3.1. Strong convergence under classical assumptions

Let the distribution function F be twice differentiable. We write $f=F'$ , $g_m=(F^m(a_m\,\cdot \,+b_m))'$ , and $f_t=F'_{\!\!t}$ . Under the classical von Mises-type conditions

(3.1) \begin{alignat}{2} \lim_{x\to\infty}\frac{xf(x)}{1-F(x)} & = \frac{1}{\gamma}, \qquad & \gamma & > 0, \nonumber \\[5pt] \lim_{x\to x^*}\frac{(x^*-x)f(x)}{1-F(x)} & = -\frac{1}{\gamma}, & \gamma & < 0, \\[5pt] \lim_{x\to x^*}\frac{f(x)\int_x^{x^*}(1-F(v))\,\textrm{d}v}{(1-F(x))^2} & = 0, & \gamma & = 0, \nonumber\end{alignat}

we know that the first-order condition in (2.3) is satisfied, and that

(3.2) \begin{equation} \lim_{v\to\infty} v a(v)f(a(v)x+U(v))=(1+\gamma x)^{-1/\gamma-1}\end{equation}

locally uniformly for $(1+\gamma x) >0$ . Since the equality $g_m(x)=F^{m-1}(a_mx+b_m) h_m(x)$ holds with $b_m=U(m)$ , $a_m=a(m)$ , and $h_m(x)=ma_mf(a_m x +b_m)$ , and since $F^{m-1}(a_mx+b_m)$ converges to $G_\gamma(x)$ locally uniformly as $m\to\infty$ , the convergence result in (3.2) thus implies that $g_m(x)$ converges to $g_\gamma(x)$ locally uniformly [Reference Resnick13, Section 2.2].

On the other hand, the density pertaining to $F_t(s(t)x)$ is

\begin{align*}l_t(x)\;:\!=\;f_t(s(t)x)s(t)=\frac{s(t)f(s(t)x+t)}{1-F(t)}\end{align*}

and, setting $v=1/(1-F(t))$ , we have $a(v)=s(t)$ and $v\to \infty$ as $t\to x^*$ . Therefore, a further implication of the convergence result in (3.2) is that $l_t(x)$ converges to $h_\gamma(x)$ locally uniformly for $x>0$ if $\gamma \geq 0$ , or for $x\in (0,-1/\gamma)$ if $\gamma <0$ .

In turn, by Scheffe’s lemma we have $\lim_{t\to x^*}{\mathscr{V\,}}({\mathcal{P}}_t, {\mathscr{P}})=0$ , where ${\mathscr{V\,}}({\mathcal{P}}_t ;\; {\mathscr{P}})=\sup_{B\in\mathbb{B}}\left|{\mathcal{P}}_t(B) - {\mathscr{P}}(B)\right|$ is the total variation distance between the probability measures ${\mathcal{P}}_t(B)\;:\!=\;\mathbb{P}(({X-t})/{s(t)}\in B \mid X>t)$ and ${\mathscr{P}}(B)\;:\!=\;\mathbb{P}(Z\in B)$ , and where Z is a random variable with distribution $H_\gamma$ and B is a set in the Borel $\sigma$ -field of $\mathbb{R}$ , denoted by $\mathbb{B}$ . Let ${\mathscr{H}\;\,}^2(l_t;\; h_\gamma)\;:\!=\;\int \big[\sqrt{l_t(x)}-\sqrt{h_\gamma(x)}\big]^2\, \mathrm{d} x$ be the square of the Hellinger distance. It is well known that the Hellinger and total variation distances are related as

(3.3) \begin{equation} {\mathscr{H}\;\,}^2(l_t;\; h_\gamma) \leq 2{\mathscr{V\,}}({\mathcal{P}}_t;\; {\mathscr{P}}) \leq 2{\mathscr{H}\;}(l_t;\; h_\gamma) \end{equation}

(see, e.g., [Reference Ghosal and van der Vaart9, Appendix B]). Therefore, the conditions in (3.1) ultimately entail that the Hellinger distance between the density of rescaled peaks over a threshold $l_t$ and the GP density $h_\gamma$ also converges to zero as $t \to x^*$ . In the next subsection we introduce a stronger assumption, allowing us to also quantify the speed of such convergence.

3.2. Convergence rates

As in [Reference Raoult and Worms12] we rely on the following assumption, in order to derive the convergence rate for the variational and Hellinger distances.

