$. □

1.2 A.2 Proof of Theorem 2

Recall that

We follow the lines of proof of Theorem 1. We introduce the sequence of classes $\widetilde {\mathcal {F}}_{n}$ on $\mathbb {R}\times \mathbb {R}^{d}$ as

$$ \widetilde {\mathcal{F}}_{n}:=\left\{(u,z)\to \widetilde f_{n,y}(u,z), y\in (0, T]\right\} $$

where

We have to verify Eqs. 9–12 in the proof of Theorem 1 for the new functions $\widetilde f_{n,y}$, and with $\rho _{x_{0}}(y,\bar y):=|y-\bar y|$. Without loss of generality, we may assume that $y>\overline y$. Thus, we have

with a o(1) −term which is uniform in y and $\overline y$ by Lemma 1. This yields Eq. 9.

Now, concerning Eq. 10 we can use the following envelope function of the class $\widetilde {\mathcal {F}}_{n}$

from which we deduce that

$$ \begin{array}{@{}rcl@{}} P\widetilde {F_{n}^{2}}&=&\frac{n}{k} {\int}_{S_{K}} K^{2}(v) \overline{F}_{2}\left( U_{2}\left( \frac{n}{kT}\left|\vphantom{\frac{n}{kT}}\right.x_{0}\right)\left|\vphantom{\frac{n}{kT}}\right.x_{0}-hv\right) f_{X}(x_{0}-hv) dv=O(1). \end{array} $$

Next Eq. 11 is also a direct consequence of the definition of the envelope since

as soon as $kh^{d}\to \infty$.

Finally, concerning Eq. 12, again without loss of generality we assume T = 1 and divide [0, 1] into m intervals of length 1/m. Then, for y ∈ [(i − 1)/m,i/m] we have the bounds

from which we deduce that

$$ \begin{array}{@{}rcl@{}} P\left( \underline{u}_{n}-\overline u_{n}\right)^{2}&=&\frac{1}{m} {\int}_{S_{K}} K^{2}(v) f_{X}(x_{0}-hv) dv\\ &&\left.\left.+{\int}_{S_{K}} K^{2}(v) \left[\frac{n}{k}\overline{F}_{2}\left( U_{2}\left( \frac{n}{k} \frac{m}{i}\right|x_{0}\right)\right|x_{0}-hv\right) -\frac{i}{m} \right] f_{X}(x_{0}-hv) dv\\ &&\left.\left.-{\int}_{S_{K}} K^{2}(v) \left[\frac{n}{k}\overline{F}_{2}\left( U_{2}\left( \frac{n}{k} \frac{m}{i-1}\right|x_{0}\right)\right|x_{0}-hv\right) -\frac{i-1}{m} \right] f_{X}(x_{0}-hv) dv\\ &\leq& \varepsilon^{3} {\int}_{S_{K}} K^{2}(v) f_{X}(x_{0}-hv) dv + 2\varepsilon^{3} \end{array} $$

when $m=\lceil \frac {1}{\varepsilon ^{3}}\rceil$. If ε is small and n large, then $P\left (\underline {u}_{n}-\overline u_{n}\right )^{2} \leq \varepsilon ^{2}$.

The pointwise convergence of the covariance function can be verified with arguments similar to those used in the proof of Theorem 1.

Consequently

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} \left[T_{n}(\infty, y_{2}|x_{0})-\mathbb{E}(T_{n}(\infty, y_{2}|x_{0}))\right]\leadsto W(\infty, y_{2}), \end{array} $$

in D((0,T]).

Now, remark that

$$ \begin{array}{@{}rcl@{}} \mathbb{E}(T_{n}(\infty, y_{2}|x_{0}))&=&y_{2} f_{X}(x_{0})+O\left( h^{\eta_{f_{X}}}\right)\\ &&\left.+f_{X}(x_{0}) {\int}_{S_{K}} K(v) \left[\frac{n}{k}\overline{F}_{2}\left( U_{2}\left( \frac{n}{ky_{2}}\right|x_{0}\right)\left|\vphantom{\frac{n}{ky_{2}}}\right.x_{0}-hv\right) -y_{2} \right]dv\\ &&\left.\left.+{\int}_{S_{K}} K(v) \left[\frac{n}{k}\overline{F}_{2}\left( U_{2}\left( \frac{n}{ky_{2}}\right|x_{0}\right)\right|x_{0}-hv\right) -y_{2} \right] \left[f_{X}(x_{0}-hv)-f_{X}(x_{0})\right] dv. \end{array} $$

Following the lines of proof of Lemma 1, we deduce that

$$ \begin{array}{@{}rcl@{}} \left.\left.\left.\left.\vphantom{\frac{n}{k}}\right|\frac{n}{ k} \overline{F}_{2}\left( U_{2}\left( \frac{n}{ky_{2}}\right|x_{0}\right)\right|x_{0}-hv\right)-y_{2}\right|\leq C\left\{h^{\eta_{A_{2}}} + h^{\eta_{\gamma_{2}}} \ln \frac{n}{k}+|\delta_{2}(U_{2}(n/k|x_{0})|x_{0})| \left (h^{\eta_{B_{2}}} + h^{\eta_{\varepsilon_{2}}}\ln \frac{n}{k}\right)\right\} \end{array} $$

from which we obtain

$$ \begin{array}{@{}rcl@{}} \mathbb{E}(T_{n}(\infty, y_{2}|x_{0}))& = & y_{2} f_{X}(x_{0})+O\left( h^{\eta_{f_{X}}\wedge\eta_{A_{2}}}\right)+O\left( h^{\eta_{\gamma_{2}}}\ln \frac{n}{ k}\right)+ O \left (|\delta_{2}(U_{2}(n/k|x_{0})|x_{0})|h^{\eta_{B_{2}}} \right ) \\ & & +O \left (|\delta_{2}(U_{2}(n/k|x_{0})|x_{0})| h^{\eta_{\varepsilon_{2}}}\ln \frac{n}{k}\right ) \end{array} $$

with O −terms which are uniform in y₂ ∈ (0,T]. This implies that, under the assumptions of Theorem 2, we have

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} \left[T_{n}(\infty, y_{2}|x_{0})-y_{2}f_{X}(x_{0})\right]\leadsto W(\infty, y_{2}), \end{array} $$

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in D((0,T]).

Finally,

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} \left( \frac{T_{n}(\infty, y_{2}|x_{0})}{ \widehat f_{n}(x_{0})}-y_{2}\right)=\sqrt{kh^{d}} \left( \frac{T_{n}(\infty, y_{2}|x_{0})}{ f_{X}(x_{0})}-y_{2}\right) -\frac{T_{n}(\infty, y_{2}|x_{0})}{ \widehat f_{n}(x_{0})f_{X}(x_{0})} \sqrt{\frac{k}{ n}} \sqrt{n h^{d}} \left( \widehat f_{n}(x_{0})-f_{X}(x_{0})\right), \end{array} $$

from which Theorem 2 follows. __

In the sequel, for convenient representation, all the limiting processes in Theorems 1 and 2 will be defined on the same probability space, via the Skorohod construction, but it should be kept in mind that they are only in distribution equal to the original processes. The Skorohod representation theorem gives then (with keeping the same notations)

$$ \begin{array}{@{}rcl@{}} \underset{y_{1},y_{2} \in (0,T]}{\sup} \left | \frac{\sqrt{kh^{d}} \left[ T_{n}(y_{1},y_{2}|x_{0})-\mathbb{E} (T_{n}(y_{1},y_{2}|x_{0})) \right] - W(y_{1},y_{2})}{{y}_{1}^{\eta}} \right | \to 0, \text{ a.s. } \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \underset{y_{2} \in (0,T]}{\sup} \left | \sqrt{kh^{d}} \left( \frac{T_{n}(\infty, y_{2}|x_{0})}{ \widehat f_{n}(x_{0})}-y_{2}\right) - \frac{W(\infty, y_{2})}{ f_{X}(x_{0})} \right | \to 0, \text{ a.s. .} \end{array} $$

1.3 A.3 Convergence result for an auxiliary statistic

In this section we give a convergence result for an auxiliary statistic. In particular, we generalize $\widetilde \theta _{n}$ to $\widetilde \theta _{n}(y_{2})$, defined as

Assuming F₁(y|x₀) strictly increasing in y, we have

$$ \begin{array}{@{}rcl@{}} \left.\widetilde \theta_{n}(y_{2})=-U_{1}\left( \frac{n}{k}\right|x_{0}\right){\int}_{0}^{\infty} T_{n}(s_{n}(u), y_{2}|x_{0}) du^{-\gamma_{1}(x_{0})}. \end{array} $$

As motivation for studying $\widetilde \theta _{n}(y_{2})$, note that $\widehat \theta _{n} = \widetilde \theta _{n}(\widehat e_{n})$, where $\widehat e_{n} :=\frac {n}{ k} \overline {F}_{2}\left (\widehat u_{n} U_{2}(\frac {n}{ k}|x_{0})|x_{0}\right )$ with $\widehat u_{n}:=\widehat U_{2}\left (\frac {n}{ k}|x_{0}\right ) / U_{2}\left (\frac {n}{ k}|x_{0}\right )$. To estimate U₂(.|x₀) we will use $\widehat U_{2}(.|x_{0}) := \inf \{ y: \widehat F_{n,2}(y|x_{0}) \ge 1-1/. \}$ with

the empirical kernel estimator of the unknown conditional distribution function of Y⁽²⁾ given X = x₀. See for instance Daouia et al. (2011). The asymptotic behavior of the quantile estimator is given in Lemma 6.

