Approximation of Sojourn Times of Gaussian Processes

Abstract

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of Berman (Commun Pure Appl Math 38(5):519–528, 1985a and Probab Theory Relat Fields 20(1):113–124, 1987) for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.

Appendix

Let K_u be an index function of u, D be a compact set in $\mathbb {R}^{n}$ and suppose without loss of generality that 0 ∈D. Further, let {ξ_u,k(t),t ∈D,k ∈ K_u} be a family of centered Gaussian random fields with a.s. continuous sample paths and variance function $\sigma ^{2}_{\xi _{u,k}}$. For t such that $\sigma ^{2}_{\xi _{u,k}}(\textit {\textbf {t}})>0$ define the standardised process

$$\tilde \xi_{u,k}(\textit{\textbf{t}}):= \frac{\xi_{u,k}(\textit{\textbf{t}})}{\sigma_{\xi_{u,k}}(\textit{\textbf{t}})},\quad \textit{\textbf{t}}\in\textit{\textbf{D}}. $$

• C0:   {g_k(u),k ∈ K_u} is a sequence of deterministic functions of u satisfying

  $$\begin{array}{@{}rcl@{}} \lim\limits_{u\to\infty}\inf\limits_{k\in K_u}g_{k}(u)=\infty. \end{array} $$

• C1:   $\sigma _{\xi _{u,k}}(\textbf {0})= 1$ for all large u and any k ∈ K_u, and there exists some bounded continuous function h on D such that

  $$\begin{array}{@{}rcl@{}} \lim\limits_{u\to\infty}\sup\limits_{\textit{\textbf{t}}\in \textit{\textbf{D}},k\in K_u}|g_{k}^2(u)\left( 1- \mathbb{E}\{ \xi_{u,k}(\textit{\textbf{t}})\xi_{u,k}(\textbf{0})\} \right) - h(\textit{\textbf{t}})| = 0. \end{array} $$

• C2:   There exists a centered Gaussian random field $\zeta (\textit {\textbf {t}}),\textit {\textbf {t}}\in \mathbb {R}^{n}$ with a.s. continuous trajectories such that for any s,t ∈D

  $$\begin{array}{@{}rcl@{}} \lim\limits_{u\to\infty}\sup\limits_{k\in K_u}|g_{k}^2(u)\left( Var(\tilde\xi_{u,k}(\textit{\textbf{t}})-\tilde\xi_{u,k}(\textit{\textbf{s}}))\right) - 2Var(\zeta(\textit{\textbf{t}})-\zeta(\textit{\textbf{s}}))| = 0. \end{array} $$

• C3:   There exist positive constants C,ν,u₀ such that

  $$\begin{array}{@{}rcl@{}} \sup\limits_{k\in K_u} g_{k}^2(u)\mathbb{E}\{(\tilde\xi_{u,k}(\textit{\textbf{t}})-\tilde\xi_{u,k}(\textit{\textbf{s}}))^2\} \leq C ||\textit{\textbf{s}}-\textit{\textbf{t}}||^{\nu} \end{array} $$

  holds for all s,t ∈D,u ≥ u₀, where $||\textit {\textbf {t}} ||^{v}={\sum }_{i = 1}^{n}|t_{i}|^{v}$.

We present below an extension of Theorem 2.1 in Dȩbicki et al. (2017c). Recalling that μ_η(dt)/ηⁿ denotes the counting measure on $\eta \mathbb {Z}^n, \eta >0$ and μ₀ is the Lebesgue measure on $\mathbb {R}^{n}$.

Theorem 5.1

Leth,g_k(u),ξ_u,k(t),t ∈D,k ∈ K_uandζbe such thatC0-C3hold. Then, forη ≥ 0

$$ \lim\limits_{u\to\infty}\sup\limits_{k\in K_u} |\frac{\mathbb{P} \left\{ {\int}_{\textit{\textbf{D}}} {\mathbb{I}\left( \xi_{u,k}(\textit{\textbf{t}})>g_{k}(u)\right)} {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} >x \right\} } {{\Psi}(g_{k}(u))} -\mathcal{B}^{h,\eta}_{\zeta}(\textit{\textbf{D}},x)| = 0 $$

(5.1)

holds at x = 0 and all x ∈ (0,μ_η(D)) continuity points of $\mathcal {B}^{h,\eta }_{\zeta }(\textit {\textbf {D}},\cdot )$, where

$$\begin{array}{@{}rcl@{}} \mathcal{B}^{h,\eta}_{\zeta}(\textit{\textbf{D}},x)= {\int}_{\mathbb{R}} \left\{ {\int}_{\textit{\textbf{D}}} \mathbb{I}_0\left( \sqrt{2}\zeta(\textit{\textbf{t}})-h(\textit{\textbf{t}}) + z \right) {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} > x\right\} e^{-z} \mathrm{d} z. \end{array} $$

Proof of Theorem

5.1 Before beginning the proof, for notational simplicity we denote by R_u,k and ρ_u,k the covariance and the correlation function of ξ_u,k. Further set

$$\chi_{u,k}(\textit{\textbf{t}}) := g_{k}(u)(\tilde\xi_{u,k}(\textit{\textbf{t}})-\rho_{u,k}(\textit{\textbf{t}},\textbf{0})\tilde\xi_{u,k}(\textbf{0})),\quad \textit{\textbf{t}}\in\textit{\textbf{D}} $$

and

$$f_{u,k}(\textit{\textbf{t}},z):=z R_{u,k}(\textit{\textbf{t}},\textbf{0})-{g_{k}^{2}}(u)\left( 1-R_{u,k}(\textit{\textbf{t}},\textbf{0}) \right) ,\ \textit{\textbf{t}}\in \textit{\textbf{D}}, z\in\mathbb{R}. $$