Condition 3.1. Assume that F is twice differentiable. Moreover, assume that there exists $\rho\leq 0$ such that

\begin{align*} A(v)\;:\!=\; \frac{vU''(v)}{U'(v)}+1-\gamma \end{align*}

defines a function of constant sign near infinity, whose absolute value $|A(v)|$ is regularly varying as $v\to\infty$ with index of variation $\rho$ .

When Condition 3.1 holds then the classical von Mises conditions in (3.1) are also satisfied for the cases where $\gamma$ is positive, negative, or equal to zero, respectively. Furthermore, Condition 3.1 implies that an appropriate scaling function for the exceedances of a high threshold $t<x^*$ , which complies with the equivalent first-order condition (2.2), is defined as $s(t)=(1-F(t))/f(t)$ . With such a choice of the scaling function s, we establish the following results.

Theorem 3.1. Assume Condition 3.1 is satisfied. Then, there exist constants $c>0$ , $\alpha_j>0$ with $j=1,2$ , $K>0$ , and $t_0 < x^*$ , depending on $\gamma$ , such that

\begin{equation*} \frac{{\mathscr{H}\;\,}^2(\tilde{l}_t ;\; h_\gamma)}{K |A(v)|^2} \leq S(v) \end{equation*}

for all $t\geq t_0$ , where $v=1/(1-F(t))$ ,

\begin{align*} \tilde{l}_t = \begin{cases} l_t, & \gamma \geq 0, \\[5pt] l_t( \cdot \, + ({x^*-t})/{s(t)} + {1}/{\gamma}), & \gamma < 0, \end{cases} \end{align*}

and $S(v) = 1-|A(v)|^{\alpha_1 }+4 \exp\!(c |A(v)|^{\alpha_2})$ .

Note that $\tilde{l}_t$ is the density of $F_t(s(t) \cdot +c(t))$ , with centring function

(3.4) \begin{equation} c(t) \;:\!=\; \begin{cases} 0, & \gamma \geq 0, \\[5pt] x^*-t +\gamma^{-1}s(t), & \gamma < 0, \end{cases}\end{equation}

for $t<x^*$ . Given the relationship between the total variation and Hellinger distances in (3.3), with obvious adaptations when a non-null recentring is considered, the following result is a direct consequence of Theorem 3.1.

Theorem 3.1 implies that when $\gamma \geq 0$ the Hellinger and variational distances of the probability density and measure of rescaled exceedances from their GP distribution counterparts are bounded from above by $C_1 |A(v)|$ , for a positive constant $C_1$ , as the threshold t approaches the end-point $x^*$ . Since, for a fixed $x \in \cap_{t\geq t_0}(0,({x^*-t})/{s(t)})$ , $|F_t(s(t)x)-H_\gamma(x)| \leq {\mathscr{V\,}}({\mathcal{P}}_t;\; {\mathscr{P}})$ , and since [Reference Raoult and Worms12, Theorem 2(i)] implies that $|F_t(s(t)x)-H_\gamma(x)|/|A(v)|$ converges to a positive constant, there also exists $C_0>0$ such that, for all large t, $C_0|A(v)|$ is a lower bound for the variational and Hellinger distances. Therefore, since $C_0|A(v)|\leq {\mathscr{V\,}}({\mathcal{P}}_t;\; {\mathscr{P}})\leq {\mathscr{H}\;}(l_t;\; h_\gamma) \leq C_1|A(v)|$ , the decay rate of the variational and Hellinger distances is precisely $|A(v)|$ as $t\to x^*$ . When $\gamma <0$ , analogous considerations apply to $\tilde{l}_t$ and $\widetilde{{\mathcal{P}}}_t$ . With the following results, we give precise indications of when a recentred version of $l_t$ is necessary to achieve the optimal rate.

Proposition 3.1. Under the assumptions of Theorem 3.1, when $\gamma <0$ there are constants $c_j$ , $j=1,2$ , and $t_1 < x^*$ , depending on $\gamma$ , such that, for all $t >t_1$ ,

\begin{align*} c_1|A(v)|^{-1/2\gamma} < {\mathscr{H}\;}(h_\gamma;\; h_\gamma(\cdot \, -\mu_t)) < c_2 |A(v)|^{\min(1, -1/2\gamma)}, \end{align*}

where $\mu_t\;:\!=\;c(t)/s(t)$ and c(t) is as in the second line of (3.4).