Proposition 1

Assume $(\mathcal {D})$, $({\mathscr{H}})$, $(\mathcal {K})$, $(\mathcal {R})$ with x → R(y₁,y₂|x) being a continuous function, x₀ ∈Int(S_X) with f_X(x₀) > 0, and y → F_j(y|x₀), j = 1, 2, are strictly increasing. Consider sequences $k\to \infty$ and h → 0 as $n \to \infty$, in such a way that k/n → 0, $kh^{d}\to \infty$ and $h^{\eta _{\gamma _{1}}\wedge \eta _{\gamma _{2}} \wedge \eta _{\varepsilon _{1}}\wedge \eta _{\varepsilon _{2}} }\ln n/k \to 0$. Then, for γ₁(x₀) < 1/2, we have

$$ \begin{array}{@{}rcl@{}} \underset{\frac{1}{2}\leq y_{2}\leq 2}{\sup}\left|\frac{\sqrt{kh^{d}}}{U_{1}(n/k|x_{0})} \left[\widetilde \theta_{n}(y_{2}) - \mathbb{E}(\widetilde\theta_{n}(y_{2}))\right] + {\int}_{0}^{\infty} W(u, y_{2})du^{-\gamma_{1}(x_{0})}\right| {\overset{\mathbb{P}}{\longrightarrow}} 0. \end{array} $$

Proof of Proposition 1

We use the decomposition

$$ \begin{array}{@{}rcl@{}} \underset{\frac{1}{2} \le y_{2} \le 2}{\sup} \left|\frac{\sqrt{kh^{d}}}{U_{1}(n/k|x_{0})} \left[\widetilde \theta_{n}(y_{2}) - \mathbb{E}(\widetilde\theta_{n}(y_{2}))\right] + {\int}_{0}^{\infty}W(u, y_{2})du^{-\gamma_{1}(x_{0})}\right|\leq I_{1}(T)+\sum\limits_{i=2}^{4} I_{i,n}(T), \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} I_{1}(T)&:=& \underset{\frac{1}{2} \le y_{2} \le 2}{\sup}\left|{\int}_{T}^{\infty} W(u, y_{2}) du^{-\gamma_{1}(x_{0})}\right|,\\ I_{2,n}(T)&:=&\underset{\frac{1}{2}\le y_{2} \le 2}{\sup} \left|{\int}_{T}^{\infty} \sqrt{kh^{d}} \left[T_{n}(s_{n}(u), y_{2}|x_{0})-\mathbb{E} \left( T_{n}(s_{n}(u), y_{2}|x_{0})\right)\right]du^{-\gamma_{1}(x_{0})}\right|,\\ I_{3,n}(T)&:=& \underset{\frac{1}{2} \le y_{2} \le 2}{\sup}\left|{{\int}_{0}^{T}} \left\{\sqrt{kh^{d}} \left[T_{n}(s_{n}(u), y_{2}|x_{0})-\mathbb{E}\left( T_{n}(s_{n}(u), y_{2}|x_{0})\right)\right] -W(s_{n}(u), y_{2})\right\} du^{-\gamma_{1}(x_{0})}\right|,\\ I_{4,n}(T)&:=& \underset{\frac{1}{2} \le y_{2} \le 2}{\sup} \left|{{\int}_{0}^{T}} \left[W(s_{n}(u), y_{2})-W(u, y_{2})\right]du^{-\gamma_{1}(x_{0})}\right|. \end{array} $$

Similarly to the proof of Proposition 2 in Cai et al. (2015), it is sufficient to show that for any ε > 0, there exists T₀ = T₀(ε) such that

$$ \begin{array}{@{}rcl@{}} \mathbb{P} (I_{1}(T_{0})>\varepsilon )<\varepsilon, \end{array} $$

(21)

and n₀ = n₀(T₀) such that, for any n > n₀

$$ \begin{array}{@{}rcl@{}} \mathbb{P}(I_{j,n}(T_{0})>\varepsilon )<\varepsilon, \text{ for $j=2, 3$ and 4.} \end{array} $$

Clearly

$$ I_{1}(T)\leq \underset{u\geq T, \frac{1}{ 2}\leq y_{2}\leq 2}{\sup} |W(u, y_{2})| T^{-\gamma_{1}(x_{0})}. $$

Since a rescaled version of our Gaussian process W(.,.) gives the one in Cai et al. (2015), according to their Lemma 2, we have $\sup _{0<u< \infty , \frac {1}{ 2}\leq y_{2}\leq 2} |W(u, y_{2})|<\infty$ with probability one. This implies that there exists T₁ = T₁(ε) such that

$$ \mathbb{P}\left( \underset{0<u< \infty, \frac{1}{ 2}\leq y_{2}\leq 2}{\sup} |W(u, y_{2})|>{T}_{1}^{\gamma_{1}(x_{0})}\varepsilon\right)<\varepsilon, $$

from which we deduce that, for any T > T₁

$$ \mathbb{P}\left( I_{1}(T)>\varepsilon\right)\leq \mathbb{P}\left( \underset{0<u<\infty, \frac{1}{ 2}\leq y_{2}\leq 2}{\sup} |W(u, y_{2})|>{T}_{1}^{\gamma_{1}(x_{0})}\varepsilon\right)<\varepsilon. $$

Consequently Eq. 21 holds for T₀ > T₁.

We continue with the term I_2,n(T). We have

Consider the class of functions

with y₁ ≥ T and 1/2 ≤ y₂ ≤ 2, and with envelope function

This class of functions satisfies the conditions of Theorem 7.3 in Wellner (2005) with σ² = O(kh^d/n) and $P{G_{n}^{2}} = O(kh^{d}/n)$ for n large, and thus, for some constant C,

$$ \begin{array}{@{}rcl@{}} \mathbb{P}\left( I_{2,n}(T)>\varepsilon\right) \le \frac{C}{\varepsilon T^{\gamma_{1}(x_{0})}} \end{array} $$

for n large enough. We have then that for every ε there is a T = T(ε) such that for n large enough

$$ \begin{array}{@{}rcl@{}} \mathbb{P}\left( I_{2,n}(T)>\varepsilon\right) \le \varepsilon. \end{array} $$

Now, to study I_3,n(T), remark that for any T > 0,∃n₁ = n₁(T) : ∀n > n₁ : s_n(T) < T + 1. Hence for n > n₁ and any $\eta _{0}\in \left (\gamma _{1}(x_{0}), 1/2\right ):$

$$ \begin{array}{@{}rcl@{}} \mathbb{P}\left( I_{3,n}(T)>\varepsilon\right)&\leq& \mathbb{P}\left( \underset{0<y_{1}\leq T+1, \frac{1}{ 2}\leq y_{2}\leq 2}{\sup} \left|\frac{\sqrt{kh^{d}} [T_{n}(y_{1},y_{2}|x_{0})-\mathbb{E}(T_{n}(y_{1},y_{2}|x_{0}))]-W(y_{1}, y_{2})}{{y}_{1}^{\eta_{0}}}\right| \right. \\ &&\left. \times \left|{{\int}_{0}^{T}} [s_{n}(u)]^{\eta_{0}} du^{-\gamma_{1}(x_{0})}\right|>\varepsilon\right). \end{array} $$

According to Lemma 3 in Cai et al. (2015)

$$ \left|{{\int}_{0}^{T}} [s_{n}(u)]^{\eta_{0}} du^{-\gamma_{1}(x_{0})}\right|\longrightarrow \frac{\gamma_{1}(x_{0})}{ \eta_{0}-\gamma_{1}(x_{0})} T^{\eta_{0}-\gamma_{1}(x_{0})}, $$

which, combining with our Theorem 1 and the Skorohod construction, entails that there exists n₂(T) > n₁(T) such that ∀n > n₂(T), $\mathbb {P}(I_{3,n}(T)>\varepsilon )<\varepsilon$.