Conditioning on ξ_u,k(0) and using that ξ_u,k(0) and ξ_u,k(t) − R_u,k(t,0)ξ_u,k(0) are mutually independent for all large u, we obtain

$$\begin{array}{@{}rcl@{}} &&\mathbb{P} \left\{ {\int}_{\textit{\textbf{D}}} {\mathbb{I}\left( \xi_{u,k}(\textit{\textbf{t}})>g_{k}(u)\right)} {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} >x \right\}\\ & = &\frac{e^{-g_{k}^2(u)/2}} { \sqrt{2\pi}g_{k}(u)} {\int}_{\mathbb{R}} \exp\left( -z-\frac{z^2}{2g_{k}^2(u)}\right)\\ && \times \mathbb{P} \left\{ {\int}_{\textit{\textbf{D}}} \left( g_{k}(u)(\xi_{u,k}(\textit{\textbf{t}})-g_{k}(u))\right) {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} >x | \xi_{u,k}(\textbf{0})=g_{k}(u)+zg_{k}^{-1}(u) \right\} \mathrm{d} z\\ & = &\frac{e^{-g_{k}^2(u)/2}} { \sqrt{2\pi}g_{k}(u)} {\int}_{\mathbb{R}} \exp\left( -z - \frac{z^2}{2g_{k}^2(u)}\right) \mathbb{P} \left\{ {\int}_{\textit{\textbf{D}}} \left( {\sigma_{\xi_{u,k}}(\textit{\textbf{t}})}\chi_{u,k}(\textit{\textbf{t}}) + f_{u,k}(\textit{\textbf{t}},z)\right) {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} \!>\!x \right\} \mathrm{d} z. \end{array} $$

Let

$$\begin{array}{@{}rcl@{}} \mathcal{I}_{u,k}(x,z):=\mathbb{P} \left\{ {\int}_{\textit{\textbf{D}}} \left( {\sigma_{\xi_{u,k}}(\textit{\textbf{t}})}\chi_{u,k}(\textit{\textbf{t}}) + f_{u,k}(\textit{\textbf{t}},z)\right) {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} >x \right\} . \end{array} $$

Consequently, in order to show the claim it suffices to prove that

$$ \lim\limits_{u\to\infty}\sup\limits_{k\in K_u}|{\int}_{\mathbb{R}} \exp\left( -z-\frac{z^2}{2g_{k}^2(u)}\right) \mathcal{I}_{u,k}(x,z) \mathrm{d} z - \mathcal{B}^{h,\eta}_{\zeta}(\textit{\textbf{D}},x)| = 0 $$

(5.2)

at x = 0 and all x ∈ (0,μ_η(D)) continuity points of $\mathcal {B}^{h,\eta }_{\zeta }(\textit {\textbf {D}},\cdot )$.

For any s,t ∈D,

$$ \sup\limits_{k\in K_u}\mathbb{E}\{\chi_{u,k}(\textit{\textbf{t}})-\chi_{u,k}(\textit{\textbf{s}})\}^2 = \sup\limits_{k\in K_u}{g_{k}^2(u)\left( \!\mathbb{E}\{(\tilde\xi_{u,k}(\textit{\textbf{t}}) - \tilde\xi_{u,k}(\textit{\textbf{s}}))^{2}\} - \left( \rho_{\xi_{u,k}}(\textit{\textbf{t}},\textbf{0}) - \rho_{\xi_{u,k}}(\textit{\textbf{s}},\textbf{0}) \right)^2 \right)}, $$

(5.3)

which together with C2-C3 yields

$$\begin{array}{@{}rcl@{}} &&\lefteqn{ \lim\limits_{u\to\infty}\sup\limits_{k\in K_u}|\text{Var}(\chi_{u,k}(\textit{\textbf{t}})-\chi_{u,k}(\textit{\textbf{s}})) - 2\text{Var}(\zeta(\textit{\textbf{t}})-\zeta(\textit{\textbf{s}}))||}\\ &\leq& \lim\limits_{u\to\infty}\sup\limits_{\textit{\textbf{t}}\in\textit{\textbf{D}},k\in K_u} 4g_{k}^2(u) |1- \rho_{\xi_{u,k}}(\textit{\textbf{t}},\textbf{0})|^2 = 0. \end{array} $$

Hence, the finite-dimensional distributions of χ_u,k converge to that of $\sqrt {2}\zeta (\textit {\textbf {t}}),\textit {\textbf {t}}\in \textit {\textbf {D}}$ uniformly with respect to k ∈ K_u. Let C(D) denote the Banach space of all continuous functions on D equipped with sup-norm. By Eq. 5.3 and C3, we know that

$$ \sup\limits_{k\in K_u}\mathbb{E}\{\chi_{u,k}(\textit{\textbf{t}})-\chi_{u,k}(\textit{\textbf{s}})\}^2 \leq C||\textit{\textbf{s}}-\textit{\textbf{t}}||^{\nu} $$

(5.4)

holds for large enough u and all s,t ∈D, thus the measures on C(D) induced by {χ_u,k(t),t ∈D,k ∈ K_u} are uniformly tight for large u. Therefore, {χ_u,k(t),t ∈D} converge weakly to $\{\sqrt {2}\zeta (\textit {\textbf {t}}),\textit {\textbf {t}}\in \textit {\textbf {D}}\}$ as u →∞ uniformly with respect to k ∈ K_u.

Further, for large u

$$1-\mathbb{E}\{\xi_{u,k}(\textit{\textbf{t}})\xi_{u,k}(\textbf{0})\}= 1-\rho_{{\xi_{u,k}}}(\textit{\textbf{t}},\textbf{0})+\rho_{{\xi_{u,k}}}(\textit{\textbf{t}},\textbf{0})(1-{\sigma_{\xi_{u,k}}(\textit{\textbf{t}})}),$$

which together with C1-C2 implies

$$ \lim\limits_{u\to\infty}\sup\limits_{\textit{\textbf{t}}\in \textit{\textbf{D}},k\in K_u}|\sigma_{\xi_{u,k}}(\textit{\textbf{t}})-1|= 0. $$

(5.5)

By C0-C1, for each $z\in \mathbb {R}$

$$ \lim_{u \to \infty} \sup\limits_{k\in K_u, \textit{\textbf{t}}\in\textit{\textbf{D}}}|f_{u,k}(\textit{\textbf{t}},z)-z +h(\textit{\textbf{t}})| = 0. $$

(5.6)