According to Corollary 3.2(ii), the density $l_t$ of rescaled exceedances $Y/s(t)$ does not achieve the optimal convergence rate $|A(V)|$ whenever $\gamma <-1/2$ , in which case the rate is only of order $|A(v)|^{-1/2\gamma}$ . In simple terms, this is due to the fact that, when $\gamma$ is negative, the supports of $l_t$ and $h_\gamma$ can be different and the approximation error is affected by the amount of probability mass in the unshared region of points. Indeed, we recall that the end-point $(x^*-t)/{s}(t)$ of $l_t$ converges to $-1/{\gamma}$ as t approaches $x^*$ at rate A(v) (see, e.g., [Reference de Haan and Ferreira4, Lemma 4.5.4]). Nevertheless, for $t<x^*$ it can be that $(x^*-t)/{s}(t)>-1/\gamma$ or $(x^*-t)/{s}(t)<-1/\gamma $ . In turn, when $\gamma$ is smaller than $-1/2$ , the approximation error due to support mismatch has a dominant effect. However, if scaled exceedances are shifted by subtracting the quantity $\mu_t$ , in this case the upper end-point of the density $\tilde{l}_t$ is the same of that of $h_\gamma$ , hence no support mismatch occurs and the optimal convergence rate is also achieved in the case where $\gamma <-1/2$ .

A further implication of Theorem 3.1 concerns the speed of convergence to zero of the Kullback–Leibler divergence ${\mathscr{K}\;}(\tilde{l}_t;\; h_\gamma)\;:\!=\; \int\ln\lbrace\tilde{l}_t(x)/h_\gamma(x)\rbrace\tilde{l}_t(x)\, \mathrm{d} x$ , and the divergences of higher order $p\geq 2$ , ${\mathscr{D}}_p(\tilde{l}_t;\; h_\gamma)\;:\!=\; \int|\ln\lbrace\tilde{l}_t(x)/h_\gamma(x)\rbrace|^p\tilde{l}_t(x)\, \mathrm{d} x$ . Using the uniform bound on density ratio provided in Lemma 4.7 we are able to translate the upper bounds on the squared Hellinger distance ${\mathscr{H}\;\,}^2(\tilde{l}_t;\; h_\gamma)$ into upper bounds on the Kullback–Leibler divergence ${\mathscr{K}\;}(\tilde{l}_t;\; h_\gamma)$ and higher-order divergences ${\mathscr{D}}_p(\tilde{l}_t;\; h_\gamma)$ .

To extend the general results in Lemma 4.7 and Corollary 3.3 to the case of $\gamma=0$ seems to be technically over-complicated. Nevertheless, there are specific examples where the properties listed in such lemmas are satisfied, such as the following.

4.1. Additional notation

For $y >0$ we write $T(y)=U(e^y)$ and, for $t < x^*$ , we define the functions

\begin{align*}p_t(y) =\begin{cases} ({T(y+T^{-1}(t))-t})/{s(t)} - ({e^{\gamma y}-1})/{\gamma}, & \gamma > 0, \\[5pt] ({T(y+T^{-1}(t))-t})/{s(t)}-y, & \gamma = 0, \\[5pt] ({T(y+T^{-1}(t))-x^* -\gamma^{-1}s(t)})/{s(t)} - ({e^{\gamma y}-1})/{\gamma}, & \gamma < 0,\end{cases} \end{align*}

with $s(t)=(1-F(t))/f(t)$ , and

\begin{align*}q_t(y) =\begin{cases} ({1}/{\gamma})\ln[1+\gamma \mathrm{e}^{-\gamma y} p_t(y)], & \gamma \neq 0, \\[5pt] p_t(y), & \gamma = 0.\end{cases} \end{align*}

Moreover, for $t<x^*$ we set

\begin{align*}\tilde{x}_t^* =\begin{cases} ({x^*-t})/{{s}(t)}, & \gamma \geq 0, \\[5pt] -{1}/{\gamma}, & \gamma < 0.\end{cases} \end{align*}

Furthermore, for $x \in (0, x^*-t)$ , we let $\phi_t(x)=T^{-1}(x+t)-T^{-1}(t)$ . Finally, for $x \in \mathbb{R}$ , $\gamma \in \mathbb{R}$ , and $\rho \leq 0$ , we set

\begin{align*}I_{\gamma,\rho}(x)=\begin{cases} \int_0^x \mathrm{e}^{\gamma s}\int_0^s \mathrm{e}^{\rho z}\, \mathrm{d} z\, \mathrm{d} s, & \gamma \geq 0, \\[5pt] -\int_x^\infty \mathrm{e}^{\gamma s} \int_0^s \mathrm{e}^{\rho z}\, \mathrm{d} z\, \mathrm{d} s, & \gamma < 0.\end{cases} \end{align*}

4.2. Auxiliary results

In this section we provide some results which are auxiliary to the proofs of the main ones, presented in Section 3. Throughout, for Lemmas 4.1–4.6, Condition 3.1 is implicitly assumed to hold. The proofs are provided in the Supplementary Material. In particular, Lemmas 4.1 and 4.2 are used directly in the proof of our main result, Theorem 3.1.