Finally, concerning I_4,n(T), we first remark that according to Lemma 2 in Cai et al. (2015), we have for η₀ ∈ (γ₁(x₀), 1/2) and any T > 0, with probability one,

$$ \underset{0<y_{1} \le T, \frac{1}{ 2}\leq y_{2}\leq 2}{\sup} \frac{|W(y_{1}, y_{2})| }{{y}_{1}^{\eta_{0}}}<\infty. $$

Then, applying Lemma 3 in Cai et al. (2015) with S = T,S₀ = T + 1 and g = W, we deduce that there exists n₃(T) such that for n > n₃(T) we have $\mathbb {P}(I_{4,n}(T)>\varepsilon )<\varepsilon$.

This achieves the proof of Proposition 1. □

In order to prove Theorem 3 we need some auxiliary results. Define for u > 0 and v ∈ S_K

$$ \begin{array}{@{}rcl@{}} \widetilde{s}_{n}(u)&:=&\frac{n}{k} \overline{F}_{1}\left( u^{-\gamma_{1}(x_{0})} U_{1}\left( \frac{n}{k}\left|\vphantom{\left( \frac{n}{k}\left|x_{0}\right.\right)}x_{0}\right.\right)\left.\vphantom{\left( \frac{n}{k}\left|x_{0}\right.\right)}\right|x_{0}-hv\right),\\ t_{n}(y_{2})&:=&\frac{n}{k} \overline{F}_{2}\left( U_{2}\left( \frac{n}{ky_{2}}\left|\vphantom{\frac{n}{ky_{2}}}x_{0}\right.\right)\left|x_{0}\vphantom{\frac{n}{ky_{2}}}\right.-hv\right). \end{array} $$

Lemma 3

Assume $(\mathcal {D})$ and $({\mathscr{H}})$ and x₀ ∈ Int(S_X). Consider sequences $k\to \infty$ and h → 0 as $n\to \infty$, in such a way that k/n → 0 and $h^{\eta _{\varepsilon _{1}}\wedge \eta _{\gamma _{1}}} \ln \frac {n}{ k}\to 0$. Then, we have, for any $u\leq T_{n}\to \infty$ such that kT_n/n → 0 and 0 < ε < β₁(x₀), that

$$ \begin{array}{@{}rcl@{}} \left|\vphantom{\frac{n}{k}}\widetilde{s}_{n}\right.(u)-u\left|\vphantom{\frac{n}{k}}\right.&\leq& C u\left\{h^{\eta_{A_{1}}} + h^{\eta_{\gamma_{1}}} \ln\frac{n}{k} + h^{\eta_{\gamma_{1}}}|\ln u| u^{\pm Ch^{\eta_{\gamma_{1}}}}\right.\\ &&+ \left|\delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right. x_{0}\right)\right|\left[1+u^{\pm Ch^{\eta_{\gamma_{1}}}}h^{\eta_{\gamma_{1}}}|\ln u|\right] \\ & & \times \left[ u^{\gamma_{1}(x_{0})\beta_{1}(x_{0})}\left (1+u^{\pm \gamma_{1}(x_{0}) \varepsilon} \right ) \left (h^{\eta_{B_{1}}} +u^{-Ch^{\eta_{\varepsilon_{1}}}}h^{\eta_{\varepsilon_{1}}} \left (|\ln u| +\ln \frac{n}{k} \right ) \right ) \right . \\ && + \left . \left . u^{\gamma_{1}(x_{0})(\beta_{1}(x_{0})\pm \varepsilon)} + \left | u^{\gamma_{1}(x_{0})\beta_{1}(x_{0})} -1 \right | \right] \right\}, \end{array} $$

where u^±∙ means u^∙ if u is greater than 1, and u^−∙ if u is smaller than 1.

Lemma 4

Assume $(\mathcal {D})$, $({\mathscr{H}})$, γ₁(x₀) < 1 and x₀ ∈ Int(S_X). For sequences $k=\left \lfloor n^{\alpha } \ell _{1}(n)\right \rfloor$ and h = n^−Δℓ₂(n), where ℓ₁ and ℓ₂ are slowly varying functions at infinity, with α ∈ (0, 1) and

$$ \begin{array}{@{}rcl@{}} &&\max\left( \frac{\alpha}{d+2\gamma_{1}(x_{0})(\eta_{A_{1}}\wedge \eta_{\gamma_{1}})}, \frac{\alpha}{d+2(1-\gamma_{1}(x_{0}))(\eta_{A_{2}}\wedge \eta_{\gamma_{2}}\wedge \eta_{B_{2}}\wedge \eta_{\varepsilon_{2}})}, \right.\\ && \left.\frac{\alpha}{ d}-\frac{2(1-\alpha){\gamma_{1}^{2}}(x_{0})\beta_{1}(x_{0})}{d+d(\beta_{1}(x_{0})+\varepsilon)\gamma_{1}(x_{0})}, \frac{\alpha-2(1-\alpha)\gamma_{1}(x_{0})}{d}\right)<{{\varDelta}} <\frac{\alpha }{ d}, \end{array} $$

one has that

$$ \underset{v \in S_{K}}{\sup} \underset{\frac{1}{2}\leq y_{2}\leq 2}{\sup} \sqrt{kh^{d}} \left|{\int}_{0}^{\infty} \left[R\left( \widetilde s_{n}(u), t_{n}(y_{2})|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\longrightarrow 0 $$

and

$$ \underset{\frac{1}{2}\leq y_{2}\leq 2}{\sup} \sqrt{kh^{d}} \left|{\int}_{0}^{\infty} \left[R\left( s_{n}(u), y_{2}|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\longrightarrow 0. $$

Lemma 5

Assume $(\mathcal {D})$, $({\mathscr{H}})$, $(\mathcal {K})$, x₀ ∈ Int(S_X) with f_X(x₀) > 0 and y → F₂(y|x₀) is strictly increasing. Consider sequences $k\to \infty$ and h → 0 as $n\to \infty$, in such a way that k/n → 0, $kh^{d}\to \infty$, $h^{\eta _{\varepsilon _{2}}}\ln n/k \to 0$, $\sqrt {kh^{d}} h^{\eta _{f_{X}}\wedge \eta _{A_{2}}}\to 0$, $\sqrt {kh^{d}} h^{\eta _{\gamma _{2}}}\ln n/k\to 0$, $\sqrt {kh^{d}} |\delta _{2}(U_{2}(n/k|x_{0})|x_{0})| h^{\eta _{B_{2}}}\to 0$, and

$\sqrt {kh^{d}} |\delta _{2}(U_{2}(n/k|x_{0})|x_{0})| h^{\eta _{\varepsilon _{2}}}\ln n/k \to 0$. Then, for any sequence u_n satisfying

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} \left (\frac{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})} -1\right ) \rightarrow c \in \mathbb{R}, \end{array} $$

as $n \to \infty$, we have

$$ \begin{array}{@{}rcl@{}} \sqrt{nh^{d}\overline{F}_{2}(u_{n}|x_{0})}\left( \frac{\widehat{\overline{F}}_{n,2}(u_{n}|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})}-1 \right) \leadsto \frac{W(\infty, 1)}{ f_{X}(x_{0})}. \end{array} $$

Lemma 6

Assume $(\mathcal {D})$, $({\mathscr{H}})$, $(\mathcal {K})$, x₀ ∈ Int(S_X) with f_X(x₀) > 0 and y → F₂(y|x₀) is strictly increasing. Consider sequences $k\to \infty$ and h → 0 as $n\to \infty$, in such a way that k/n → 0, $kh^{d}\to \infty$, $h^{\eta _{\varepsilon _{2}}}\ln n/k \to 0$, $\sqrt {kh^{d}} h^{\eta _{f_{X}}\wedge \eta _{A_{2}}}\to 0$, $\sqrt {kh^{d}} h^{\eta _{\gamma _{2}}}\ln n/k\to 0$, $\sqrt {kh^{d}} |\delta _{2}(U_{2}(n/k|x_{0})|x_{0})|\to 0$. Then, as $n \to \infty$, we have