Thus, for each $z\in \mathbb {R}$, the probability measures on C(D) induced by $\{{\sigma _{\xi _{u,k}}(\textit {\textbf {t}})}\chi _{u,k}(\textit {\textbf {t}}) + f_{u,k}(\textit {\textbf {t}},z),\textit {\textbf {t}}\in \textit {\textbf {D}}\}$ converge weakly, as u →∞, to that induced by $\{\sqrt {2}\zeta (\textit {\textbf {t}})-h(\textit {\textbf {t}}) + z,\textit {\textbf {t}}\in \textit {\textbf {D}}\}$ uniformly with respect to k ∈ K_u. Therefore, for any $\eta >0,z\in \mathbb {Z}$

$$ \lim\limits_{u\to\infty}\sup\limits_{k\in K_u}|\mathcal{I}_{u,k}(x,z) - \mathcal{I}(x,z)| = 0 $$

(5.7)

holds at all x ∈ (0,μ_η(D)) continuity points (depending on z) of $\mathcal {I}(\cdot ,z)$ defined by

$$\mathcal{I}(x,z) := \mathbb{P} \left\{ {\int}_{\textit{\textbf{D}}} \left( \sqrt{2}\zeta(\textit{\textbf{t}})-h(\textit{\textbf{t}}) + z\right) {\mu_{\eta}(\mathrm{d}\textit{\textbf{t}})} > x \right\} . $$

For η = 0, by Berman (1973)[Lemma 4.2] the set of discontinuity points under the uniform convergence on D of

$${\int}_{\textit{\textbf{D}}} \mathbb{I}_{0}(f(\textit{\textbf{t}})) \mathrm{d}\textit{\textbf{t}},\quad f\in C(\textit{\textbf{D}})$$

is of measure 0 under the probability measure induced by $\{ {\sqrt {2}\zeta (\textit {\textbf {t}})-h(\textit {\textbf {t}})} + z,\textit {\textbf {t}}\in \textit {\textbf {D}}\}$. Consequently, by the continuous mapping theorem we also have Eq. 5.7 for η = 0.

Next, we borrow an argument from Berman (1992) [Theorem 1.3.1] to verify Eq. 5.2 for all positive continuity points of $\mathcal {B}^{h,\eta }_{\zeta }(\textit {\textbf {D}},\cdot )$. Let x₀ ∈ (0,μ_η(D)) satisfy

$$\lim\limits_{\varepsilon\to0}{\int}_{\mathbb{R}} \left( \mathcal{I}(x_{0}+\varepsilon,z) -\mathcal{I}(x_{0}-\varepsilon,z)\right)e^{-z} \mathrm{d} z = 0.$$

Since for large M and all x ≥ 0 by Borell-TIS inequality

$$ e^{-z}\mathcal{I}(x,z) \leq C_1 e^{-C_2z^2-C_3z},\quad z<-M, $$

(5.8)

it follows from the dominated convergence theorem that

$${\int}_{\mathbb{R}} \left( \mathcal{I}(x_{0}+,z) -\mathcal{I}(x_{0}-,z)\right) e^{-z} \mathrm{d} z = 0,$$

and thus by the monotonicity of $\mathcal {I}(x,z)$ in x for each fixed z, x₀ is a continuity point of $\mathcal {I}(\cdot ,z)$ for almost all $z\in \mathbb {R}$. Hence by Eq. 5.7 for almost all $z\in \mathbb {R}$

$$ \lim\limits_{u\to\infty}\sup\limits_{k\in K_u}| \mathcal{I}_{u,k}(x_0,z) - \mathcal{I}(x_0,z)| = 0. $$

(5.9)

Further, by Eqs. 5.5 and 5.6 and Borell-TIS inequality, for sufficiently large u and M

$$\begin{array}{@{}rcl@{}} \sup\limits_{k\in K_u}e^{-z}{\mathcal{I}_{u,k}(x,z)} &\leq& \sup_{k\in K_u}\mathbb{P} \left\{ \sup_{\textit{\textbf{t}}\in\textit{\textbf{D}}}\{{\sigma_{\xi_{u,k}}(\textit{\textbf{t}})}\chi_{u,k}(\textit{\textbf{t}}) + f_{u,k}(\textit{\textbf{t}},z)\}>0 \right\} e^{-z}\\ &\leq& \sup_{k\in K_u}\mathbb{P} \left\{ \sup_{\textit{\textbf{t}}\in\textit{\textbf{D}}} \chi_{u,k}(\textit{\textbf{t}})>C_4|z|-C_5 \right\} e^{-z}\\ &\leq& \sup_{k\in K_u} e^{-z} \exp\left( - \frac{\left( C_4|z|-C_5-\mathbb{E}\{\sup_{\textit{\textbf{t}}\in \textit{\textbf{D}}} \chi_{u,k}(\textit{\textbf{t}})\} \right)^2} {2\text{Var}_{\textit{\textbf{t}}\in\textit{\textbf{D}}} \chi_{u,k}(\textit{\textbf{t}})} \right)\\&\!\leq\!& e^{-z} \exp\left( - \frac{\left( C_4|z| - C_5 - \mathbb{E}\{\sup_{\textit{\textbf{t}}\in \textit{\textbf{D}}} {\sum}_{i = 1}^{n}\sqrt{C} B_{\nu}^{i}(t_i)\} \right)^2} {2\sup_{\textit{\textbf{t}}\in\textit{\textbf{D}}} C||\textit{\textbf{t}}||^{\nu}} \right) \end{array} $$

(5.10)

$$\begin{array}{@{}rcl@{}} &=& C_6 e^{-C_7z^2-C_8|z|},\quad z<-M, \end{array} $$

(5.11)

where Eq. 5.10 follows from Sudakov-Fernique inequality (see, e.g., Adler (1990)[Theorem 2.9]), with conditions required therein being ensured by Eq. 5.4 and $B_{\nu }^{(i)}$’s being independent fBm’s with Hurst index ν/2.