Lemma 4.1. For every $\varepsilon>0$ and every $\alpha>0$ , there exist $x_1 < x^*$ and $\kappa_1>0$ (depending on $\gamma$ ) such that, for all $t \geq x_1$ and $y \in (0, -\alpha \ln \vert A(\mathrm{e}^{T^{-1}(t)})\vert)$ ,

\begin{align*} \exp\!\big\{ {-}\kappa_1 \vert A(\mathrm{e}^{T^{-1}(t)}) \vert \mathrm{e}^{2\varepsilon y}\big\} < \mathrm{e}^{-q_t(y)} < \exp\!\big\{ \kappa_1 \vert A(\mathrm{e}^{T^{-1}(t)}) \vert \mathrm{e}^{2\varepsilon y}\big\}. \end{align*}

Lemma 4.2. For every $\varepsilon>0$ and every $\alpha>0$ , there exist $x_2 < x^*$ and $\kappa_2>0$ (depending on $\gamma$ ) such that, for all $t \geq x_2$ and $y \in (0,-\alpha \ln \vert A(\mathrm{e}^{T^{-1}(t)}) \vert )$ ,

\begin{align*} \exp\!\big\{{-}\kappa_2 \vert A(\mathrm{e}^{T^{-1}(t)}) \vert \mathrm{e}^{2\varepsilon y}\big\} < 1 + q'_{\!\!t}(y) < \exp\!\big\{ \kappa_2 \vert A(\mathrm{e}^{T^{-1}(t)}) \vert \mathrm{e}^{2\varepsilon y} \big\}. \end{align*}

Lemma 4.3. If $\gamma>0$ and $\rho<0$ , there exists a regularly varying function $\mathcal{R}$ with negative index $\varrho$ such that, defining the function

\begin{align*} \eta(t)\;:\!=\;\frac{(1+\gamma t)f(t)}{1-F(t)} -1, \end{align*}

as $v\to\infty$ , $\eta(U(v))=O(\mathcal{R}(v))$ .

Lemma 4.5. If $\gamma<0$ and $\rho<0$ , there exists a a regularly varying function $\tilde{\mathcal{R}}$ with negative index $\tilde{\varrho}=(\!-\!1)\vee(\!-\!\rho/\gamma)$ such that, defining the function

\begin{align*} \tilde{\eta}(y)\;:\!=\;\frac{(1-\gamma y)f(x^*-1/y)}{[1-F(x^*-1/y)]y^2} -1, \end{align*}

as $y\to\infty$ , $\tilde{\eta}(y)=O(\tilde{\mathcal{R}}(y))$ .

Lemma 4.6. If $\gamma<0$ and $\rho<0$ , there exist $\tilde{\delta}>0$ such that, as $y\to \infty$ ,

\begin{align*} \frac{f(x^*-1/y)}{y^2}=(1-\gamma y)^{1/\gamma-1}[1+O(\{1-H_{-\gamma}(y)\}^{\tilde{\delta}})]. \end{align*}

Finally, in order to exploit Theorem 3.1 to give bounds on Kullback–Leibler and higher-order divergences, we introduce by the next lemma a uniform bound on density ratios.

Lemma 4.7. Under the assumptions of Theorem 3.1, if $\rho <0$ and $\gamma \neq 0$ , then there exist a $t^* <x^*$ and a constant $M \in (0,\infty)$ such that

\begin{align*} \sup_{t \geq t^*}\,\sup_{ 0<x<\tilde{x}_t^*}\frac{\tilde{l}_t(x)}{h_\gamma(x)}<M. \end{align*}