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} \left (\widehat u_{n} -1 \right ) \leadsto \frac{\gamma_{2}(x_{0}) W(\infty,1)}{f_{X}(x_{0})}. \end{array} $$

1.4 A.4 Proof of Theorem 3

Let $\mathbb {E}_{n}(y) := \mathbb {E}\left (\widetilde \theta _{n}(y)/U_{1}(n/k|x_{0})\right )$. We have the following decomposition:

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} \left( \frac{\widehat \theta_{n} }{ f_{X}(x_{0})\theta_{k/n}} - 1\right)&=& \frac{U_{1}(n/k|x_{0}) }{ \theta_{k/n}} \frac{\sqrt{kh^{d}} }{ f_{X}(x_{0})} \left( \frac{\widehat \theta_{n}}{U_{1}(n/k|x_{0})} - \mathbb{E}_{n}(1)\right)\\ &&+\frac{U_{1}(n/k|x_{0})}{\theta_{k/n}} \frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \mathbb{E}_{n}(1) - \frac{f_{X}(x_{0})\theta_{k/n}}{U_{1}(n/k|x_{0}) }\right)\\ &=& \frac{U_{1}(n/k|x_{0})}{\theta_{k/n}} \frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \frac{\widetilde \theta_{n}(\widehat e_{n})}{U_{1}(n/k|x_{0})} - \mathbb{E}_{n}(\widehat e_{n})\right)\\ &&+\frac{U_{1}(n/k|x_{0})}{\theta_{k/n}} \frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \mathbb E_{n}(\widehat e_{n}) - \mathbb E_{n}(1)\right)\\ &&+\frac{U_{1}(n/k|x_{0})}{\theta_{k/n}} \frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \mathbb E_{n}(1) - \frac{f_{X}(x_{0})\theta_{k/n}}{U_{1}(n/k|x_{0}) }\right)\\ &=:& T_{1} + T_{2} + T_{3}. \end{array} $$

First, remark that the common factor of the three terms, U₁(n/k|x₀)/𝜃_k/n can be handled in a similar way as in Proposition 1 in Cai et al. (2015), i.e., as $n\to \infty$

$$ \frac{U_{1}(n/k|x_{0})}{\theta_{k/n}} \longrightarrow {-1\over {\int}_{0}^{\infty} R(s, 1|x_{0}) ds^{-\gamma_{1}(x_{0})}}. $$

Thus the three terms without this factor need to be studied.

We start with T₁. Note that

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} (\widehat e_{n}-1) = -\frac{f_{2}(\tilde u_{n}U_{2}(n/k|x_{0})|x_{0})U_{2}(n/k|x_{0})}{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})}\sqrt{kh^{d}} (\widehat u_{n}-1), \end{array} $$

where $\tilde u_{n}$ is a random value between $\widehat u_{n}$ and 1. By the continuous mapping theorem we have then

$$ \begin{array}{@{}rcl@{}} \frac{f_{2}(\tilde u_{n}U_{2}(n/k|x_{0})|x_{0})U_{2}(n/k|x_{0})}{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})} \stackrel{\mathbb P}{\rightarrow} \frac{1}{\gamma_{2}(x_{0})}, \end{array} $$

and hence by Lemma 6

$$ \begin{array}{@{}rcl@{}} \sqrt{kh^{d}} (\widehat e_{n}-1) \leadsto -W(\infty,1)/f_{X}(x_{0}). \end{array} $$

(22)

This implies that

$$ \mathbb P\left( |\widehat e_{n}-1|>(kh^{d})^{-1/4}\right) \to 0. $$

Hence, with probability tending to one,

$$ \begin{array}{@{}rcl@{}} &&\left|\frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \frac{\widetilde \theta_{n}(\widehat e_{n})}{U_{1}(n/k|x_{0})} - \mathbb E_{n}(\widehat e_{n})\right) +\frac{1}{f_{X}(x_{0})} {\int}_{0}^{\infty} W(s, 1) ds^{-\gamma_{1}(x_{0})}\right|\\ &\leq& \underset{|y-1|\leq (kh^{d})^{-1/4}}{\sup} \left|\frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \frac{\widetilde \theta_{n}(y)}{U_{1}(n/k|x_{0})} - \mathbb E_{n}(y)\right) +\frac{1}{f_{X}(x_{0})} {\int}_{0}^{\infty} W(s, y) ds^{-\gamma_{1}(x_{0})}\right|\\ &&+ \frac{1}{f_{X}(x_{0})} \underset{|y-1|\leq (kh^{d})^{-1/4}}{\sup} \left|{\int}_{0}^{\infty} [W(s, y)-W(s,1)] ds^{-\gamma_{1}(x_{0})}\right|. \end{array} $$

The first term of the right-hand side tends to 0 in probability by our Proposition 1, whereas the second term can be handled similarly as in the proof of Proposition 3 in Cai et al. (2015). Consequently

$$ \begin{array}{@{}rcl@{}} T_{1} \leadsto \frac{1}{{\int}_{0}^{\infty} R(s, 1|x_{0}) ds^{-\gamma_{1}(x_{0})}}\frac{1}{f_{X}(x_{0})} {\int}_{0}^{\infty} W(s, 1) ds^{-\gamma_{1}(x_{0})}. \end{array} $$

(23)

Next step consists to look at T₂. To this aim, remark that for y equal either to 1 or $\widehat e_{n}$, we have

$$ \begin{array}{@{}rcl@{}} &&{}{\int}_{0}^{\infty} \mathbb E\left( T_{n}(s_{n}(u), y|x_{0})\right)du^{-\gamma_{1}(x_{0})}\\ &=&{\int}_{0}^{\infty} {\int}_{S_{K}} K(v) R_{\frac{n}{k}}\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}-hv\right) f_{X}(x_{0}-hv)dv du^{-\gamma_{1}(x_{0})}\\ &=&{\int}_{0}^{\infty} {\int}_{S_{K}} K(v) R\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}\right) f_{X}(x_{0}-hv)dv du^{-\gamma_{1}(x_{0})}\\ &&+{\int}_{0}^{\infty} {\int}_{S_{K}} K(v) \left[R_{\frac{n}{k}}\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}-hv\right) - R\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}\right)\right] f_{X}(x_{0}-hv)dv du^{-\gamma_{1}(x_{0})}\\ &=& {\int}_{0}^{\infty} R\left( u, y|x_{0}\right) du^{-\gamma_{1}(x_{0})} {\int}_{S_{K}} K(v)f_{X}(x_{0}-hv) dv\\ &&+{\int}_{S_{K}} K(v) {\int}_{0}^{\infty} \left[R\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}\right) - R(u, y|x_{0})\right] du^{-\gamma_{1}(x_{0})} f_{X}(x_{0}-hv)dv\\ &&+ {\int}_{0}^{\infty} {\int}_{S_{K}} K(v) \left[R_{\frac{n}{k}}\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}-hv\right) - R\left( \widetilde s_{n}(u), t_{n}(y)|x_{0}\right)\right] f_{X}(x_{0}-hv)dv du^{-\gamma_{1}(x_{0})} \\ & =:& \widetilde T_{2,1}+\widetilde T_{2,2}+\widetilde T_{2,3}. \end{array} $$

By Lemma 4, Assumptions $(\mathcal {S})$ and $({\mathscr{H}})$ we obtain

$$ \begin{array}{@{}rcl@{}} \widetilde{T}_{2,1 }&=&f_{X}(x_{0}) {\int}_{0}^{\infty} R\left( u, y|x_{0}\right) du^{-\gamma_{1}(x_{0})}+O_{\mathbb{P}}\left( h^{\eta_{f_{X}}}\right), \\ \widetilde T_{2,2}&=& o_{\mathbb{P}}\left( \frac{1}{\sqrt{kh^{d}}}\right),\\ | \widetilde{T}_{2,3} | & \le & - \underset{x\in B(x_{0}, h)}{\sup}\underset{0<y_{1}<\infty, \frac{1}{2}\leq y_{2}\leq 2}{\sup} \frac{|R_{n/k}(y_{1},y_{2}|x)-R(y_{1}, y_{2}|x_{0})|}{{y}_{1}^{\beta} \wedge 1} \\ & & \times {\int}_{S_{K}} K(v){\int}_{0}^{\infty} \left( [\widetilde s_{n}(u)]^{\beta}\wedge 1\right) du^{-\gamma_{1}(x_{0})} f_{X}(x_{0}-hv)dv \\ & = & O_{\mathbb P}\left( \left( \frac{n}{k}\right)^{\tau}\right). \end{array} $$