Consequently, by Eqs. 5.8, 5.9 and 5.11, a direct application of the dominated convergence theorem yields that

$$\begin{array}{@{}rcl@{}} &&\lefteqn{\lim_{u\to\infty}\sup_{k\in K_u}|{\int}_{\mathbb{R}} \exp\left( -z-\frac{z^2}{2g_{k}^2(u)} \right) \mathcal{I}_{u,k}(x_0,z) \mathrm{d} z -\mathcal{B}^{h,\eta}_{\zeta}(\textit{\textbf{D}},x_0)|}\\ &\leq& \lim_{u\to\infty}{\int}_{\mathbb{R}} \sup_{k\in K_u} \left( e^{-z} \mathcal{I}_{u,k}(x_0,z)\right) |1- e^{-z^2/(2g_{k}^2(u))}|\mathrm{d} z\\ &&+ \lim_{u\to\infty} {\int}_{\mathbb{R}} \sup_{k\in K_u}|\mathcal{I}_{u,k}(x_0,z) - \mathcal{I}(x_0,z)| e^{-z} \mathrm{d} z\\ &=&0 \end{array} $$

verifying Eq. 5.1 for any x₀ ∈ (0,μ_η(D)) a continuity point of $\mathcal {B}^{h,\eta }_{\zeta }(\textit {\textbf {D}},\cdot )$.

The case x = 0,η = 0 is shown in Dȩbicki et al. (2017c). Since the case x = 0,η > 0 can be established by arguments similar to the presented above, we omit the details. This completes the proof. □

Let for any η ≥ 0,b > 0,S > 0, x ∈ [0,μ_η([−S,S]))

$$ \mathcal{P}_{\alpha}^{b,\eta}(S,x) := {\int}_{\mathbb{R}} \mathbb{P} \left\{ {\int}_{-S}^S \left( W_{\alpha}(s)-b|s|^{\alpha}+ z\right) \mu_{\eta}(\mathrm{d} s) >x \right\} e^{-z}\mathrm{d} z $$

(5.12)

and

$$ \mathcal{T}_{\beta}^{b,\eta}(S,x) := {\int}_{0}^{\infty} \left\{{\int}_{-S}^S \left( -b|s|^{\beta}+ z\right) \mu_{\eta}(\mathrm{d} s) >x\right\} e^{-z}\mathrm{d} z. $$

(5.13)

Lemma 5.1

• i) Let X be as in Theorem 2.1 andv(u) be as in Eq. ??. For anyη ≥ 0, S > ηandt_usuch that$\lim _{u \to \infty } t_{u}=t_{0}\in [0,T]$,we have

  $$\begin{array}{@{}rcl@{}} \lim_{u\to\infty} {\Psi}^{-1}(u) \mathbb{P} \left\{ v(u){\int}_0^{S/v(u)} \left( X(t_u+s)\right) \mu_{\eta_u}(\mathrm{d} s)>x \right\} = \mathcal{B}_{\alpha,H(t_0)}^{\eta} (S,x) \end{array} $$

  atx = 0 and anyx ∈ (0,μ_η([0,S])) a continuitypoint of$\mathcal {B}_{\alpha ,H(t_{0})}^{\eta } (S,\cdot )$.

• ii) Let X be as in Theorem 3.1,v(u),$\mathcal {P}_{\alpha }^{b,\eta }(\cdot ,\cdot )$and$\mathcal {T}_{\beta }^{b,\eta }(\cdot ,\cdot )$bedefined in Eqs. ??, 5.12and 5.13, respectively. Then for anyη ≥ 0 andS > 0,asu →∞

  $$\begin{array}{@{}rcl@{}} \mathbb{P} \left\{ L^{\ast}_{\eta,u}[-S/v(u),S/v(u)] >x \right\} \sim {\Psi}(u)\times \left\{ \begin{array}{ll} \mathcal{P}_{\alpha}^{b,\eta}(S,x) & \text{if}\ \alpha=\beta\\ \mathcal{T}_{\beta}^{b,\eta}(S,x) & \text{if}\ \alpha>\beta \end{array} \right. \end{array} $$

  holds forx = 0 and anyx ∈ (0,μ_η([−S,S])) a continuity pointof$\mathcal {P}_{\alpha }^{b,\eta }(S,\cdot )$or$\mathcal {T}_{\beta }^{b,\eta }(S,\cdot )$respectively.

Proof of Lemma

5.1 i) For any x ≥ 0

$$\begin{array}{@{}rcl@{}} \mathbb{P} \left\{ v(u){\int}_{0}^{S/v(u)} \left( X(t_{u} + s)\right) \mu_{\eta_{u}}(\mathrm{d} s)>x \right\} = \mathbb{P} \left\{ {{\int}_{0}^{S}} {\mathbb{I}\left( X(t_{u}+t/v(u))>u\right)} \mu_{\eta}(\mathrm{d} t)>x \right\} . \end{array} $$

Set D = [0,S],g_k(u) = u,K_u = 1 and ξ_u,k(t) = X(t_u + t/v(u)). We now check that C1-C3 are fulfilled with h(t) = H(t₀)t^α,t ∈ [0,S] and $\zeta (t)=\sqrt {H(t_{0})}B_{\alpha }(t)$. First note that Eq. ?? implies

$$\begin{array}{@{}rcl@{}} &&\lefteqn{\limsup_{u\to\infty}\sup_{t\in[0,S]}|\frac{1-\rho(t_{u},t_{u}+t/v(u))}{K(t/v(u))}-H(t_{0})}|\\ &\leq& \limsup_{u\to\infty}\sup_{t\in[0,S]}|\frac{1-\rho(t_{u},t_{u}+t/v(u))}{K(t/v(u))}-H(t_{u})|+\lim_{u\to\infty}|H(t_{u})-H(t_{0})|\\ &=&0. \end{array} $$

Further, by Eq. 2.2 and the Uniform Convergence Theorem (see, e.g., Bingham et al. (1989)[Theorem 1.5.2])

$$\begin{array}{@{}rcl@{}} &&\lefteqn{ \limsup_{u\to\infty}\sup_{t\in[0,S]}|u^{2}K(t/v(u))-t^{\alpha}|}\\ &\leq& \limsup_{u\to\infty}\sup_{t\in[0,S]}|\frac{K(t/v(u))}{K(1/v(u))}-t^{\alpha}| + \limsup_{u\to\infty}\sup_{t\in[0,S]}\frac{K(t/v(u))}{K(1/v(u))}|u^{2}K(1/v(u))-1|\\ & =&0, \end{array} $$