4.3. Proof of Theorem 3.1

Proof of Theorem 3.1. For every $x_t>0$ ,

\begin{align*} & {\mathscr{H}\;\,}^2(l_t;\; h_\gamma) \\ & = \int_0^{x_t} + \int_{x_t}^\infty \Big[\sqrt{f_t}(x)-\sqrt{h_\gamma({(x-c(t))}/s(t))/s(t)}\Big]^2\,\mathrm{d} x \\[5pt] & \leq \int_{0}^{\phi_t(x_t)}\mathrm{e}^{-y} \Big[1-\sqrt{\mathrm{e}^{-q_t(y)}(1+q'_{\!\!t}(y))}\Big]^2\,\mathrm{d} y + \Bigg[\sqrt{1-F_t(x_t)} + \sqrt{1-{H_\gamma\bigg(\frac{x_t-c(t)}{s(t)}\bigg)}}\Bigg]^2 \\[5pt] & \;=\!:\; \mathcal{I}_1(t)+\mathcal{I}_2(t). \end{align*}

Let $x_t$ be such that $\phi_t(x_t)=-\alpha \ln \big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert$ for a positive constant $\alpha$ to be specified later. Then, by Lemmas 4.1 and 4.2, for a suitably small $\varepsilon>0$ there exists $\kappa_3>0$ such that, for all sufficiently large t,

\begin{equation*} \mathcal{I}_1(t) \leq \int_0^{-\alpha \ln \vert A(\mathrm{e}^{T^{-1}(t)})\vert} \kappa_3\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^2\mathrm{e}^{(4\varepsilon-1) y}\,\mathrm{d} y \leq \kappa_3\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^2 \big[1-\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^{\alpha_1}\big], \end{equation*}

where $\alpha_1\;:\!=\;\alpha(1-4\varepsilon)$ is positive. Moreover, on one hand we have the identity $1-F_t(x_t)=\vert A(\mathrm{e}^{T^{-1}(t)})\vert^\alpha$ . On the other hand, for some constant $\kappa_5>0$ we have the inequality

\begin{align*} 1- {H_\gamma\bigg(\frac{x_t-c(t)}{s(t)}\bigg)} & = \big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^\alpha \exp\big\lbrace -q_t\big({-}\alpha\ln\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert\big)\big\rbrace \\[5pt] & \leq \big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^\alpha \exp\big\lbrace\kappa_5\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^{1-2\varepsilon\alpha}\big\rbrace. \end{align*}

Consequently,

\begin{align*} \mathcal{I}_2(t) \leq \big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^\alpha \bigg[1+\exp\bigg\lbrace\frac{\kappa_5}{2} \big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^{1-2\varepsilon\alpha}\bigg\rbrace\bigg]. \end{align*}

Now, we can choose $\alpha >2$ and $\varepsilon$ small enough that $\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^\alpha < \big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert^2$ and $\alpha_2 \;:\!=\; 1-2\varepsilon\alpha > 0$ . The conclusion then follows noting that $T^{-1}(t)=-\!\ln\!(1-F(t))$ and, in turn, $\big\vert A(\mathrm{e}^{T^{-1}(t)})\big\vert = \vert A(v)\vert$ .

4.4. Proof of Proposition 3.1

Proof of Proposition 3.1. Assume first that, for all large t, $\mu_t=({x^*-t})/{s(t)}+{1}/{\gamma}$ is positive. In this case, we have the following identities:

\begin{align*} {\mathscr{H}\;\,}^2(h_\gamma;\; h_\gamma(\cdot \, - \mu_t)) & = \int_0^{-1/\gamma}\Big[\sqrt{h_\gamma(x)}-\sqrt{h_\gamma(x-\mu_t)}\Big]^2\,\mathrm{d} x + 1 - H_\gamma\bigg({-}\frac{1}{\gamma}-\mu_t\bigg) \\[5pt] & = \int_0^\infty\mathrm{e}^{-s}\Bigg[\sqrt{\frac{\exp\{-({1}/{\gamma})\ln\!(1-\gamma\mathrm{e}^{-\gamma s}\mu_t)\}} {1-\gamma\mathrm{e}^{-\gamma s}\mu_t}} - 1\Bigg]^2\, \mathrm{d} s + (\!-\!\gamma \mu_t)^{-1/\gamma}. \end{align*}