Note that the integral appearing in the bound for $|\widetilde T_{2,3}|$ is finite for n large, as $\widetilde s_{n}(u) \le C u^{1-\xi }$ for u ∈ (0, 1/2],ξ ∈ (0, (β − γ₁(x₀))/β) and n large. Consequently, under our assumptions and using the homogeneity of the R −function and the mean value theorem combining with Eq. 22, we have

$$ \begin{array}{@{}rcl@{}} &&\frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \mathbb{E}_{n}(\widehat e_{n}) - \mathbb{E}_{n}(1)\right)\\ &=&\frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( {\int}_{0}^{\infty} \mathbb E\left( T_{n}(s_{n}(u), 1|x_{0})\right)du^{-\gamma_{1}(x_{0})}- {\int}_{0}^{\infty} \mathbb E\left( T_{n}(s_{n}(u), \widehat e_{n}|x_{0})\right)du^{-\gamma_{1}(x_{0})}\right)\\ &=&\sqrt{kh^{d}} \left( {\int}_{0}^{\infty} R\left( u, 1|x_{0}\right) du^{-\gamma_{1}(x_{0})} - {\int}_{0}^{\infty} R\left( u, \widehat e_{n}|x_{0}\right) du^{-\gamma_{1}(x_{0})}\right)+o_{\mathbb{P}}(1)\\ &=&\sqrt{kh^{d}} \left( 1-\widehat e_{n}^{1-\gamma_{1}(x_{0})}\right){\int}_{0}^{\infty} R\left( u, 1|x_{0}\right) du^{-\gamma_{1}(x_{0})}+o_{\mathbb{P}}(1)\\ & \leadsto & (1-\gamma_{1}(x_{0})) \frac{W(\infty, 1)}{f_{X}(x_{0})} {\int}_{0}^{\infty} R\left( u, 1|x_{0}\right) du^{-\gamma_{1}(x_{0})}. \end{array} $$

This implies that

$$ \begin{array}{@{}rcl@{}} T_{2} \leadsto -(1-\gamma_{1}(x_{0})) \frac{W(\infty, 1)}{f_{X}(x_{0})}. \end{array} $$

(24)

Finally, for T₃ we have,

$$ \begin{array}{@{}rcl@{}} &&\frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( \mathbb E_{n}(1) - \frac{f_{X}(x_{0})\theta_{k/n}}{U_{1}(n/k|x_{0}) }\right)\\ &=&\frac{\sqrt{kh^{d}}}{f_{X}(x_{0})} \left( -{\int}_{0}^{\infty} \mathbb E(T_{n}(s_{n}(u), 1|x_{0})) du^{-\gamma_{1}(x_{0})} - \frac{f_{X}(x_{0})\theta_{k/n}}{U_{1}(n/k|x_{0}) }\right)\\ &=&\sqrt{kh^{d}} {\int}_{0}^{\infty} \left[R_{n/k}(s_{n}(u), 1|x_{0})-R(u,1|x_{0})\right] du^{-\gamma_{1}(x_{0})} + o(1)\\ &=&\sqrt{kh^{d}} {\int}_{0}^{\infty} \left[R_{n/k}(s_{n}(u), 1|x_{0})-R(s_{n}(u),1|x_{0})\right] du^{-\gamma_{1}(x_{0})}\\ &&+\sqrt{kh^{d}} {\int}_{0}^{\infty} \left[R(s_{n}(u), 1|x_{0})-R(u,1|x_{0})\right] du^{-\gamma_{1}(x_{0})}+ o (1)\\ & =: & \widetilde T_{3,1}+\widetilde T_{3,2}+o(1), \\ \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} |\widetilde T_{3,1}| &\leq& \sqrt{kh^{d}} \underset{x\in B(x_{0}, h)}{\sup}\underset{0<y_{1}<\infty, \frac{1}{2}\leq y_{2}\leq 2}{\sup} \frac{|R_{n/k}(y_{1},y_{2}|x)-R(y_{1}, y_{2}|x_{0})|}{{y}_{1}^{\beta} \wedge 1}\\ &&\times \left|{\int}_{0}^{\infty} \left( [ s_{n}(u)]^{\beta}\wedge 1\right) du^{-\gamma_{1}(x_{0})}\right|\\ &=&O\left( \sqrt{kh^{d}}\left( \frac{n}{k}\right)^{\tau}\right), \\ \widetilde T_{3,2} & = & o(1). \end{array} $$

Overall, we have then

$$ \begin{array}{@{}rcl@{}} T_{3}=o(1). \end{array} $$

(25)

Combining Eqs. 23, 24 and 25, and following the argument as at the end of the proof of Theorem 2, we can establish the result of Theorem 3.

1.5 A.5 Proofs of the auxiliary results

Proof of Lemma 1

First note that, by continuity of y → F_j(y|x),

$$ \begin{array}{@{}rcl@{}} t_{n}\overline{F}_{j}(U_{j}(t_{n}/y|x_{0})|x) = y \frac{\overline{F}_{j}(U_{j}(t_{n}/y|x_{0})|x)}{\overline{F}_{j}(U_{j}(t_{n}/y|x_{0})|x_{0})}. \end{array} $$

Then, from condition $(\mathcal {D})$, and a straightforward decomposition,

$$ \begin{array}{@{}rcl@{}} \lefteqn{\left | \frac{t_{n} \overline{F}_{j}(U_{j}(t_{n}/y|x_{0})|x)}{y^{\eta}} -y^{1-\eta} \right | } \\ &\le & y^{1-\eta} \left \{ \left | \frac{A_{j}(x)}{A_{j}(x_{0})}-1 \right | (U_{j}(t_{n}/y|x_{0}))^{1/\gamma_{j}(x_{0})-1/\gamma_{j}(x)}\frac{1+\frac{1}{\gamma_{j}(x)} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x)}{1+\frac{1}{\gamma_{j}(x_{0})} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x_{0})} \right . \\ & & + \left | (U_{j}(t_{n}/y|x_{0}))^{1/\gamma_{j}(x_{0})-1/\gamma_{j}(x)}-1 \right | \frac{1+\frac{1}{\gamma_{j}(x)} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x)}{1+\frac{1}{\gamma_{j}(x_{0})} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x_{0})} \\ & & + \left . \left | \frac{1+\frac{1}{\gamma_{j}(x)} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x)}{1+\frac{1}{\gamma_{j}(x_{0})} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x_{0})}-1 \right | \right \}. \end{array} $$

Each of the absolute differences in the right-hand side of the above display can be handled by condition $(\mathcal H)$. Obviously, for some constant C,

$$ \begin{array}{@{}rcl@{}} \left | \frac{A_{j}(x)}{A_{j}(x_{0})}-1 \right | \le C h_{n}^{\eta_{A_{j}}}, \text{ for } x \in B(x_{0},h_{n}). \end{array} $$

Next, using the inequality |e^z − 1|≤ e^|z||z|, we have, for some constant C (not necessarily equal to the one introduced above), and x ∈ B(x₀,h_n),

$$ \begin{array}{@{}rcl@{}} \left | (U_{j}(t_{n}/y|x_{0}))^{1/\gamma_{j}(x_{0})-1/\gamma_{j}(x)}-1 \right | \le e^{Ch_{n}^{\eta_{\gamma_{j}}} \ln U_{j}(t_{n}/y|x_{0})} Ch_{n}^{\eta_{\gamma_{j}}} \ln U_{j}(t_{n}/y|x_{0}). \end{array} $$

For distributions satisfying $(\mathcal D)$, one easily verifies that

$$ U_{j}(t_{n}|x_{0}) = (A_{j}(x_{0}))^{\gamma_{j}(x_{0})}t_{n}^{\gamma_{j}(x_{0})}(1+a_{j}(t_{n}|x_{0})) $$

where |a_j(.|x₀)| is regularly varying with index equal to − γ_j(x₀)β_j(x₀). Hence, for some constants C₁ and C₂, not depending on y, one gets for x ∈ B(x₀,h_n) and n large,

$$ \begin{array}{@{}rcl@{}} \left | (U_{j}(t_{n}/y|x_{0}))^{1/\gamma_{j}(x_{0})-1/\gamma_{j}(x)}-1 \right | \le C_{1}t_{n}^{C_{2}h_{n}^{\eta_{\gamma_{j}}}}y^{-C_{2}h_{n}^{\eta_{\gamma_{j}}}}\left ({h}_{n}^{\eta_{\gamma_{j}}} \ln t_{n} - {h}_{n}^{\eta_{\gamma_{j}}} \ln y \right ). \end{array} $$