(5.14)

and thus

$$\begin{array}{@{}rcl@{}} &&\lefteqn{ \limsup_{u\to\infty}\sup_{t\in[0,S]}|u^{2}(1-\mathbb{E}\{X(t_{u}+t/v(u))X(t_{u})\})-H(t_{0})t^{\alpha}|}\\ &=&\limsup_{u\to\infty}\sup_{t\in[0,S]}|u^{2}(1-\rho(t_{u},t_{u}+t/v(u)))-H(t_{0})t^{\alpha}|\\ &\leq& \limsup_{u\to\infty} \sup_{t\in[0,S]} |\frac{1-\rho(t_{u},t_{u}+t/v(u))}{K(t/v(u))}-H(t_{0})||t|^{\alpha}\\ &&+\ \limsup_{u\to\infty} \sup_{t\in[0,S]} |\frac{1-\rho(t_{u},t_{u}+t/v(u))}{K(t/v(u))}||u^{2}K(t/v(u))-t^{\alpha}|\\ &=&0. \end{array} $$

Similarly, for any s,t ∈ [0,S],

$$\begin{array}{@{}rcl@{}} &&\lefteqn{\limsup_{u\to\infty}2^{-1}|u^{2}\text{Var}\left( X(t_{u}+t/v(u))-X(t_{u}+s/v(u))\right)-2H(t_{0})|t-s|^{\alpha}|}\\ &=&\limsup_{u\to\infty}|u^{2}(1-\rho(t_{u}+s/v(u),t_{u}+t/v(u)))-H(t_{0})|t-s|^{\alpha}|\\ &\leq& \limsup_{u\to\infty} |\frac{1-\rho(t_{u}+s/v(u),t_{u}+t/v(u))}{K((t-s)/v(u))}-H(t_{u}+s/v(u))||t-s|^{\alpha}\\ &&+\ \limsup_{u\to\infty} |H(t_{u}+s/v(u))-H(t_{0})||t-s|^{\alpha}\\ &&+\ \limsup_{u\to\infty} |\frac{1-\rho(t_{u}+s/v(u),t_{u}+t/v(u))}{K((t-s)/v(u))}||u^{2}K((t-s)/v(u))-|t-s|^{\alpha}|\\ &=&0. \end{array} $$

Next, according to Eq. 2.2 and Potter’s Theorem (see, e.g., Bingham et al. (1989)[Theorem 1.5.6]), for large enough u there exists some constant C > 1 such that

$$ u^{2}K(|s-t|/v(u))=u^{2} K(1/v(u))\frac{K(|s-t|/v(u))}{K(1/v(u))}\leq C|| s-t ||^{\alpha/2} $$

(5.15)

holds for all s,t ∈ [0,S]. Hence

$$\begin{array}{@{}rcl@{}} &&\lefteqn{u^{2}\mathbb{E}\{\left( X(t_{u}+t/v(u))-X(t_{u}+s/v(u))\right)^{2}\}}\\ &=&2u^{2}(1-\rho(t_{u}+s/v(u),t_{u}+t/v(u)))\\ &=& 2u^{2}K(|| s-t ||/v(u))\frac{1-\rho(t_{u}+s/v(u),t_{u}+t/v(u))}{K(|| s-t ||/v(u))}\\ &\leq& 4C\overline{h}|| s-t ||^{\alpha/2}, \end{array} $$

where in the last inequality we have used Eq. 2.1 with $\overline {h}=\max _{t\in [0,T]}H(t)$. Therefore, C0-C3 hold and thus our claim i) follows by Theorem 5.1 with $\mathcal {B}^{h,\eta }_{\zeta }(\textit {\textbf {D}},x)=\mathcal {B}_{\alpha ,H(t_{0})}^{\eta } (S,x)$. The claim in ii) follows with similar arguments. □

Lemma 5.2

If$\zeta (\textit {\textbf {t}})={\sum }_{i = 1}^{n} \sqrt {\lambda _{i}}B_{\alpha }^{(i)}(t_{i})$for somepositive constantsλ₁…λ_n,where$B_{\alpha }^{(i)}$’s are independentfBm’s with Hurst indexα/2,then for anyx,η ≥ 0,h(t) = Varζ(t) and$\textit {\textbf {D}}={\prod }_{i = 1}^{n}[0,T_{i}]$wehave

$$\mathcal{B}_{\zeta}^{h,\eta} (\textit{\textbf{D}},x)\leq {\prod}_{i = 1}^{n}\lceil\lambda_{i}^{1/\alpha}\rceil\lceil T_{i}\rceil H_{\alpha}^{n}([0,1]).$$

Proof of Lemma

5.2 For any x,η ≥ 0 and $\textit {\textbf {D}}={\prod }_{i = 1}^{n}[0,T_{i}]$,

$$\begin{array}{@{}rcl@{}} \mathcal{B}_{\zeta}^{h,\eta} (\textit{\textbf{D}},x) &\leq& \mathbb{E}\{\exp\left( \sup_{\textit{\textbf{t}}\in{\prod}_{i = 1}^n[0,T_i]} {\sum}_{i = 1}^n \sqrt{2\lambda_i}B_{\alpha}^{(i)}(t_i) - {\sum}_{i = 1}^n \lambda_i t_i^{\alpha} \right)\} \\ &=& {\prod}_{i = 1}^n \mathbb{E}\{\sup_{t_i\in[0,\lambda_i^{1/\alpha}T_i]} e^{\sqrt{2}B_{\alpha}^{(i)}(t_i)-|| t_i ||^{\alpha}} \}\\ &\leq& {\prod}_{i = 1}^n\lceil\lambda_i^{1/\alpha}\rceil\lceil T_i\rceil H_{\alpha}^n([0,1]), \end{array} $$

where in the last inequality we have used H_α([0,T]) ≤⌈T⌉H_α([0, 1]). □

Lemma 5.3

If X is a centered Gaussian process with unit variance fulfilling (??),v(u) is defined in Eq. ?? andK_uis an index function of u, then for a sequence ofnumbers$\{0<{S_{1}^{k}}<{S_{2}^{k}}<{T_{1}^{k}}<{T_{2}^{k}}<\infty , k\in K_{u}\}$suchthat for someu₀ > 0

$$ \inf_{u>u_0}\inf_{k\in K_u}(T_1^k-S_2^k)\geq1, $$

(5.16)

$$ \limsup_{u\to\infty}\frac{\sup_{k\in K_u}(T_2^k-S_1^k)}{v(u)}= 0, $$

(5.17)

$$ \overline{T}:=\sup_{u>u_0}\sup_{k\in K_u}(T_2^k-T_1^k)<\infty \quad \textmd{and}\quad \overline{S}:=\sup_{u>u_0}\sup_{k\in K_u}(S_2^k-S_1^k)<\infty, $$