Concerning the first term on the right-hand side, for all $s>0$ as $t\to x^*$ we have $0 \leq \ln\!(1-\gamma\mathrm{e}^{-\gamma s}\mu_t) \leq \ln\!(1-\gamma\mu_t) = O(\mu_t)$ and

\begin{align*} 1 \geq \frac{1}{1- \gamma \mathrm{e}^{-\gamma s} \mu_t} \geq \frac{1}{1- \gamma\mu_t}=1+O(\mu_t), \end{align*}

where, for a positive constant $\tau$ , ${\mu_t}$ satisfies ${\mu_t}/{|A(v)|} = (1+o(1))\tau$ (see, e.g., [Reference de Haan and Ferreira4, Lemma 4.5.4]). Then, as $t \to x^*$ ,

\begin{equation*} \int_0^\infty\mathrm{e}^{-s}\Bigg[\sqrt{\frac{\exp\{-({1}/{\gamma})\ln\!(1-\gamma\mathrm{e}^{-\gamma s}\mu_t)\}} {1-\gamma\mathrm{e}^{-\gamma s}\mu_t}}-1\Bigg]^2\, \mathrm{d} s = O(\mu_t^2) = O(|A(v)|^2). \end{equation*}

Concerning the second term, as $t \to x^*$ , $(\!-\!\gamma\mu_t)^{-1\gamma} = (\!-\!\gamma)^{-1/\gamma}(1+o(1))\tau^{-1/\gamma}|A(v)|^{-1/\gamma}$ . The result now follows for the case where $\mu_t$ is ultimately positive. When it is ultimately negative, simply note that

(4.1) \begin{equation} {\mathscr{H}\;\,}^2(h_\gamma;\; h_\gamma(\cdot \, - \mu_t)) = {\mathscr{H}\;\,}^2(h_\gamma(\cdot \, + \mu_t);\; h_\gamma) = {\mathscr{H}\;\,}^2(h_\gamma(\cdot \, -(\!-\! \mu_t));\; h_\gamma), \end{equation}

and proceed as above, but replacing $\mu_t$ with $-\mu_t$ .

4.5. Proof of Corollary 3.2

Proof of Corollary 3.2. Observe that, for $\gamma <0$ , ${\mathscr{H}\;}(l_t;\; h_\gamma) = {\mathscr{H}\;}(\tilde{l}_t;\; h_\gamma(\cdot\, +\mu_t))$ . Moreover, note that, by the triangular inequality and the first identity in (4.1),

\begin{align*} {\mathscr{H}\;}(h_\gamma;\; h_\gamma(\cdot \, - \mu_t)) - {\mathscr{H}\;}(\tilde{l}_t;\; h_\gamma(\cdot\, +\mu_t)) & \leq {\mathscr{H}\;}(\tilde{l}_t;\; h_\gamma(\cdot\, +\mu_t)) \\[5pt] & \leq {\mathscr{H}\;}(h_\gamma;h_\gamma(\cdot \, - \mu_t)) + {\mathscr{H}\;}(\tilde{l}_t;\; h_\gamma(\cdot\, +\mu_t)). \end{align*}

Whenever $\gamma \geq -1/2$ , by Theorem 3.1 and Proposition 3.1, the term on the second line is of order $O(|A(v)|)$ as $t \to x^*$ . Instead, if $\gamma <-1/2$ , by Theorem 3.1 and Proposition 3.1, as $t\to x^*$ the term on the left-hand side of the first line satisfies

\begin{align*} {\mathscr{H}\;}(h_\gamma;\; h_\gamma(\cdot \, - \mu_t)) - {\mathscr{H}\;}(\tilde{l}_t;\; h_\gamma(\cdot\, +\mu_t)) & \geq c_3 |A(v)|^{-1/2\gamma} - O(|A(v)|) \\[5pt] & = c_3 |A(v)|^{-1/2\gamma}(1+o(1)). \end{align*}

The result now follows.

4.6. Proof of Corollary 3.3

Proof of Corollary 3.3. By [Reference Ghosal, Ghosh and van der Vaart8, Lemma 8.2],

\begin{align*} {\mathscr{K}\;}(\tilde{l}_t;\; h_\gamma) \leq 2\bigg[\sup_{0<x<{\tilde{x}_t^*}}\frac{\tilde{l}_t(x)}{h_\gamma(x)}\bigg] {\mathscr{H}\;\,}^2(\tilde{l}_t;\; h_\gamma). \end{align*}

Moreover, by [Reference Ghosal and van der Vaart9, Lemma B.3], for $p\geq 2$ ,

\begin{align*} {\mathscr{D}}_p(\tilde{l}_t;\; h_\gamma) \leq 2p!\bigg[\sup_{0<x<{\tilde{x}_t^*}}\frac{\tilde{l}_t(x)}{h_\gamma(x)}\bigg] {\mathscr{H}\;\,}^2(\tilde{l}_t;\; h_\gamma). \end{align*}

The conclusion now follows by combining the above inequalities and applying Theorem 3.1 and Lemma 4.7.