Finally, for n large,

$$ \begin{array}{@{}rcl@{}} \lefteqn{\left | \frac{1+\frac{1}{\gamma_{j}(x)} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x)}{1+\frac{1}{\gamma_{j}(x_{0})} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x_{0})}-1 \right | } \\ &\le & C |\delta_{j}(U_{j}(t_{n}/y|x_{0})|x_{0})| \left \{ \left | \frac{\delta_{j}(U_{j}(t_{n}/y|x_{0})|x)}{\delta_{j}(U_{j}(t_{n}/y|x_{0})|x_{0})} -1 \right | + \left | \frac{1}{\gamma_{j}(x)}-\frac{1}{\gamma_{j}(x_{0})} \right | \right \}. \end{array} $$

By the assumptions on δ_j we obtain

$$ \begin{array}{@{}rcl@{}} \left | \frac{\delta_{j}(U_{j}(t_{n}/y|x_{0})|x)}{\delta_{j}(U_{j}(t_{n}/y|x_{0})|x_{0})} -1 \right | & \le & \left | \frac{B_{j}(x)}{B_{j}(x_{0})}-1 \right | e^{{\int}_{1}^{U_{j}(t_{n}/y|x_{0})} \frac{\varepsilon_{j}(u|x)-\varepsilon_{j}(u|x_{0})}{u} du} \\ & & + \left | e^{{\int}_{1}^{U_{j}(t_{n}/y|x_{0})} \frac{\varepsilon_{j}(u|x)-\varepsilon_{j}(u|x_{0})}{u} du}-1 \right |, \end{array} $$

and, hence, using $(\mathcal H)$, for x ∈ B(x₀,h_n) and n large,

$$ \begin{array}{@{}rcl@{}} \lefteqn{\left | \frac{1+\frac{1}{\gamma_{j}(x)} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x)}{1+\frac{1}{\gamma_{j}(x_{0})} \delta_{j}(U_{j}(t_{n}/y|x_{0}) |x_{0})}-1 \right |} \\ & & \le C_{1} \left [ {h}_{n}^{\eta_{\gamma_{j}} \wedge \eta_{B_{j}}} +t_{n}^{C_{2}h_{n}^{\eta_{\varepsilon_{j}}}}y^{-C_{2}h_{n}^{\eta_{\varepsilon_{j}}}}\left (h_{n}^{\eta_{\varepsilon_{j}}} \ln t_{n} - h_{n}^{\eta_{\varepsilon_{j}}} \ln y \right ) \right ]. \end{array} $$

Combining the above results establishes the lemma. □

Proof of Lemma 2

We have

Concerning T_1,n, by the continuity of f_X(x) and R(y₁,y₂|x) at x₀, we have that f_X and R are bounded in a neighborhood of x₀, and hence, by Lebesgue’s dominated convergence theorem

$$ \begin{array}{@{}rcl@{}} T_{1,n} \to f_{X}(x_{0})R(y_{1},y_{2}|x_{0}), \text{ as } n \to \infty. \end{array} $$

As for T_2,n,

$$ \begin{array}{@{}rcl@{}} |T_{2,n}| & \le & \underset{v \in S_{K}}{\sup} \left | \frac{n}{k} \mathbb P\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right) \le (k/n) y_{1}, \overline{F}_{2}\left( Y^{(2)}|x_{0}\right) \le (k/n) y_{2} |X=x_{0}-hv\right) \right . \\ & & -R(y_{1},y_{2}|x_{0}-hv) \left | {\int}_{S_{K}} K(v) f_{X}(x_{0}-hv) dv,\right. \end{array} $$

and note that

$$ \begin{array}{@{}rcl@{}} &&{}{\mathbb{P}\left( \overline{F}_{1}\left( Y^{(1)}|x_{0}\right) \le (k/n) y_{1}, \overline{F}_{2}\left( Y^{(2)}|x_{0}\right) \le (k/n) y_{2} |X=x_{0}-hv\right) } \\ & = & \mathbb P \left (\overline{F}_{1}\left( Y^{(1)}|x_{0}-hv\right) \le \frac{k}{n} \frac{n}{k}\overline{F}_{1}(U_{1}(n/(ky_{1})|x_{0})|x_{0}-hv), \right . \\ & & \left . \overline{F}_{2}\left( Y^{(2)}|x_{0}-hv\right) \le \frac{k}{n} \frac{n}{k}\overline{F}_{2}(U_{2}(n/(ky_{2})|x_{0})|x_{0}-hv) | X=x_{0}-hv \right ). \end{array} $$

Then, by the result of Lemma 1 and the uniformity of the convergence in Assumption $(\mathcal R)$, we have that T_2,n → 0 as $n \to \infty$.

Now, consider the variance. We have

from which the result follows. □

Proof of Lemma 3

Using Assumption $(\mathcal D)$, we have

$$ \begin{array}{@{}rcl@{}} \widetilde s_{n}(u)&=& \frac{\overline{F}_{1}\left( u^{-\gamma_{1}(x_{0})} U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}-hv\right)}{\overline{F}_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)}\\ &=&\frac{A_{1}(x_{0}-hv)}{A_{1}(x_{0})} \left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\right)^{\frac{1}{\gamma_{1}(x_{0})}-\frac{1}{\gamma_{1}(x_{0}-hv)}} u^{\frac{\gamma_{1}(x_{0})}{\gamma_{1}(x_{0}-hv)}} \\ &&\times \frac{1+\frac{1}{\gamma_{1}(x_{0}-hv)} \delta_{1}\left( u^{-\gamma_{1}(x_{0})} U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}-hv\right) }{ 1+\frac{1}{\gamma_{1}(x_{0})} \delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)}. \end{array} $$

This implies that

$$ \begin{array}{@{}rcl@{}} &&\left|\widetilde s_n(u)-u^{\frac{\gamma_1(x_0)}{\gamma_1(x_0-hv)}}\right| \\ &\leq& u^{\frac{\gamma_1(x_0)}{\gamma_1(x_0-hv)}} \left\{\left|\frac{A_1(x_0-hv)}{A_1(x_0)}-1\right| \left( U_1\left( \frac{n}{k}\left|x_0\right.\right)\right)^{\frac{1}{\gamma_1(x_0)}-\frac{1}{\gamma_1(x_0-hv)}}\right.\\ &&\times\left|\frac{1+\frac{1}{\gamma_1(x_0-hv)} \delta_1\left( u^{-\gamma_1(x_0)} U_1\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}x_0\right.\right)\left.\vphantom{\frac{n}{k}}\right|x_0-hv\right)}{1+\frac{1}{\gamma_1(x_0)} \delta_1\left( U_1\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}x_0\right.\right)\left.\vphantom{\frac{n}{k}}\right|\vphantom{\frac{n}{k}}x_0\right)}\right|\\ &&+\left|\left( U_1\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_0\right)\right)^{\frac{1}{\gamma_1(x_0)}-\frac{1}{\gamma_1(x_0-hv)}}-1\right|\\ &&\times\left|\frac{1+\frac{1}{\gamma_1(x_0-hv)} \delta_1\left( u^{-\gamma_1(x_0)} U_1\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}x_0\right.\right)\left.\vphantom{\frac{n}{k}}\right|x_0-hv\right)}{1+\frac{1}{\gamma_1(x_0)} \delta_1\left( U_1\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}x_0\right.\right)\left.\vphantom{\frac{n}{k}}\right|x_0\right)}\right|\\ &&\left.+\left|\frac{1+\frac{1}{\gamma_1(x_0-hv)} \delta_1\left( u^{-\gamma_1(x_0)} U_1\left( \frac{n}{k}\left|x_0\right.\right)\left.\vphantom{\frac{n}{k}}\right|x_0-hv\right)}{1+\frac{1}{\gamma_1(x_0)} \delta_1\left( U_1\left( \frac{n}{k}\left|x_0\right.\right)\left.\vphantom{\frac{n}{k}}\right|x_0\right)}-1\right|\right\}\\ &=:&u^{\frac{\gamma_1(x_0)}{\gamma_1(x_0-hv)}} \{T_1+T_2+T_3\}. \end{array} $$