(5.18)

we have

$$\begin{array}{@{}rcl@{}} \sup_{k\in K_u}\frac{\mathbb{P} \left\{ \sup_{t\in[S_1^k,S_2^k]/v(u)}X(t)>u,\sup_{t\in[T_1^k,T_2^k]/v(u)}X(t)>u \right\}} {\exp(-\frac{1}{16}\underline{h}|T_1^k-S_2^k|^{\alpha/2}){\Psi}(u)} \leq C(\alpha,\overline{S},\overline{T}) \end{array} $$

holds for sufficiently large u, where $\underline {h}=\inf _{t\in [0,T]}H(t)>0$, $\overline {h}=\sup _{t\in [0,T]}H(t)<\infty$ and

$C(\alpha ,\overline {S},\overline {T})= 2\lceil (16\overline {h})^{1/\alpha }\rceil ^{2} \lceil \overline {S}\rceil \lceil \overline {T}\rceil H_{\alpha }^{2}([0,1])$.

Proof of Lemma

5.3 We borrow some arguments from the proof of Michna (2017)[Lemma 5]. Define

$$\begin{array}{@{}rcl@{}} {A_{u}^{k}}&=&[{S_{1}^{k}},{S_{2}^{k}}]/v(u),\qquad\qquad\quad {B_{u}^{k}}=[{T_{1}^{k}},{T_{2}^{k}}]/v(u),\quad k\in K_{u},\\ Y(s,t)&=&X(s)+X(t),\qquad\qquad\sigma^{2}(s,t)=\text{Var} (Y(s,t)). \end{array} $$

In view of Eq. 2.1, for sufficiently small ε we have

$$\frac{1}{2}\underline{h}K(|| s-t ||)\leq 1-\rho(s,t)\leq2\overline{h}K(|| s-t ||)\leq1,\quad \forall s,t\in[0,T],|| s-t ||<\varepsilon,$$

where the last inequality follows from the regularity of K. Therefore, by Eq. 5.17 for large enough u and then each k ∈ K_u

$$\inf_{(s,t)\in {A_{u}^{k}}\times {B_{u}^{k}}}\sigma^{2}(s,t)\geq 4-4\overline{h}\sup_{(s,t)\in {A_{u}^{k}}\times {B_{u}^{k}}}K(|| s-t ||)\geq2$$

and

$$\sup_{(s,t)\in {A_{u}^{k}}\times {B_{u}^{k}}}\sigma^{2}(s,t)\leq 4-\underline{h}\inf_{(s,t)\in {A_{u}^{k}}\times {B_{u}^{k}}}K(|| s-t ||).$$

Then for every k ∈ K_u we have

$$\begin{array}{@{}rcl@{}} \mathbb{P} \left\{ \sup_{t\in A_u^k}X(t)>u,\ \sup_{t\in B_u^k}X(t)>u \right\}&\leq& \mathbb{P} \left\{ \sup_{(s,t)\in A_u^k\times B_u^k} Y(s,t)>2u \right\} \end{array} $$

$$\begin{array}{@{}rcl@{}} &\leq& \mathbb{P} \left\{ \sup_{(s,t)\in A_u^k\times B_u^k} \tilde{Y}(s,t)>u^{*}_k \right\}, \end{array} $$

(5.19)

where $\tilde {Y}(s,t)=Y(s,t)/\sigma (s,t)$ and

$$\begin{array}{@{}rcl@{}} u_k^{*}=\frac{2u}{\sqrt{4-\underline{h}\inf_{(s,t)\in A_u^k \times B_u^k}K(|| s-t ||)}}. \end{array} $$

Moreover, as in (Michna 2017), for any $(s_{i},t_{i})\in {A_{u}^{k}}\times {B_{u}^{k}}, i = 1,2, k\in K_{u}$ we get

$$\begin{array}{@{}rcl@{}} \text{Cov}(\tilde{Y}(s_1,t_1),\tilde{Y}(s_2,t_2))&\geq& 1-8\overline{h}K(|| s_2-s_1 ||)-8\overline{h}K(|| t_2-t_1 ||). \end{array} $$

Let $Z(s,t):=\frac {1}{\sqrt {2}}(\vartheta _{1}(s)+\vartheta _{2}(t)),$where 𝜗_i,i = 1, 2 are mutually independent copies of a mean zero stationary Gaussian process 𝜗 with unit variance and covariance function satisfying

$$ 1-\text{Cov}(\vartheta(s),\vartheta(t))\sim 32\overline h K(|| s-t ||),\quad || t-s ||\to0. $$

(5.20)

As mentioned in the proof of Theorem 2.1, the existence of such a Gaussian process is guaranteed by the Assertion in (Hüsler and Piterbarg 1999b)[p.265]. Hence by Slepian inequality and Eq. 5.17, for sufficiently large u(> u₀) and then each k ∈ K_u we have

$$\begin{array}{@{}rcl@{}} \mathbb{P} \left\{ \sup_{(s,t)\in A_u^k\times B_u^k}\tilde{Y}(s,t)>u^{*}_k \right\} &\leq& \mathbb{P} \left\{ \sup_{(s,t)\in A_u^k\times B_u^k} Z(s,t)>u^{*}_k \right\}\\ &\leq& \mathbb{P} \left\{ \sup_{(s,t)\in[0,\overline{S}/v(u)]\times [0,\overline{T}/v(u)]} Z(s,t)>u^{*}_k \right\}, \end{array} $$

(5.21)

where the last inequality follows from stationarity of Z with $\overline {S}$ and $\overline {T}$ defined in Eq. 5.18.

Next, we apply Theorem 5.1 to give an uniform upper bound for the right hand side of Eq. 5.21. Setting $\textit {\textbf {D}}=[0,\overline {S}]\times [0,\overline {T}], g_{k}(u)=u_{k}^{*}, k\in K_{u}$ and ξ_u,k(s,t) = Z(s/v(u),t/v(u)), we need to check that C0-C3 are fulfilled. First note $\inf _{k\in K_{u}}u^{\ast }_{k}\geq u$ then C0 is naturally satisfied. Moreover, due to Eq. 5.17 and the regularity of K at 0

$$\begin{array}{@{}rcl@{}} \lim_{u\to\infty}\sup_{k\in K_u} || \frac{(u^{\ast}_k)^2}{u^2}-1 || &=& \lim_{u\to\infty}\sup_{k\in K_u} \frac{\underline{h}\inf_{(s,t)\in A_u^k \times B_u^k}K(|| s-t ||)}{4-\underline{h}\inf_{(s,t)\in A_u^k \times B_u^k}K(|s-t|)}\\ &\leq& \lim_{u\to\infty}\sup_{k\in K_u} \frac{\underline{h}K\left( (T_2^k-S_1^k)/v(u)\right)}{4-\underline{h}K\left( (T_2^k-S_1^k)/v(u)\right)}= 0, \end{array} $$

(5.22)

and thus

$$\begin{array}{@{}rcl@{}} &&{} (32\overline{h})^{-1}\limsup\limits_{u\to\infty}\sup\limits_{k\in K_u,{(s,t)\in[0,\overline{S}]\times[0,\overline{T}]}}|(u_k^{\ast})^2(1 - \mathbb{E}\{Z(s/v(u),t/v(u))Z(0,0)\}) - 16\overline{h}(s^{\alpha} + t^{\alpha})|\\ &\leq& \limsup\limits_{u\to\infty}\sup\limits_{k\in K_u,t\in[0,{\overline{S}\vee\overline{T}}]}|\frac{(u^{\ast}_k)^2}{u^2}\frac{1-\text{Cov}(\vartheta(t/v(u)), \vartheta(0))}{32\overline{h}K(t/v(u))}u^2K(t/v(u)) - t^{\alpha}|\\ &\leq& \limsup\limits_{u\to\infty}\sup\limits_{t\in[0,{\overline{S}\vee\overline{T}}]}\frac{1-\text{Cov}(\vartheta(t/v(u)),\vartheta(0))}{32\overline{h}K(t/v(u))}u^2K(t/v(u)) \sup\limits_{k\in K_u}|\frac{(u^{\ast}_k)^2}{u^2}-1|\\ &&{}+ \limsup\limits_{u\to\infty}\sup\limits_{t\in[0,{\overline{S}\vee\overline{T}}]}|1-\frac{1-\text{Cov}(\vartheta(t/v(u)),\vartheta(0))}{32\overline{h}K(t/v(u))}|\sup\limits_{t\in{\overline{S} \vee\overline{T}}}u^2K(t/v(u))\\ &&{}+ \limsup\limits_{u\to\infty}\sup\limits_{t\in[0,{\overline{S}\vee\overline{T}}]}|| u^2K(t/v(u))-t^{\alpha} ||\\ & =&0, \end{array} $$

where in the last equality we have used Eqs. 5.14 and 5.20. Therefore, C1 holds with $h(s,t)= 16\overline {h}(s^{\alpha }+t^{\alpha }), {(s,t)\in [0,\overline {S}]\times [0,\overline {T}]}$. Similarly, for any $(s_{i},t_{i})\in [0,\overline {S}]\times [0,\overline {T}],i = 1,2$

$$\begin{array}{@{}rcl@{}} &&{} (32\overline{h})^{-1}\!\limsup\limits_{u\to\infty}|| (u_k^{\ast})^2\text{Var}\left( Z(s_1\!/v(u),\!t_1/v(u)) - Z(s_2/v(u),t_2/v(u))\right) - 32\overline{h} (|s_2 - s_1|^{\alpha} + |t_2 - t_1|^{\alpha})||\\ &\leq& \limsup\limits_{u\to\infty}|| (u^{\ast}_k)^2\frac{1-\text{Cov}(\vartheta(s_1/v(u)),\vartheta(s_2/v(u)))}{32\overline{h}} -|s_2-s_1|^{\alpha} ||\\ && + \limsup\limits_{u\to\infty}|| (u^{\ast}_k)^2\frac{1-\text{Cov}(\vartheta(t_1/v(u)),\vartheta(st_2/v(u)))}{32\overline{h}} -|t_2-t_1|^{\alpha} ||\\ & =&0 \end{array} $$

verifying C2 with $\zeta (s,t)=\sqrt {16\overline {h}}(B_{\alpha }^{(1)}(s)+B_{\alpha }^{(2)}(t))$, where $B_{\alpha }^{(i)},i = 1,2$ are independent fBm’s with Hurst index α/2. Next, in view of Eqs. 5.15 and 5.22, for large enough u there exists some constant C > 1 such that

$$\sup\limits_{k\in K_{u}}(u_{k}^{\ast})^{2}K(|| s-t ||/v(u))=\sup_{k\in K_{u}}\frac{(u_{k}^{\ast})^{2}}{u^{2}}u^{2} K(|| s-t ||/v(u))\leq C|s-t|^{\alpha/2}$$

holds for all $s,t\in [0,{\overline {S}\vee \overline {T}}]$. This together with Eq. 5.20 implies that for any two points $(s_{i},t_{i})\in [0,\overline {S}]\times [0,\overline {T}],i = 1,2$ and sufficiently large u