Using Assumption $(\mathcal H)$ and the inequality |e^x − 1|≤|x|e^|x|, we deduce that, for n large,

$$ \begin{array}{@{}rcl@{}} |\frac{A_{1}(x_{0}-hv)}{A_{1}(x_{0})}-1 | & \leq & C h^{\eta_{A_{1}}} \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} | \left( U_{1}\left( \frac{n}{k}|x_{0}\right)\right)^{\frac{1}{\gamma_{1}(x_{0})}-\frac{1}{\gamma_{1}(x_{0}-hv)}}-1 | &\leq & C h^{\eta_{\gamma_{1}}} \ln\frac{n}{k}. \end{array} $$

(27)

Now, direct computations yield, for n large,

$$ \begin{array}{@{}rcl@{}} T_{3}& \leq & C \left|\delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\right| \left\{\left|\frac{\gamma_{1}(x_{0})}{\gamma_{1}(x_{0}-hv)}-1\right| \right . \\ & & \left. \times \left| \delta_{1}\left( u^{-\gamma_{1}(x_{0})} U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}-hv\right) \over \delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\right|\right.\\ &&\left.+\left|\frac{\delta_{1}\left( u^{-\gamma_{1}(x_{0})} U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}-hv\right)}{\delta_{1}\left( u^{-\gamma_{1}(x_{0})}U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)}\frac{\delta_{1}\left( u^{-\gamma_{1}(x_{0})} U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)}{\delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)} -1\right|\right\}. \end{array} $$

(28)

Using the assumed form for δ₁(y|x), $(\mathcal H)$, and the uniform bound from Proposition B.1.10 in de Haan and Ferreira (2006) with 0 < ε < β₁(x₀), we obtain, for n large, that

$$ \begin{array}{@{}rcl@{}} T_{3} & \le & C \left|\delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}x_{0}\right)\right| \left\{ h^{\eta_{\gamma_{1}}} + u^{\gamma_{1}(x_{0})\beta_{1}(x_{0})} \left( 1+u^{\pm \gamma_{1}(x_{0}) \varepsilon} \right)\right.\right. \\ && \left. \times \left[ h^{\eta_{B_{1}}} +u^{-Ch^{\eta_{\varepsilon_{1}}}}h^{\eta_{\varepsilon_{1}}} \left( |\ln u| +\ln \frac{n}{k} \right)\right]\right. \\ && \left. + u^{\gamma_{1}(x_{0})(\beta_{1}(x_{0})\pm \varepsilon)} + \left| u^{\gamma_{1}(x_{0})\beta_{1}(x_{0})} -1 \right| \right\}. \qquad \end{array} $$

(29)

Since

$$ \begin{array}{@{}rcl@{}} \left|\widetilde{s}_{n}(u)-u\right| & \leq & \left|\widetilde s_{n}(u)-u^{\frac{\gamma_{1}(x_{0})}{\gamma_{1}(x_{0}-hv)}}\right|+u\left|u^{\frac{\gamma_{1}(x_{0})-\gamma_{1}(x_{0}-hv)}{\gamma_{1}(x_{0}-hv)}}-1\right|\\ & \leq & \left|\widetilde s_{n}(u)-u^{\frac{\gamma_{1}(x_{0})}{\gamma_{1}(x_{0}-hv)}}\right| + Cu^{1\pm Ch^{\eta_{\gamma_{1}}}} h^{\eta_{\gamma_{1}}} |\ln u|, \end{array} $$

(30)

combining Eqs. 26, 27, 29 with Eq. 30, Lemma 3 is established. □

Proof of Lemma 4

We use the following decomposition along with the Lipschitz property of the function R:

$$ \begin{array}{@{}rcl@{}} &&{}\sqrt{kh^{d}} \left|{\int}_{0}^{\infty} \left[R\left( \widetilde s_{n}(u), t_{n}(y_{2})|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ & \leq & \sqrt{kh^{d}} \left|{\int}_{0}^{\delta_{n}} \left[R\left( \widetilde s_{n}(u), t_{n}(y_{2})|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ &&+ \sqrt{kh^{d}} \left|{\int}_{\delta_{n}}^{T_{n}} \left[R\left( \widetilde s_{n}(u), t_{n}(y_{2})|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&+ \sqrt{kh^{d}} \left|{\int}_{T_{n}}^{\infty} \left[R\left( \widetilde s_{n}(u), t_{n}(y_{2})|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ & \leq & \sqrt{kh^{d}} \left|{\int}_{0}^{\delta_{n}} R\left( \widetilde{s}_{n}(u), t_{n}(y_{2})|x_{0}\right) du^{-\gamma_{1}(x_{0})}\right|\\ &&+ \sqrt{kh^{d}} \left|{\int}_{0}^{\delta_{n}} R(u,y_{2}|x_{0}) du^{-\gamma_{1}(x_{0})}\right|\\ &&- \sqrt{kh^{d}} {\int}_{\delta_{n}}^{T_{n}} \left[\left|\widetilde{s}_{n}(u)-u\right|+ \left| t_{n}(y_{2})-y_{2}\right|\right] du^{-\gamma_{1}(x_{0})}\\ &&+2 \underset{u\geq 0, \frac{1}{2}-\zeta \leq y_{2}\leq 2+\zeta}{\sup} R(u,y_{2}|x_{0}) \sqrt{kh^{d}} T_{n}^{-\gamma_{1}(x_{0})}\\ &=:&T_{1}+T_{2}+T_{3}+T_{4}, \end{array} $$

for ζ > 0 small and where δ_n → 0 and $T_{n}\to \infty$, as $n\to \infty$.

Now, since R(y₁,y₂|x₀) ≤ y₁ ∧ y₂, using Lemma 3, and assuming $h^{\eta _{\varepsilon _{1}}\wedge \eta _{\gamma _{1}}}|\ln \delta _{n}|\to 0$, we obtain after tedious calculations, for n large,

$$ \begin{array}{@{}rcl@{}} T_{1}+T_{2}& \leq & - 2\sqrt{kh^{d}} {\int}_{0}^{\delta_{n}} u du^{-\gamma_{1}(x_{0})} - \sqrt{kh^{d}} {\int}_{0}^{\delta_{n}} \left|\widetilde s_{n}(u)-u\right| du^{-\gamma_{1}(x_{0})} \\ & \leq & C \sqrt{kh^{d}} \delta_{n}^{1-\gamma_{1}(x_{0})}. \end{array} $$

(31)

As for T₃, using again Lemma 3 and following the lines of proof of Lemma 1, we have, for n large,

$$ \begin{array}{@{}rcl@{}} T_{3}& \leq &- \sqrt{kh^{d}} {\int}_{0}^{T_{n}} \left|\widetilde{s}_{n}(u)-u\right| du^{-\gamma_{1}(x_{0})}- \sqrt{kh^{d}} {\int}_{\delta_{n}}^{T_{n}}\left| t_{n}(y_{2})-y_{2}\right| du^{-\gamma_{1}(x_{0})}\\ & \leq & C\sqrt{kh^{d}} {T}_{n}^{1-\gamma_{1}(x_{0})} \left\{h^{\eta_{A_{1}}}+h^{\eta_{\gamma_{1}}}\ln\frac{n}{k} +h^{\eta_{\gamma_{1}}}\ln T_{n} \right . \\ & & \left . + \left|\delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\right| T_{n}^{(\beta_{1}(x_{0})+\varepsilon)\gamma_{1}(x_{0})} \right \} \\ &&+ C\sqrt{kh^{d}} \delta_{n}^{-\gamma_{1}(x_{0})} \left\{h^{\eta_{A_{2}}}+h^{\eta_{\gamma_{2}}}\ln \frac{n}{k}+ \left|\delta_{2}\left( U_{2}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\right| \right . \\ & & \left . \times \left[h^{\eta_{B_{2}}}+h^{\eta_{\varepsilon_{2}}}\ln\frac{n}{k}\right]\right\} \end{array} $$

(32)

assuming $h^{\eta _{\varepsilon _{1}}\wedge \eta _{\gamma _{1}}}\ln T_{n}\to 0$.

Finally

$$ \begin{array}{@{}rcl@{}} T_{4}& \leq & C \sqrt{kh^{d}} T_{n}^{-\gamma_{1}(x_{0})}. \end{array} $$

(33)

Take δ_n = h^ξ and T_n = n^κ, with ξ and κ positive numbers, and 0 < ε < β₁(x₀). Combining Eqs. 31, 32 and 33, the first part of Lemma 4 follows if the sequences δ_n and T_n are chosen such that

$$ \begin{array}{@{}rcl@{}} \alpha-{{\varDelta}}\left[d-2\xi \gamma_{1}(x_{0})+2(\xi\wedge \eta_{A_{2}}\wedge \eta_{B_{2}}\wedge \eta_{\gamma_{2}}\wedge \eta_{\varepsilon_{2}}) \right]&<&0,\\ \alpha-{{\varDelta}} d - 2 \kappa \gamma_{1}(x_{0})&<&0,\\ \alpha-{{\varDelta}} d + 2 \kappa(1- \gamma_{1}(x_{0})) -2{{\varDelta}} (\eta_{A_{1}}\wedge \eta_{\gamma_{1}})&<&0, \\ \alpha-{{\varDelta}} d -2(1-\alpha) \gamma_{1}(x_{0})\beta_{1}(x_{0}) +2 \kappa[1+(\beta_{1}(x_{0})+\varepsilon)\gamma_{1}(x_{0}) -\gamma_{1}(x_{0})] & < & 0. \end{array} $$

Note that this is possible if we proceed as follows:

• α and Δ are chosen as stated in Lemma 4;

• κ is chosen such that

  $$ \begin{array}{@{}rcl@{}} &&\frac{\alpha-{{\varDelta}} d}{2\gamma_{1}(x_{0})}<\kappa < \\ & & \min\left( 1-\alpha, \frac{2{{\varDelta}}(\eta_{A_{1}}\wedge \eta_{\gamma_{1}})-(\alpha-{{\varDelta}} d)}{2(1-\gamma_{1}(x_{0}))}, \frac{2(1-\alpha)\gamma_{1}(x_{0})\beta_{1}(x_{0}) - (\alpha-{{\varDelta}} d)}{2[1-\gamma_{1}(x_{0})+(\beta_{1}(x_{0})+\varepsilon)\gamma_{1}(x_{0})]}\right); \end{array} $$

• ξ is chosen such that

  $$ \frac{\alpha-{{\varDelta}} d}{2{{\varDelta}}(1-\gamma_{1}(x_{0}))}<\xi<\eta_{A_{2}}\wedge \eta_{\gamma_{2}}\wedge \eta_{B_{2}}\wedge \eta_{\varepsilon_{2}}. $$

  Note that the choices of κ and ξ only depend on those of α and Δ.

The second part of Lemma 4 is similar, although simpler. Indeed, a decomposition of the quantity of interest this time into two parts yields

$$ \begin{array}{@{}rcl@{}} &&{}\sqrt{kx^{d}} \left|{\int}_{0}^{\infty} \left[R\left( s_{n}(u), y_{2}|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ & \leq & \sqrt{kh^{d}} \left|{\int}_{0}^{T_{n}} \left[R\left( s_{n}(u), y_{2}|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ &&+ \sqrt{kh^{d}} \left|{\int}_{T_{n}}^{\infty} \left[R\left( s_{n}(u), y_{2}|x_{0}\right) - R(u,y_{2}|x_{0})\right] du^{-\gamma_{1}(x_{0})}\right|\\ & \leq &- \sqrt{kh^{d}} {\int}_{0}^{T_{n}} \left|s_{n}(u)-u\right| du^{-\gamma_{1}(x_{0})}+2 \underset{u\geq 0, \frac{1}{2}\leq y_{2}\leq 2}{\sup} R(u,y_{2}|x_{0}) \sqrt{kh^{d}} T_{n}^{-\gamma_{1}(x_{0})}\\ & \leq &- \sqrt{kh^{d}} \frac{|\delta_{1}\left( U_{1}\left( \frac{n}{k}|x_{0}\right)|x_{0}\right)|}{|\gamma_{1}(x_{0})+ \delta_{1}\left( U_{1}\left( \frac{n}{k}|x_{0}\right)|x_{0}\right)|} {\int}_{0}^{T_{n}} u \left|\frac{\delta_{1}\left( u^{-\gamma_{1}(x_{0})}U_{1}\left( \frac{n}{k}|x_{0}\right)|x_{0}\right)}{\delta_{1}\left( U_{1}\left( \frac{n}{k}|x_{0}\right)|x_{0}\right)} - 1\right| du^{-\gamma_{1}(x_{0})} \\ & & +C \sqrt{kh^{d}} T_{n}^{-\gamma_{1}(x_{0})}\\ & \leq & C \sqrt{kh^{d}} \left|\delta_{1}\left( U_{1}\left( \frac{n}{k}\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\left|\vphantom{\frac{n}{k}}\right.x_{0}\right)\right| T_{n}^{1-\gamma_{1}(x_{0})+(\beta_{1}(x_{0})+\varepsilon)\gamma_{1}(x_{0})} +C \sqrt{kh^{d}} T_{n}^{-\gamma_{1}(x_{0})}. \end{array} $$

This achieves the proof of Lemma 4. □

Proof of Lemma 5

In this proof, as mentioned above, we will use the Skorohod representation with keeping the same notation. First remark that

We have, with $r_{n}:= \sqrt {nh^{d}\overline {F}_{2}(u_{n}|x_{0})}$,

$$ \begin{array}{@{}rcl@{}} &&{}\left | r_{n} \left [ \frac{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})} T_{n}\left (\infty, \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})\left|\vphantom{\frac{n}{k}}\right.x_{0} \right ) - f_{X}(x_{0})\right ] - W(\infty,1) \right | \\ & \le & \left | \sqrt{kh^{d}} \left [ T_{n}\left (\infty, \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})\left|\vphantom{\frac{n}{k}}\right.x_{0} \right ) - \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})f_{X}(x_{0}) \right ] - W\left (\infty, \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0}) \right ) \right|\\ & & + \sqrt{kh^{d}} \left| \sqrt{\frac{\overline{F}_{2}(u_{n}|x_{0})}{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})} } -1 \right| \left| T_{n}\left (\infty, \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})\left|\vphantom{\frac{n}{k}}\right.x_{0} \right ) - \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})f_{X}(x_{0}) \right | \\ & & + \left | W\left (\infty, \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0}) \right ) - W(\infty,1) \right| \\ & & + r_{n} \left | \frac{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})}-1 \right | \left | T_{n}\left (\infty, \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})\left|\vphantom{\frac{n}{k}}\right.x_{0} \right ) - \frac{n}{k} \overline{F}_{2}(u_{n}|x_{0})f_{X}(x_{0}) \right |. \end{array} $$

(34)

From Eq. 20 combined with the Skorohod construction, we have

Finally

□

Proof of Lemma 6

To prove the lemma we will use the idea of Wretman (1978), applied to our situation. We have, for $z\in \mathbb R$, and u_n from Lemma 5 taken as $U_{2}(n/k|x_{0})(1+z/\sqrt {kh^{d}})$, that

$$ \begin{array}{@{}rcl@{}} &&{}\mathbb{P} \left( \sqrt{kh^{d}} \left( \widehat u_{n} -1 \right) \le z \right) \\ && = \mathbb{P} \left( \sqrt{nh^{d}\overline{F}_{2}(u_{n}|x_{0})} \left( \frac{\widehat{\overline{F}}_{n,2}(u_{n}|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})}-1 \right)\right.\\ && \qquad ~~\left. \le \sqrt{nh^{d}\overline{F}_{2}(u_{n}|x_{0})} \left( \frac{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})}-1 \right)\right). \end{array} $$

We have that in the present context

$$ \begin{array}{@{}rcl@{}} a_{n} :=\sqrt{nh^{d}\overline{F}_{2}(u_{n}|x_{0})} \left( \frac{\overline{F}_{2}(U_{2}(n/k|x_{0})|x_{0})}{\overline{F}_{2}(u_{n}|x_{0})}-1 \right) \to \frac{z}{\gamma_{2}(x_{0})}. \end{array} $$

Let H_n denote the distribution function of $\sqrt {nh^{d}\overline {F}_{2}(u_{n}|x_{0})} (\widehat {\overline {F}}_{n,2}(u_{n}|x_{0})/\overline {F}_{2}(u_{n}|x_{0})-1 )$, and H is the distribution function of $W(\infty ,1)/f_{X}(x_{0})$. Then by Lemma 5 and by continuity of H one has that H_n(a_n) → H(z/γ₂(x₀)), as $n \to \infty$, hence the result of the lemma. □