$$\begin{array}{@{}rcl@{}} &&\lefteqn{\sup\limits_{k\in K_u}(u_k^{\ast})^2\mathbb{E}\{\left( Z(s_1/v(u),t_1/v(u))-Z(s_2/v(u),t_2/v(u))\right)^2\}}\\ &=&\sup\limits_{k\in K_u}(u_k^{\ast})^2 || 2-\text{Cov}(\vartheta(s_1/v(u)),\vartheta(s_2/v(u)))-\text{Cov}(\vartheta(t_1/v(u)),\vartheta(t_2/v(u))) ||\\ &\leq& 32\overline{h}\sup_{k\in K_u}(u_k^{\ast})^2K(|s_2-s_1|/v(u))\frac{1-\text{Cov}(\vartheta(s_1/v(u)),\vartheta(s_2/v(u)))}{32\overline{h}K(|s_2-s_1|//v(u))}\\ &&+ 32\overline{h}\sup\limits_{k\in K_u}(u_k^{\ast})^2K(|t_2-t_1|/v(u))\frac{1-\text{Cov}(\vartheta(t_1/v(u)),\vartheta(t_2/v(u)))}{32\overline{h}K(|t_2-t_1|//v(u))}\\ &\leq& C_1(|| s_2-s_1 ||^{\alpha/2}+|| t_2-t_1 ||^{\alpha/2}), \end{array} $$

i.e., C3 is fulfilled with ν = α/2. Therefore, by Theorem 5.1 and Lemma 4.2,

$$ \sup_{k\in K_u} \frac{\mathbb{P} \left\{ \sup_{(s,t)\in[0,\overline{S}/v(u)]\times [0,\overline{T}/v(u)]} Z(s,t)>u^{*}_k \right\} } {{\Psi}(u_k^{\ast})} \leq \frac{3}{2}\lceil (16\overline{h})^{1/\alpha}\rceil^2 \lceil \overline{S}\rceil\lceil \overline{T}\rceil H_{\alpha}^2([0,1]) $$

(5.23)

holds for sufficiently large u. Further, by Potter’s theorem, there exists a small enough ε such that

$$\frac{K(\lambda x)}{\lambda^{\alpha/2} K(x)}\geq\frac{4}{5},\quad\forall 0<x<\lambda x<\varepsilon,$$

which together with Eqs. 5.16 and 5.17 yields for sufficiently large u(> u₀)

$$\inf_{k\in K_{u}}\frac{\inf_{(s,t)\in {A_{u}^{k}}\times {B_{u}^{k}}}K(|| s-t ||)}{K(1/v(u))|| {T_{1}^{k}}-{S_{2}^{k}} ||^{\alpha/2}}\geq \frac{4}{5}.$$

Hence

$$\begin{array}{@{}rcl@{}} \inf_{k\in K_u} \frac{(u^{\ast}_k)^2-u^2}{\underline{h}(T_1^k-S_2^k)^{\alpha/2}} &\geq& \inf_{k\in K_u} \frac{u^2 \inf_{(s,t)\in A_u^k\times B_u^k}K(|| s-t ||)}{4(T_1^k-S_2^k)^{\alpha/2}}\\ &=& \frac{1}{4} u^2K(1/v(u)) \inf_{k\in K_u}\frac{\inf_{(s,t)\in A_u^k\times B_u^k}K(|| s-t ||)}{K(1/v(u))|| T_1^k-S_2^k ||^{\alpha/2}}\\ &\geq&\frac{1}{5} u^2K(1/v(u)) \geq \frac{1}{8}, \end{array} $$

where the last inequality follows from Eq. 2.2. Therefore for large enough u and each k ∈ K_u

$$\begin{array}{@{}rcl@{}} {\Psi}(u^{\ast}_k)\leq \frac{1}{\sqrt{2\pi}u^{\ast}_k}e^{-\frac{(u^{\ast}_k)^2}{2}}\leq \frac{4}{3}{\Psi}(u)\exp\left( -\frac{1}{16}\underline{h}|| T_1^k-S_2^k ||^{\alpha/2}\right), \end{array} $$

which combined with Eq. 5.23 yileds

$$ \sup_{k\in K_u} \frac{\mathbb{P} \left\{ \sup_{(s,t)\in[0,\overline{S}/v(u)]\times [0,\overline{T}/v(u)]} Z(s,t)>u^{*}_k \right\} } {\exp(-\frac{1}{16}\underline{h}|| T_1^k-S_2^k ||^{\alpha/2}){\Psi}(u)} \leq 2\lceil (16\overline{h})^{1/\alpha}\rceil^2 \lceil \overline{S}\rceil\lceil \overline{T}\rceil H_{\alpha}^2([0,1]). $$

(5.24)

Consequently, substitution of Eqs. 5.21 and 5.24 to Eq. 5.19 completes the proof. □

Lemma 5.4

Let W be anN(0, 1) random variable independent of$\mathcal {Z}$whichis exponentially distributed with parameter 1. For anyc > 0 we have

$$\begin{array}{@{}rcl@{}} \left\{ cW- c^2/2 + Z> 0\right\}= 2 \mathbb{P} \left\{ W > c/2 \right\}. \end{array} $$

Proof of Lemma

5.4 Since Z > 0 almost surely, then

$$\mathbb{P} \left\{ cW- c^{2}/2 + Z> 0, cW- c^{2}/2 \ge 0 \right\} = \mathbb{P} \left\{ W> c/2 \right\}.$$

Let the random variable V be such that

$$\mathbb{P} \left\{ V \le x \right\}= \mathbb{E}\{ e^{c W- c^{2}/2} \mathbb{I}(cW- c^{2}/2 \le x)\}, \quad x\in \mathbb{R}.$$

It is well-known, see e.g., Hashorva (2018)[Lemma 7.1] that V has an N(c²/2,c²) distribution. Hence by the independence of Z and W

$$\begin{array}{@{}rcl@{}} \mathbb{P} \left\{ cW- c^2/2 + Z> 0, cW- c^2/2 \le 0 \right\} &=&\mathbb{E}\{ e^{cW- c^2/2} \mathbb{I}(cW- c^2/2\le 0)\}\\ &=& \mathbb{P} \left\{ V \le 0 \right\}= \mathbb{P} \left\{ W \le - c/2 \right\} \end{array} $$

establishing the proof. □